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2612 y(2.2)100 b(Examples)32 b(of)g(Matrix)g(Lie)g(Groups)70 b(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(16)293 2736 y(2.3)100 b(Compactness)c(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.) g(.)g(.)g(.)g(.)139 b(22)293 2860 y(2.4)100 b(Connectedness)33 b(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(24)293 2984 y(2.5)100 b(Simple-connectedness)45 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(27)293 3108 y(2.6)100 b(Homomorphisms)30 b(and)i(Isomorphisms)57 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.) g(.)g(.)139 b(28)293 3232 y(2.7)100 b(Lie)32 b(Groups)26 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.) h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(29)293 3356 y(2.8)100 b(Exercises)35 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f (.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(31)147 3580 y FI(3)90 b(Lie)37 b(Algebras)g(and)i(the)e(Exp)s(onen)m(tial)f (Mapping)1214 b(37)293 3704 y FH(3.1)100 b(The)33 b(Matrix)f(Exp)s (onen)m(tial)76 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(37)293 3828 y(3.2)100 b(Computing)31 b(the)i(Exp)s(onen)m(tial)f(of)g(a)g (Matrix)99 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g (.)g(.)139 b(40)293 3952 y(3.3)100 b(The)33 b(Matrix)f(Logarithm)65 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(42)293 4076 y(3.4)100 b(F)-8 b(urther)32 b(Prop)s(erties)h(of)f(the)h(Matrix)f(Exp) s(onen)m(tial)96 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g (.)139 b(45)293 4200 y(3.5)100 b(The)33 b(Lie)f(Algebra)g(of)g(a)g (Matrix)g(Lie)g(Group)47 b(.)j(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f (.)h(.)g(.)g(.)g(.)g(.)139 b(49)293 4324 y(3.6)100 b(Prop)s(erties)32 b(of)g(the)h(Lie)f(Algebra)99 b(.)50 b(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(53)293 4448 y(3.7)100 b(The)33 b(Exp)s(onen)m(tial)f(Mapping)69 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(59)293 4572 y(3.8)100 b(Lie)32 b(Algebras)41 b(.)50 b(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) g(.)g(.)g(.)139 b(61)293 4696 y(3.9)100 b(The)33 b(Complexi\034cation)e (of)h(a)g(Real)g(Lie)f(Algebra)97 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f (.)h(.)g(.)g(.)g(.)g(.)139 b(65)293 4820 y(3.10)51 b(Exercises)35 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(67)147 5044 y FI(4)90 b(The)38 b(Bak)m(er-Campb)s(ell-Hausdor\033)e (F)-9 b(orm)m(ula)1357 b(71)293 5168 y FH(4.1)100 b(The)33 b(Bak)m(er-Campb)s(ell-Hausdor\033)e(F)-8 b(orm)m(ula)30 b(for)i(the)h(Heisen)m(b)s(erg)h(Group)48 b(.)i(.)139 b(71)293 5292 y(4.2)100 b(The)33 b(General)f(Bak)m(er-Campb)s (ell-Hausdor\033)f(F)-8 b(orm)m(ula)31 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)g(.)g(.)g(.)139 b(75)p eop %%Page: 4 4 4 3 bop 147 48 a FK(4)3079 b FG(CONTENTS)293 298 y FH(4.3)100 b(The)33 b(Series)g(F)-8 b(orm)31 b(of)h(the)h(Bak)m(er-Campb)s (ell-Hausdor\033)e(F)-8 b(orm)m(ula)44 b(.)50 b(.)g(.)g(.)g(.)g(.)139 b(83)293 418 y(4.4)100 b(Subgroups)33 b(and)g(Subalgebras)92 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) f(.)h(.)g(.)g(.)g(.)g(.)139 b(84)293 539 y(4.5)100 b(Exercises)35 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(86)147 756 y FI(5)90 b(Basic)37 b(Represen)m(tation)g(Theory)1911 b(89)293 877 y FH(5.1)100 b(Represen)m(tations)55 b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(89)293 997 y(5.2)100 b(Wh)m(y)33 b(Study)h(Represen)m(tations?)27 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) f(.)h(.)g(.)g(.)g(.)g(.)139 b(92)293 1118 y(5.3)100 b(Examples)32 b(of)g(Represen)m(tations)39 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(93)293 1238 y(5.4)100 b(The)33 b(Irreducible)g(Represen)m(tations)g (of)f FF(su)q FE(\(2\))83 b FH(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)g(.)g(.)139 b(99)293 1358 y(5.5)100 b(Direct)32 b(Sums)h(of)f(Represen)m(tations)i(and)e(Complete)g (Reducibilit)m(y)52 b(.)e(.)g(.)g(.)g(.)g(.)90 b(104)293 1479 y(5.6)100 b(T)-8 b(ensor)33 b(Pro)s(ducts)h(of)e(Represen)m (tations)56 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.) h(.)g(.)g(.)g(.)g(.)90 b(108)293 1599 y(5.7)100 b(Sc)m(h)m(ur's)34 b(Lemma)78 b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)90 b(114)293 1719 y(5.8)100 b(Group)32 b(V)-8 b(ersus)34 b(Lie)e(Algebra)f(Represen)m(tations)65 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)90 b(116)293 1840 y(5.9)100 b(Co)m(v)m(ering)33 b(Groups)89 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g (.)g(.)g(.)90 b(124)293 1960 y(5.10)51 b(Exercises)35 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)90 b(126)147 2178 y FI(6)g(The)38 b(Represen)m(tations)f(of)h FF(SU)p FE(\(3\))p FI(,)f(and)i(Bey)m(ond)1206 b(133)293 2299 y FH(6.1)100 b(Preliminaries)91 b(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) f(.)h(.)g(.)g(.)g(.)g(.)90 b(133)293 2419 y(6.2)100 b(W)-8 b(eigh)m(ts)32 b(and)h(Ro)s(ots)75 b(.)49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.) g(.)g(.)90 b(135)293 2539 y(6.3)100 b(Highest)32 b(W)-8 b(eigh)m(ts)33 b(and)f(the)h(Classi\034cation)f(Theorem)54 b(.)c(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)90 b(139)293 2660 y(6.4)100 b(Pro)s(of)32 b(of)g(the)h(Classi\034cation)e(Theorem)55 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.) g(.)g(.)90 b(140)293 2780 y(6.5)100 b(An)33 b(Example:)43 b(Highest)32 b(W)-8 b(eigh)m(t)32 b FE(\(1)p FD(;)17 b FE(1\))68 b FH(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f (.)h(.)g(.)g(.)g(.)g(.)90 b(145)293 2900 y(6.6)100 b(The)33 b(W)-8 b(eyl)33 b(Group)89 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.) g(.)90 b(147)293 3021 y(6.7)100 b(Complex)32 b(Semisimple)e(Lie)i (Algebras)45 b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)g(.)g(.)90 b(150)293 3141 y(6.8)100 b(Exercises)35 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) g(.)g(.)g(.)90 b(154)147 3359 y FI(7)g(Final)37 b(Examination)2369 b(155)147 3577 y(8)90 b(Bibliograph)m(y)2651 b(157)p eop %%Page: 5 5 5 4 bop 1688 1047 a FL(Chapter)37 b(1)1696 1222 y(GR)m(OUPS)147 1491 y FI(1.1)112 b(De\034nition)36 b(of)i(a)g(Group,)g(and)g(Basic)f (Prop)s(erties)147 1659 y(De\034nition)f(1)49 b FC(A)41 b FB(gr)-6 b(oup)42 b FC(is)f(a)f(set)h FD(G)p FC(,)h(to)-5 b(gether)41 b(with)g(a)f(map)g(of)h FD(G)26 b FA(\002)h FD(G)41 b FC(into)f FD(G)h FC(\(denote)-5 b(d)147 1779 y FD(g)194 1794 y Fz(1)255 1779 y FA(\003)22 b FD(g)374 1794 y Fz(2)414 1779 y FC(\))34 b(with)h(the)g(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)446 1900 y(First,)35 b(asso)-5 b(ciativity:)44 b(for)34 b(al)5 b(l)35 b FD(g)1650 1915 y Fz(1)1689 1900 y FD(;)17 b(g)1780 1915 y Fz(2)1846 1900 y FA(2)29 b FD(G)p FC(,)1342 2110 y FD(g)1389 2125 y Fz(1)1450 2110 y FA(\003)22 b FE(\()p FD(g)1607 2125 y Fz(2)1668 2110 y FA(\003)g FD(g)1787 2125 y Fz(3)1826 2110 y FE(\))28 b(=)f(\()p FD(g)2080 2125 y Fz(1)2142 2110 y FA(\003)21 b FD(g)2260 2125 y Fz(2)2300 2110 y FE(\))h FA(\003)g FD(g)2479 2125 y Fz(3)2518 2110 y FC(.)994 b FH(\(1.1\))147 2321 y FC(Se)-5 b(c)g(ond,)34 b(ther)-5 b(e)35 b(exists)f(an)g(element)h FD(e)g FC(in)f FD(G)h FC(such)g(that)g(for)f(al)5 b(l)35 b FD(g)c FA(2)d FD(G)p FC(,)1584 2531 y FD(g)d FA(\003)d FD(e)28 b FE(=)g FD(e)22 b FA(\003)g FD(g)31 b FE(=)c FD(g)t FC(.)1236 b FH(\(1.2\))147 2742 y FC(and)34 b(such)h(that)g(for)g(al)5 b(l)34 b FD(g)d FA(2)d FD(G)p FC(,)35 b(ther)-5 b(e)35 b(exists)f FD(h)28 b FA(2)g FD(G)35 b FC(with)1575 2953 y FD(g)26 b FA(\003)c FD(h)28 b FE(=)f FD(h)22 b FA(\003)g FD(g)31 b FE(=)d FD(e)p FC(.)1228 b FH(\(1.3\))446 3163 y FC(If)31 b FD(g)19 b FA(\003)d FD(h)28 b FE(=)f FD(h)16 b FA(\003)g FD(g)35 b FC(for)d(al)5 b(l)31 b FD(g)t(;)17 b(h)27 b FA(2)h FD(G)p FC(,)33 b(then)e(the)h(gr)-5 b(oup)32 b(is)g(said)f(to)h (b)-5 b(e)32 b FB(c)-6 b(ommutative)38 b FC(\(or)147 3283 y FB(ab)-6 b(elian)p FC(\).)147 3542 y FI(Remarks)147 3727 y FH(The)41 b(elemen)m(t)f FD(e)h FH(is)f(\(as)g(w)m(e)i(shall)c (see)k(momen)m(tarily\))37 b(unique,)43 b(and)d(is)g(called)f(the)i FI(iden)m(tit)m(y)147 3847 y(elemen)m(t)34 b FH(of)j(the)h(group,)g(or) g(simply)e(the)h FI(iden)m(tit)m(y)p FH(.)56 b(P)m(art)38 b(of)f(the)h(de\034nition)e(of)h(a)g(group)g(is)147 3967 y(that)27 b(m)m(ultiplying)d(a)k(group)f(elemen)m(t)g FD(g)k FH(b)m(y)d(the)g(iden)m(tit)m(y)f(on)h FC(either)i(the)g(right)g (or)g(the)g(left)37 b FH(m)m(ust)147 4088 y(giv)m(e)c(bac)m(k)g FD(g)t FH(.)446 4208 y(The)41 b(map)d(of)h FD(G)27 b FA(\002)h FD(G)39 b FH(in)m(to)g FD(G)g FH(is)g(called)g(the)h FI(pro)s(duct)45 b(op)s(eration)39 b FH(for)g(the)h(group.)147 4328 y(P)m(art)32 b(of)f(the)h(de\034nition)e(of)h(a)g(group)g FD(G)h FH(is)f(that)g(the)h(pro)s(duct)f(op)s(eration)f(map)h FD(G)20 b FA(\002)g FD(G)31 b FH(in)m(to)g FD(G)p FH(,)147 4449 y(i.e.,)i(that)h(the)f(pro)s(duct)h(of)f(t)m(w)m(o)h(elemen)m(ts)g (of)f FD(G)g FH(b)s(e)h(again)d(an)j(elemen)m(t)f(of)g FD(G)p FH(.)46 b(This)33 b(prop)s(ert)m(y)147 4569 y(is)f(refered)i(to) e(as)h FI(closure)p FH(.)446 4690 y(Giv)m(en)39 b(a)g(group)g(elemen)m (t)f FD(g)t FH(,)i(a)f(group)g(elemen)m(t)g FD(h)g FH(suc)m(h)h(that)f FD(g)30 b FA(\003)c FD(h)39 b FE(=)g FD(h)26 b FA(\003)h FD(g)42 b FE(=)c FD(e)h FH(is)147 4810 y(called)j(an)h FI(in)m(v)m(erse)f FH(of)h FD(g)t FH(.)74 b(W)-8 b(e)44 b(shall)d(see)j(momen)m(tarily)d(that)h(eac)m(h)i(group)f(elemen)m(t)g (has)g(a)147 4930 y FC(unique)d FH(in)m(v)m(erse.)446 5051 y(Giv)m(en)c(a)g(set)h(and)g(an)f(op)s(eration,)g(there)h(are)f (four)g(things)g(that)g(m)m(ust)g(b)s(e)h(c)m(hec)m(k)m(ed)i(to)147 5171 y(sho)m(w)27 b(that)e(this)g(is)g(a)h(group:)39 b FC(closur)-5 b(e)p FH(,)27 b FC(asso)-5 b(ciativity)p FH(,)26 b(existence)h(of)e(an)h FC(identity)p FH(,)h(and)f(existence) 147 5292 y(of)32 b FC(inverses)p FH(.)p eop %%Page: 6 6 6 5 bop 147 48 a FK(6)3286 b FG(Groups)147 298 y FI(Prop)s(osition)36 b(2)h(\(Uniqueness)h(of)f(the)h(Iden)m(tit)m(y\))46 b FC(L)-5 b(et)36 b FD(G)f FC(b)-5 b(e)35 b(a)g(gr)-5 b(oup,)35 b(and)g(let)g FD(e;)17 b(f)39 b FA(2)29 b FD(G)147 418 y FC(b)-5 b(e)35 b(such)f(that)i(for)e(al)5 b(l)35 b FD(g)c FA(2)d FD(G)1584 616 y(e)22 b FA(\003)g FD(g)31 b FE(=)d FD(g)d FA(\003)d FD(e)28 b FE(=)f FD(g)1570 761 y(f)33 b FA(\003)22 b FD(g)31 b FE(=)d FD(g)d FA(\003)d FD(f)38 b FE(=)28 b FD(g)t FC(.)147 958 y(Then)34 b FD(e)28 b FE(=)g FD(f)11 b FC(.)147 1215 y FI(Pro)s(of)147 1399 y FH(Since)33 b FD(e)g FH(is)f(an)g(iden)m(tit)m(y)-8 b(,)32 b(w)m(e)i(ha)m(v)m(e)1737 1597 y FD(e)22 b FA(\003)g FD(f)39 b FE(=)27 b FD(f)11 b FH(.)147 1794 y(On)33 b(the)g(other)f (hand,)h(since)g FD(f)43 b FH(is)33 b(an)f(iden)m(tit)m(y)-8 b(,)32 b(w)m(e)i(ha)m(v)m(e)1744 1992 y FD(e)22 b FA(\003)g FD(f)39 b FE(=)27 b FD(e)p FH(.)147 2189 y(Th)m(us)34 b FD(e)28 b FE(=)g FD(e)22 b FA(\003)g FD(f)38 b FE(=)28 b FD(f)11 b FH(.)43 b Fy(\003)147 2392 y FI(Prop)s(osition)36 b(3)h(\(Uniqueness)h(of)f(In)m(v)m(erses\))49 b FC(L)-5 b(et)37 b FD(G)g FC(b)-5 b(e)37 b(a)f(gr)-5 b(oup,)37 b FD(e)h FC(the)e(\(unique\))h(iden-)147 2512 y(tity)f(of)f FD(G)p FC(,)f(and)g FD(g)t(;)17 b(h;)g(k)37 b FC(arbitr)-5 b(ary)35 b(elements)f(of)h FD(G)p FC(.)44 b(Supp)-5 b(ose)34 b(that)1578 2710 y FD(g)25 b FA(\003)d FD(h)28 b FE(=)g FD(h)22 b FA(\003)g FD(g)31 b FE(=)c FD(e)1580 2855 y(g)e FA(\003)d FD(k)31 b FE(=)d FD(k)d FA(\003)d FD(g)31 b FE(=)c FD(e:)147 3052 y FC(Then)34 b FD(h)28 b FE(=)g FD(k)s FC(.)147 3309 y FI(Pro)s(of)147 3493 y FH(W)-8 b(e)33 b(kno)m(w)h(that)e FD(g)26 b FA(\003)21 b FD(h)28 b FE(=)g FD(g)d FA(\003)d FD(k)36 b FE(\(=)27 b FD(e)p FE(\))p FH(.)44 b(Multiplying)30 b(on)i(the)h(left)f(b)m(y)h FD(h)g FH(giv)m(es)1440 3691 y FD(h)22 b FA(\003)g FE(\()p FD(g)j FA(\003)d FD(h)p FE(\))28 b(=)g FD(h)22 b FA(\003)g FE(\()p FD(g)j FA(\003)d FD(k)s FE(\))p FH(.)147 3888 y(By)33 b(asso)s(ciativit)m(y)-8 b(,)31 b(this)i(giv)m(es)1440 4086 y FE(\()p FD(h)22 b FA(\003)g FD(g)t FE(\))f FA(\003)h FD(h)28 b FE(=)g(\()p FD(h)22 b FA(\003)g FD(g)t FE(\))f FA(\003)h FD(k)s FH(,)147 4283 y(and)33 b(so)1685 4481 y FD(e)22 b FA(\003)g FD(h)28 b FE(=)f FD(e)c FA(\003)f FD(k)1824 4626 y(h)28 b FE(=)f FD(k)s FH(.)446 4823 y(This)33 b(is)f(what)h(w)m(e)g(w)m(an)m(ted)h(to)e(pro)m(v)m(e.)45 b Fy(\003)147 5026 y FI(Prop)s(osition)36 b(4)48 b FC(L)-5 b(et)43 b FD(G)e FC(b)-5 b(e)42 b(a)g(gr)-5 b(oup,)43 b FD(e)f FC(the)g(identity)g(element)f(of)g FD(G)p FC(,)j(and)d FD(g)k FC(an)d(arbitr)-5 b(ary)147 5146 y(element)37 b(of)h FD(G)p FC(.)54 b(Supp)-5 b(ose)37 b FD(h)d FA(2)g FD(G)k FC(satis\034es)f(either)h FD(h)24 b FA(\003)h FD(g)36 b FE(=)e FD(e)k FC(or)g FD(g)28 b FA(\003)c FD(h)33 b FE(=)h FD(e)p FC(.)54 b(Then)37 b FD(h)h 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FI(Prop)s(osition)i(5)h(\(Prop)s(erties)f(of)h(In)m(v)m(erses\)) 49 b FC(L)-5 b(et)31 b FD(G)g FC(b)-5 b(e)31 b(a)g(gr)-5 b(oup,)31 b FD(e)g FC(its)h(identity,)f(and)g FD(g)t(;)17 b(h)147 4330 y FC(arbitr)-5 b(ary)35 b(elements)f(of)h FD(G)p FC(.)44 b(Then)1566 4463 y Fv(\000)1612 4543 y FD(g)1663 4502 y Fw(\000)p Fz(1)1757 4463 y Fv(\001)1802 4485 y Fw(\000)p Fz(1)1924 4543 y FE(=)28 b FD(g)1620 4704 y FE(\()p FD(g)t(h)p FE(\))1802 4656 y Fw(\000)p Fz(1)1924 4704 y FE(=)g FD(h)2084 4663 y Fw(\000)p Fz(1)2178 4704 y FD(g)2229 4663 y Fw(\000)p Fz(1)1757 4849 y FD(e)1802 4808 y Fw(\000)p Fz(1)1924 4849 y FE(=)g FD(e)p FC(.)147 5107 y FI(Pro)s(of)147 5292 y FH(Exercise.)p eop %%Page: 8 8 8 7 bop 147 48 a FK(8)3286 b FG(Groups)147 298 y FI(1.2)112 b(Some)36 b(Examples)e(of)k(Groups)147 477 y(Notation)147 684 y FH(F)-8 b(rom)30 b(no)m(w)i(on,)g(w)m(e)h(will)c(denote)j(the)g (pro)s(duct)g(of)f(t)m(w)m(o)i(group)e(elemen)m(ts)h FD(g)2951 699 y Fz(1)3022 684 y FH(and)f FD(g)3257 699 y Fz(2)3328 684 y FH(simply)f(b)m(y)147 804 y FD(g)194 819 y Fz(1)233 804 y FD(g)280 819 y Fz(2)319 804 y FH(,)j(instead)g(of) f(the)h(more)e(cum)m(b)s(ersome)i FD(g)1844 819 y Fz(1)1905 804 y FA(\003)22 b FD(g)2024 819 y Fz(2)2063 804 y FH(.)44 b(Moreo)m(v)m(er,)34 b(since)f(w)m(e)h(ha)m(v)m(e)f(asso)s(ciativit)m (y)-8 b(,)147 925 y(w)m(e)34 b(will)c(write)i(simply)f FD(g)1083 940 y Fz(1)1122 925 y FD(g)1169 940 y Fz(2)1209 925 y FD(g)1256 940 y Fz(3)1327 925 y FH(in)h(place)g(of)g FE(\()p FD(g)1886 940 y Fz(1)1925 925 y FD(g)1972 940 y Fz(2)2012 925 y FE(\))p FD(g)2097 940 y Fz(3)2168 925 y FH(or)g FD(g)2334 940 y Fz(1)2374 925 y FE(\()p FD(g)2459 940 y Fz(2)2498 925 y FD(g)2545 940 y Fz(3)2584 925 y FE(\))p FH(.)147 1253 y FI(The)38 b(trivial)d(group)147 1461 y FH(The)k(set)f(with)f(one)h(elemen)m(t,)g FD(e)p FH(,)h(is)e(a)g(group,)i(with)e(the)h(group)f(op)s(eration)f(b)s(eing)h (de\034ned)i(as)147 1581 y FD(ee)28 b FE(=)g FD(e)p FH(.)43 b(This)33 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b(group)f(is)g(also)g(comm)m(utativ)m(e.)446 4011 y(The)i(v)m(eri\034cation)d(is)i(the)f(same)h(as)g(for)f(the)h(in) m(tegers.)147 4339 y FI(Non-zero)38 b(real)e(n)m(um)m(b)s(ers)g(under)i (m)m(ultiplic)o(ation)147 4547 y FH(The)46 b(set)f(of)f(non-zero)h (real)e(n)m(um)m(b)s(ers)j(forms)d(a)i(group)f(with)g(resp)s(ect)i(to)e (the)h(op)s(eration)e(of)147 4667 y(m)m(ultiplication.)c(This)33 b(group)f(is)g(comm)m(utativ)m(e.)446 4799 y(Again)k(w)m(e)i(c)m(hec)m (k)i(closure:)53 b(the)37 b(pro)s(duct)h(of)e(t)m(w)m(o)i(non-zero)f (real)g(n)m(um)m(b)s(ers)h(is)e(a)h(non-)147 4919 y(zero)25 b(real)f(n)m(um)m(b)s(er.)41 b(Multiplication)21 b(is)j(asso)s(ciativ)m (e;)j(one)e(is)g(the)g(m)m(ultiplicativ)m(e)c(iden)m(tit)m(y;)27 b(eac)m(h)147 5040 y(non-zero)34 b(real)e(n)m(um)m(b)s(er)i FD(x)f FH(has)h(a)f(m)m(ultiplicativ)m(e)d(in)m(v)m(erse,)35 b(namely)-8 b(,)2773 5000 y Fz(1)p 2771 5017 40 4 v 2771 5074 a Ft(x)2820 5040 y FH(.)46 b(Since)34 b(m)m(ultiplication)147 5160 y(of)e(real)g(n)m(um)m(b)s(ers)h(is)f(comm)m(utativ)m(e,)g(this)g (is)g(a)g(comm)m(utativ)m(e)g(group.)446 5292 y(This)h(group)f(is)g (denoted)i Fu(R)1478 5255 y Fw(\003)1523 5292 y FH(.)p eop %%Page: 9 9 9 8 bop 147 48 a Fx(Some)28 b(Examples)f(of)g(Groups)2596 b FG(9)147 298 y FI(Non-zero)38 b(complex)c(n)m(um)m(b)s(ers)h(under)j (m)m(ultiplicati)o(on)147 484 y FH(The)32 b(set)g(of)e(non-zero)h (complex)f(n)m(um)m(b)s(ers)i(forms)e(a)g(group)h(with)f(resp)s(ect)i (to)f(the)g(op)s(eration)e(of)147 604 y(complex)j(m)m(ultiplication.)39 b(This)33 b(group)f(is)g(comm)m(utativ)m(e.)446 725 y(This)h(group)f (in)g(denoted)h Fu(C)1493 689 y Fw(\003)1539 725 y FH(.)147 988 y FI(Complex)i(n)m(um)m(b)s(ers)g(of)j(absolute)f(v)-6 b(alue)37 b(one)h(under)g(m)m(ultiplicati)o(on)147 1174 y FH(The)i(set)f(of)f(complex)g(n)m(um)m(b)s(ers)h(with)f(absolute)g(v) -5 b(alue)38 b(one)h(\(i.e.,)g(of)f(the)h(form)e FD(e)3279 1137 y Ft(i\022)3342 1174 y FH(\))i(forms)e(a)147 1294 y(group)c(under)g(complex)f(m)m(ultiplication.)39 b(This)32 b(group)h(is)f(comm)m(utativ)m(e.)446 1415 y(This)h(group)f(is)g(the)h (unit)f(circle,)g(denoted)h FD(S)2133 1379 y Fz(1)2172 1415 y FH(.)147 1678 y FI(In)m(v)m(ertible)j(matrices)147 1864 y FH(F)-8 b(or)35 b(eac)m(h)i(p)s(ositiv)m(e)e(in)m(teger)g FD(n)p FH(,)i(the)f(set)h(of)e(all)e FD(n)25 b FA(\002)g FD(n)36 b FH(in)m(v)m(ertible)f(matrices)f(with)i(real)e(en)m(tries)147 1984 y(forms)f(a)h(group)f(with)g(resp)s(ect)i(to)f(the)g(op)s(eration) e(of)h(matrix)f(m)m(ultiplication.)42 b(This)34 b(group)g(in)147 2104 y(non-comm)m(utativ)m(e,)d(for)h FD(n)c FA(\025)g FE(2)p FH(.)446 2225 y(W)-8 b(e)40 b(c)m(hec)m(k)i(closure:)58 b(the)41 b(pro)s(duct)f(of)f(t)m(w)m(o)h(in)m(v)m(ertible)f(matrices)g (is)h(in)m(v)m(ertible,)h(since)147 2346 y FE(\()p FD(AB)5 b FE(\))375 2297 y Fw(\000)p Fz(1)509 2346 y FE(=)39 b FD(B)703 2309 y Fw(\000)p Fz(1)798 2346 y FD(A)871 2309 y Fw(\000)p Fz(1)965 2346 y FH(.)64 b(Matrix)39 b(m)m(ultiplication)c(is)j(asso)s(ciativ)m(e;)43 b(the)d(iden)m(tit)m (y)f(matrix)e(\(with)147 2466 y(ones)h(do)m(wn)f(the)g(diagonal,)e(and) i(zeros)g(elsewhere\))h(is)f(the)g(iden)m(tit)m(y)f(elemen)m(t;)i(b)m (y)g(de\034nition,)147 2586 y(an)i(in)m(v)m(ertible)g(matrix)f(has)h (an)h(in)m(v)m(erse.)68 b(Simple)38 b(examples)i(sho)m(w)i(that)e(the)h (group)f(is)f(non-)147 2707 y(comm)m(utativ)m(e,)32 b(except)i(in)e (the)h(trivial)c(case)34 b FD(n)28 b FE(=)f(1)p FH(.)44 b(\(See)33 b(Exercise)h(7.\))446 2828 y(This)26 b(group)f(is)g(called)f (the)i FI(general)j(linear)f(group)d FH(\(o)m(v)m(er)i(the)e(reals\),)i (and)e(is)g(denoted)147 2948 y FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))h FH(.)147 3211 y FI(Symmetric)31 b(group)38 b(\(p)s(erm)m(utation)33 b(group\))147 3397 y FH(The)g(set)f(of)f(one-to-one,)h(on)m(to)f(maps)h(of)f(the)h(set)g FA(f)p FE(1)p FD(;)17 b FE(2)p FD(;)g FA(\001)g(\001)g(\001)d FD(n)p FA(g)32 b FH(to)f(itself)f(forms)h(a)h(group)f(under)147 3517 y(the)i(op)s(eration)e(of)h(comp)s(osition.)41 b(This)33 b(group)f(is)g(non-comm)m(utativ)m(e)g(for)g FD(n)27 b FA(\025)i FE(3)p FH(.)446 3638 y(W)-8 b(e)31 b(c)m(hec)m(k)i (closure:)42 b(the)31 b(comp)s(osition)e(of)h(t)m(w)m(o)h(one-to-one,)f (on)m(to)h(maps)f(is)g(again)f(one-)147 3758 y(to-one)40 b(and)g(on)m(to.)67 b(Comp)s(osition)38 b(of)i(functions)g(is)g(asso)s (ciativ)m(e;)k(the)d(iden)m(tit)m(y)f(map)g(\(whic)m(h)147 3879 y(sends)j(1)e(to)f(1,)j(2)e(to)f(2,)j(etc.\))70 b(is)40 b(the)i(iden)m(tit)m(y)e(elemen)m(t;)45 b(a)c(one-to-one,)i(on) m(to)e(map)f(has)h(an)147 3999 y(in)m(v)m(erse.)j(Simple)29 b(examples)h(sho)m(w)i(that)e(the)h(group)f(is)g(non-comm)m(utativ)m (e,)g(as)h(long)e(as)i FD(n)f FH(is)g(at)147 4119 y(least)i(3.)44 b(\(See)33 b(Exercise)h(9.\))446 4240 y(This)29 b(group)f(is)g(called)f (the)i FI(symmetri)o(c)e(group)p FH(,)i(and)g(is)f(denoted)h FD(S)3053 4255 y Ft(n)3100 4240 y FH(.)42 b(A)29 b(one-to-one,)147 4361 y(on)m(to)i(map)g(of)g FA(f)o FE(1)p FD(;)17 b FE(2)p FD(;)g FA(\001)g(\001)g(\001)d FD(n)p FA(g)31 b FH(is)g(a)g(p)s(erm)m (utation,)f(and)i(so)f FD(S)2325 4376 y Ft(n)2403 4361 y FH(is)g(also)g(called)f(the)h FI(p)s(erm)m(utation)147 4481 y(group)p FH(.)44 b(The)33 b(group)g FD(S)1037 4496 y Ft(n)1116 4481 y FH(has)g FD(n)p FE(!)g FH(elemen)m(ts.)147 4744 y FI(In)m(tegers)k(mo)s(d)e FD(n)147 4930 y FH(The)e(set)f FA(f)p FE(0)p FD(;)17 b FE(1)p FD(;)g FA(\001)g(\001)g(\001)d FD(n)22 b FA(\000)h FE(1)p FA(g)31 b FH(forms)g(a)h(group)f(under)i (the)f(op)s(eration)e(of)h(addition)f FI(mo)s(d)k FD(n)p FH(.)44 b(This)147 5050 y(group)33 b(is)f(comm)m(utativ)m(e.)446 5171 y(Explicitly)-8 b(,)22 b(the)g(group)g(op)s(eration)f(is)g(the)h (follo)m(wing.)37 b(Consider)23 b FD(a;)17 b(b)28 b FA(2)g(f)p FE(0)p FD(;)17 b FE(1)g FA(\001)g(\001)g(\001)c FD(n)23 b FA(\000)f FE(1)p FA(g)p FH(.)147 5292 y(If)32 b FD(a)22 b FE(+)f FD(b)28 b(<)f(n)p FH(,)33 b(then)f FD(a)22 b FE(+)f FD(b)32 b Fs(mo)s(d)g FD(n)c FE(=)g FD(a)21 b FE(+)g FD(b)p FH(,)33 b(if)e FD(a)21 b FE(+)g FD(b)28 b FA(\025)h FD(n)p FH(,)j(then)h FD(a)21 b FE(+)g FD(b)33 b Fs(mo)s(d)f FD(n)27 b FE(=)h FD(a)21 b FE(+)g FD(b)h FA(\000)g FD(n)p FH(.)p eop %%Page: 10 10 10 9 bop 147 48 a FK(10)3241 b FG(Groups)147 298 y FH(\(Since)28 b FD(a)g FH(and)g FD(b)g FH(are)g(less)g(than)g FD(n)p FH(,)h FD(a)12 b FE(+)g FD(b)g FA(\000)g FD(n)31 b FH(is)c(less)h(than) g FD(n)p FH(;)h(th)m(us)g(w)m(e)g(ha)m(v)m(e)g(closure.\))42 b(T)-8 b(o)28 b(sho)m(w)147 418 y(asso)s(ciativit)m(y)-8 b(,)28 b(note)h(that)f(b)s(oth)h FE(\()p FD(a)14 b FE(+)g FD(b)29 b Fs(mo)s(d)g FD(n)p FE(\))14 b(+)g FD(c)29 b Fs(mo)s(d)f FD(n)h FH(and)g FD(a)14 b FE(+)g(\()p FD(b)g FE(+)g FD(c)29 b Fs(mo)s(d)g FD(n)p FE(\))f Fs(mo)s(d)h FD(n)147 539 y FH(are)44 b(equal)h(to)f FD(a)30 b FE(+)g FD(b)g FE(+)h FD(c)p FH(,)47 b(min)m(us)d(some)g(m)m(ultiple)d(of)j FD(n)p FH(,)k(and)c(hence)i(di\033er)d(b)m(y)j(a)e(m)m(ultiple)147 659 y(of)e FD(n)p FH(.)72 b(But)43 b(since)f(b)s(oth)g(are)g(in)g(the)g (set)h FA(f)p FE(0)p FD(;)17 b FE(1)p FD(;)g FA(\001)g(\001)g(\001)d FD(n)22 b FA(\000)h FE(1)p FA(g)o FH(,)45 b(the)d(only)g(p)s(ossible)g (m)m(ultiple)d(on)147 779 y FD(n)g FH(is)f(zero.)61 b(Zero)38 b(is)g(still)e(the)i(iden)m(tit)m(y)h(for)e(addition)g FI(mo)s(d)f FD(n)p FH(.)61 b(The)39 b(in)m(v)m(erse)h(of)e(an)g(elemen) m(t)147 900 y FD(a)c FA(2)h(f)o FE(0)p FD(;)17 b FE(1)p FD(;)g FA(\001)g(\001)g(\001)d FD(n)23 b FA(\000)f FE(1)p FA(g)36 b FH(is)g FD(n)25 b FA(\000)g FD(a)p FH(.)54 b(\(Exercise:)f(c)m(hec)m(k)38 b(that)e FD(n)25 b FA(\000)g FD(a)36 b FH(is)g(in)g FA(f)o FE(0)p FD(;)17 b FE(1)p FD(;)g FA(\001)g(\001)g(\001)d FD(n)23 b FA(\000)f FE(1)p FA(g)p FH(,)37 b(and)147 1020 y(that)c FD(a)23 b FE(+)g(\()p FD(n)g FA(\000)g FD(a)p FE(\))33 b Fs(mo)s(d)h FD(n)29 b FE(=)g(0)p FH(.\))46 b(The)34 b(group)f(is)g(comm)m(utativ)m(e)f(b)s (ecause)j(ordinary)e(addition)147 1140 y(is)f(comm)m(utativ)m(e.)446 1261 y(This)h(group)f(is)g(refered)i(to)e(as)g(\020)p Fu(Z)e Fs(mo)s(d)j FD(n)p FH(,\021)f(and)h(is)f(denoted)h Fu(Z)2891 1276 y Ft(n)2936 1261 y FH(.)147 1519 y FI(1.3)112 b(Subgroups,)39 b(the)e(Cen)m(ter,)h(and)g(Direct)e(Pro)s(ducts)147 1687 y(De\034nition)g(6)49 b FC(A)34 b FB(sub)-6 b(gr)g(oup)36 b FC(of)e(a)g(gr)-5 b(oup)34 b FD(G)g FC(is)g(a)g(subset)g FD(H)41 b FC(of)34 b FD(G)g FC(with)g(the)g(fol)5 b(lowing)33 b(pr)-5 b(op-)147 1807 y(erties:)263 2007 y(1.)48 b(The)34 b(identity)h(is)g(an)f(element)g(of)h FD(H)8 b FC(.)263 2204 y(2.)48 b(If)34 b FD(h)28 b FA(2)g FD(H)8 b FC(,)35 b(then)f FD(h)1097 2168 y Fw(\000)p Fz(1)1219 2204 y FA(2)28 b FD(H)8 b FC(.)263 2400 y(3.)48 b(If)34 b FD(h)549 2415 y Fz(1)589 2400 y FD(;)17 b(h)689 2415 y Fz(2)756 2400 y FA(2)28 b FD(H)8 b FC(,)34 b(then)h FD(h)1276 2415 y Fz(1)1315 2400 y FD(h)1371 2415 y Fz(2)1439 2400 y FA(2)28 b FD(H)42 b FC(.)147 2656 y FI(Remark)147 2841 y FH(The)34 b(conditions)e(on)h FD(H)40 b FH(guaran)m(tee)34 b(that)e FD(H)41 b FH(is)32 b(a)h(group,)g(with)f(the)i(same)e(pro)s (duct)h(op)s(eration)147 2961 y(as)46 b FD(G)g FH(\(but)g(restricted)g (to)g FD(H)8 b FH(\).)83 b(Closure)46 b(is)f(assured)j(b)m(y)e(\(3\),)j (asso)s(ciativit)m(y)c(follo)m(ws)f(from)147 3082 y(asso)s(ciativit)m (y)34 b(in)g FD(G)p FH(,)i(and)f(the)g(existence)i(of)d(an)h(iden)m (tit)m(y)g(and)g(of)g(in)m(v)m(erses)i(is)d(assured)i(b)m(y)g(\(1\))147 3202 y(and)d(\(2\).)147 3458 y FI(Examples)147 3643 y FH(Ev)m(ery)39 b(group)e FD(G)g FH(has)g(at)g(least)f(t)m(w)m(o)i (subgroups:)53 b FD(G)37 b FH(itself,)g(and)g(the)h(one-elemen)m(t)e (subgroup)147 3763 y FA(f)p FD(e)p FA(g)p FH(.)53 b(\(If)36 b FD(G)g FH(itself)e(is)h(the)i(trivial)c(group,)j(then)h(these)g(t)m (w)m(o)f(subgroups)h(coincide.\))52 b(These)38 b(are)147 3884 y(called)32 b(the)g FI(trivial)j(subgroups)f FH(of)e FD(G)p FH(.)446 4004 y(The)38 b(set)g(of)f(ev)m(en)h(in)m(tegers)g(is)e (a)h(subgroup)h(of)f Fu(Z)p FH(:)50 b(zero)37 b(is)g(ev)m(en,)j(the)d (negativ)m(e)g(of)g(an)147 4124 y(ev)m(en)d(in)m(teger)f(is)f(ev)m(en,) i(and)f(the)g(sum)f(of)g(t)m(w)m(o)h(ev)m(en)h(in)m(tegers)f(is)f(ev)m (en.)446 4245 y(The)49 b(set)h FD(H)55 b FH(of)48 b FD(n)34 b FA(\002)f FD(n)49 b FH(real)e(matrices)h(with)g(determinan)m(t)g(one) g(is)g(a)g(subgroup)h(of)147 4365 y FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))p FH(.)66 b(The)39 b(set)g FD(H)46 b FH(is)38 b(a)g(sub)p FC(set)48 b FH(of)38 b FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))44 b FH(b)s(ecause)39 b(an)m(y)g(matrix)e(with)h(determinan)m(t)147 4486 y(one)k(is)g(in)m(v) m(ertible.)71 b(The)43 b(iden)m(tit)m(y)e(matrix)g(has)h(determinan)m (t)f(one,)k(so)d(1)g(is)f(satis\034ed.)72 b(The)147 4606 y(determinan)m(t)33 b(of)g(the)h(in)m(v)m(erse)h(is)e(the)h(recipro)s (cal)f(of)g(the)h(determinan)m(t,)f(so)h(2)f(is)g(satis\034ed;)i(and) 147 4726 y(the)25 b(determinan)m(t)g(of)f(a)g(pro)s(duct)h(is)g(the)g (pro)s(duct)g(of)f(the)h(determinan)m(ts,)i(so)e(3)f(is)h(satis\034ed.) 41 b(This)147 4847 y(group)33 b(is)f(called)f(the)i FI(sp)s(ecial)j (linear)h(group)32 b FH(\(o)m(v)m(er)i(the)f(reals\),)f(and)h(is)f (denoted)h FF(SL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))p FH(.)446 4967 y(Additional)30 b(examples,)j(as)f(w)m(ell)g(as)h (some)f(non-examples,)g(are)h(giv)m(en)f(in)g(Exercise)i(1.)147 5167 y FI(De\034nition)i(7)49 b FC(The)33 b FB(c)-6 b(enter)34 b FC(of)f(a)g(gr)-5 b(oup)33 b FD(G)h FC(is)f(the)g(set)h(of)f(al)5 b(l)33 b FD(g)e FA(2)d FD(G)34 b FC(such)f(that)h FD(g)t(h)27 b FE(=)g FD(hg)37 b FC(for)147 5288 y(al)5 b(l)35 b FD(h)27 b FA(2)i FD(G)p FC(.)p eop %%Page: 11 11 11 10 bop 147 48 a Fx(Homomorphisms)26 b(and)i(Isomorphisms)2216 b FG(11)147 298 y FI(Exercise)147 482 y FH(Sho)m(w)34 b(that)e(the)h(cen)m(ter)h(of)e(an)m(y)h(group)f FD(G)h FH(is)f(a)g(subgroup)h FD(G)p FH(.)147 693 y FI(De\034nition)j(8)49 b FC(L)-5 b(et)36 b FD(G)f FC(and)g FD(H)43 b FC(b)-5 b(e)36 b(gr)-5 b(oups,)35 b(and)g(c)-5 b(onsider)34 b(the)i(Cartesian)e (pr)-5 b(o)g(duct)36 b(of)f FD(G)h FC(and)147 813 y FD(H)8 b FC(,)34 b(i.e.,)g(the)h(set)f(of)h(or)-5 b(der)g(e)g(d)34 b(p)-5 b(airs)34 b FE(\()p FD(g)t(;)17 b(h)p FE(\))33 b FC(with)i FD(g)c FA(2)d FD(G;)17 b(h)27 b FA(2)h FD(H)8 b FC(.)44 b(De\034ne)34 b(a)g(pr)-5 b(o)g(duct)35 b(op)-5 b(er)g(ation)147 933 y(on)35 b(this)f(set)h(as)g(fol)5 b(lows:)1321 1138 y FE(\()p FD(g)1406 1153 y Fz(1)1445 1138 y FD(;)17 b(h)1545 1153 y Fz(1)1585 1138 y FE(\)\()p FD(g)1708 1153 y Fz(2)1747 1138 y FD(;)g(h)1847 1153 y Fz(2)1886 1138 y FE(\))28 b(=)f(\()p FD(g)2140 1153 y Fz(1)2179 1138 y FD(g)2226 1153 y Fz(2)2265 1138 y FD(;)17 b(h)2365 1153 y Fz(1)2405 1138 y FD(h)2461 1153 y Fz(2)2500 1138 y FE(\))p FC(.)147 1342 y(This)47 b(op)-5 b(er)g(ation)46 b(makes)h(the)g(Cartesian)f(pr)-5 b(o)g(duct)48 b(of)f FD(G)g FC(and)f FD(H)55 b FC(into)47 b(a)h(gr)-5 b(oup,)50 b(c)-5 b(al)5 b(le)-5 b(d)46 b(the)147 1462 y FB(dir)-6 b(e)g(ct)41 b(pr)-6 b(o)g(duct)38 b FC(of)c FD(G)h FC(and)f FD(H)43 b FC(and)34 b(denote)-5 b(d)34 b FD(G)22 b FA(\002)h FD(H)8 b FC(.)147 1720 y FI(Remark)147 1904 y FH(It)37 b(is)f(a)g(simple)e(matter)i(to)g(c)m(hec)m(k)j(that)d (this)g(op)s(eration)f(truly)h(mak)m(es)h FD(G)24 b FA(\002)i FD(H)43 b FH(in)m(to)36 b(a)g(group.)147 2025 y(F)-8 b(or)26 b(example,)h(the)g(iden)m(tit)m(y)f(elemen)m(t)g(of)g FD(G)9 b FA(\002)g FD(H)35 b FH(is)26 b(the)g(pair)g FE(\()p FD(e)2514 2040 y Fz(1)2553 2025 y FD(;)17 b(e)2642 2040 y Fz(2)2682 2025 y FE(\))p FH(,)27 b(where)h FD(e)3095 2040 y Fz(1)3161 2025 y FH(is)d(the)i(iden)m(tit)m(y)147 2145 y(for)32 b FD(G)p FH(,)h(and)f FD(e)667 2160 y Fz(2)740 2145 y FH(is)g(the)h(iden)m(tit)m(y)f(for)g FD(H)8 b FH(.)147 2405 y FI(1.4)112 b(Homomorphism)n(s)31 b(and)39 b(Isomorphisms)147 2573 y(De\034nition)d(9)49 b FC(L)-5 b(et)41 b FD(G)g FC(and)f FD(H)49 b FC(b)-5 b(e)41 b(gr)-5 b(oups.)62 b(A)42 b(map)e FD(\036)f FE(:)g FD(G)g FA(!)f FD(H)49 b FC(is)40 b(c)-5 b(al)5 b(le)-5 b(d)40 b(a)h FB(homomor-)147 2693 y(phism)36 b FC(if)f FD(\036)p FE(\()p FD(g)709 2708 y Fz(1)748 2693 y FD(g)795 2708 y 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b Fu(R)5 b FE(\))p FH(.)391 2436 y FC(Note)i FH(:)65 b(a)43 b(group)g FD(G)g FH(with)f(no)h(normal) e(subgroups)j(other)f(than)g FD(G)g FH(and)g FA(f)p FD(e)p FA(g)g FH(is)f(called)391 2556 y FI(simple)p FH(.)p eop %%Page: 15 15 15 14 bop 1688 1047 a FL(Chapter)37 b(2)1316 1222 y(MA)-10 b(TRIX)39 b(LIE)g(GR)m(OUPS)147 1491 y FI(2.1)112 b(De\034nition)36 b(of)i(a)g(Matrix)f(Lie)f(Group)147 1659 y FH(Recall)29 b(that)g(the)i FI(general)j(linear)f(group)e FH(o)m(v)m(er)g(the)f (reals,)g(denoted)h FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))p FH(,)36 b(is)30 b(the)g(group)147 1779 y(of)f(all)e FD(n)16 b FA(\002)g FD(n)30 b FH(in)m(v)m(ertible)f (matrices)f(with)h(real)f(en)m(tries.)43 b(W)-8 b(e)30 b(ma)m(y)f(similarly)d(de\034ne)31 b FF(GL)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))35 b FH(to)147 1900 y(b)s(e)26 b(the)h(group)f(of)f(all)f FD(n)9 b FA(\002)g FD(n)26 b FH(in)m(v)m(ertible)g(matrices)f(with)g(complex)h(en)m(tries.)42 b(Of)25 b(course,)k FF(GL)o FE(\()p FD(n)p FE(;)17 b 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1373 y(0)108 b FD(e)1981 1336 y Ft(ita)2114 1173 y Fv(\023)2203 1313 y FA(j)p FD(t)28 b FA(2)g Fu(R)2470 1173 y Fv(\033)147 1603 y FH(Clearly)-8 b(,)37 b FD(H)45 b FH(is)37 b(a)f(subgroup)i(of)f FF(GL)p FE(\(2)p FD(;)17 b Fu(C)i FE(\))p FH(.)63 b(Because)38 b FD(a)g FH(is)e(irrational,)f(the)j(matrix)d FA(\000)p FD(I)46 b FH(is)36 b(not)147 1723 y(in)c FD(H)8 b FH(,)32 b(since)h(to)f(mak)m(e)h FD(e)1067 1687 y Ft(it)1153 1723 y FH(equal)f(to)h FA(\000)p FE(1)p FH(,)f(w)m(e)i(m)m(ust)f(tak)m (e)g FD(t)g FH(to)f(b)s(e)g(an)h(o)s(dd)f(in)m(teger)h(m)m(ultiple)c (of)147 1843 y FD(\031)t FH(,)37 b(in)f(whic)m(h)h(case)g FD(ta)g FH(cannot)f(b)s(e)h(an)f(o)s(dd)g(in)m(teger)g(m)m(ultiple)e (of)i FD(\031)t FH(.)55 b(On)36 b(the)h(other)f(hand,)i(b)m(y)147 1964 y(taking)30 b FD(t)e FE(=)f(\(2)p FD(n)17 b FE(+)h(1\))p FD(\031)34 b FH(for)29 b(a)h(suitably)g(c)m(hosen)i(in)m(teger)e FD(n)p FH(,)h(w)m(e)g(can)f(mak)m(e)h FD(ta)f FH(arbitrarily)e FC(close)147 2084 y FH(to)j(an)f(o)s(dd)h(in)m(teger)f(m)m(ultiple)e (of)j FD(\031)t FH(.)42 b(\(It)31 b(is)f(left)g(to)h(the)g(reader)g(to) f(v)m(erify)i(this.\))42 b(Hence)32 b(w)m(e)g(can)147 2205 y(\034nd)38 b(a)g(sequence)i(of)d(matrices)g(in)g FD(H)45 b FH(whic)m(h)38 b(con)m(v)m(erges)i(to)d FA(\000)p FD(I)8 b FH(,)39 b(and)f(so)g FD(H)45 b FH(is)37 b(not)g(a)h(matrix)147 2325 y(Lie)32 b(group.)43 b(See)34 b(Exercise)g(1.)147 2601 y FI(2.2)112 b(Examples)35 b(of)j(Matrix)e(Lie)h(Groups)147 2772 y FH(Mastering)28 b(the)g(sub)5 b(ject)29 b(of)e(Lie)g(groups)h (in)m(v)m(olv)m(es)g(not)g(only)f(learning)f(the)i(general)f(theory)-8 b(,)29 b(but)147 2892 y(also)39 b(familiarizing)c(oneself)40 b(with)f(examples.)67 b(In)40 b(this)g(section,)i(w)m(e)f(in)m(tro)s (duce)f(some)g(of)g(the)147 3013 y(most)32 b(imp)s(ortan)m(t)f (examples)h(of)g(\(matrix\))f(Lie)h(groups.)147 3296 y FI(The)38 b(general)f(linear)f(groups)i FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))43 b FI(and)38 b FF(GL)p FE(\()p FD(n)p 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FD(A)2821 3988 y Ft(n)2900 3973 y FA(2)e FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)t FE(\))p FH(,)43 b(and)35 b FD(A)3695 3988 y Ft(n)147 4094 y FH(con)m(v)m(erges)43 b(to)e FD(A)p FH(,)i(then)e(the)g(en)m(tries)g(of)g FD(A)g FH(are)f(real.)68 b(Th)m(us)42 b(either)e FD(A)h FH(is)g(not)f(in)m(v)m(ertible,)i(or)147 4214 y FD(A)28 b FA(2)g FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))p FH(.)147 4497 y FI(The)38 b(sp)s(ecial)e(linear)h (groups)h Fq(SL)p FE(\()o FD(n)p FE(;)17 b Fu(R)5 b FE(\))43 b FI(and)c Fq(SL)p FE(\()o FD(n)p FE(;)17 b Fu(C)j FE(\))147 4690 y FH(The)39 b FI(sp)s(ecial)j(linear)f(group)d FH(\(o)m(v)m(er)h Fu(R)48 b FH(or)37 b Fu(C)20 b FH(\))43 b(is)37 b(the)h(group)f(of)g FD(n)26 b FA(\002)g FD(n)37 b FH(in)m(v)m(ertible)g(matrices)147 4810 y(\(with)d(real)e(or)i(complex)f(en)m(tries\))i(ha)m(ving)e (determinan)m(t)g(one.)48 b(Both)34 b(of)f(these)i(are)f(subgroups)147 4930 y(of)28 b FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(,)35 b(as)29 b(noted)g(in)f(Chapter)h(1.)42 b(F)-8 b(urthermore,)29 b(if)e FD(A)2411 4945 y Ft(n)2487 4930 y FH(is)h(a)g(sequence)k(of)c(matrices)f(with)147 5051 y(determinan)m(t)34 b(one,)h(and)g FD(A)1172 5066 y Ft(n)1254 5051 y FH(con)m(v)m(erges)h(to)e FD(A)p FH(,)h(then)h FD(A)e FH(also)g(has)h(determinan)m(t)e(one,)j(b)s(ecause)147 5171 y(the)30 b(determinan)m(t)f(is)g(a)g(con)m(tin)m(uous)h(function.) 41 b(Th)m(us)31 b Fp(SL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))35 b FH(and)30 b Fp(SL)p FE(\()o FD(n)p FE(;)17 b Fu(C)j FE(\))36 b FH(are)29 b(matrix)f(Lie)147 5292 y(groups.)p eop %%Page: 17 17 17 16 bop 147 48 a Fx(Examples)28 b(of)f(Matrix)g(Lie)h(Groups)2358 b FG(17)147 298 y FI(The)38 b(orthogonal)f(and)h(sp)s(ecial)f (orthogonal)g(groups,)h FF(O)p FE(\()p FD(n)p FE(\))f FI(and)h Fq(SO)p FE(\()p FD(n)p FE(\))147 487 y FH(An)30 b FD(n)16 b FA(\002)g FD(n)31 b FH(real)e(matrix)f FD(A)h FH(is)g(said)g(to)h(b)s(e)g FI(orthogonal)f FH(if)f(the)i(column)f(v)m (ectors)i(that)e(mak)m(e)h(up)147 607 y FD(A)j FH(are)f(orthonormal,)f (that)h(is,)g(if)1653 780 y Ft(n)1603 810 y Fv(X)1618 1020 y Ft(i)p Fz(=1)1763 905 y FD(A)1836 920 y Ft(ij)1897 905 y FD(A)1970 920 y Ft(ik)2065 905 y FE(=)27 b FD(\016)2211 920 y Ft(j)t(k)147 1219 y FH(Equiv)-5 b(alen)m(tly)d(,)43 b FD(A)e FH(is)g(orthogonal)e(if)i(it)f(preserv)m(es)k(the)e(inner)f (pro)s(duct,)i(namely)-8 b(,)43 b(if)d FA(h)p FD(x;)17 b(y)t FA(i)41 b FE(=)147 1340 y FA(h)p FD(Ax;)17 b(Ay)t FA(i)31 b FH(for)h(all)e(v)m(ectors)k FD(x;)17 b(y)36 b FH(in)31 b Fu(R)1530 1304 y Ft(n)1583 1340 y FH(.)44 b(\()17 b(Angled)31 b(brac)m(k)m(ets)j(denote)f(the)g(usual)f(inner)g (pro)s(duct)147 1460 y(on)f Fu(R)347 1424 y Ft(n)400 1460 y FH(,)h FA(h)p FD(x;)17 b(y)t FA(i)26 b FE(=)818 1386 y Fv(P)923 1489 y Ft(i)968 1460 y FD(x)1023 1475 y Ft(i)1052 1460 y FD(y)1100 1475 y Ft(i)1127 1460 y FH(.\))43 b(Still)29 b(another)i(equiv)-5 b(alen)m(t)31 b(de\034nition)g(is)f(that)h FD(A)h FH(is)f(orthogonal)e(if)147 1581 y FD(A)220 1544 y Ft(tr)284 1581 y FD(A)h FE(=)g FD(I)8 b FH(,)34 b(i.e.,)g(if)f FD(A)955 1544 y Ft(tr)1049 1581 y FE(=)d FD(A)1228 1544 y Fw(\000)p Fz(1)1322 1581 y FH(.)48 b(\()p FD(A)1508 1544 y Ft(tr)1606 1581 y FH(is)33 b(the)i(transp)s(ose)f(of)g FD(A)p FH(,)g FE(\()p FD(A)2671 1544 y Ft(tr)2735 1581 y FE(\))2773 1610 y Ft(ij)2863 1581 y FE(=)c FD(A)3042 1596 y Ft(j)t(i)3103 1581 y FH(.\))48 b(See)35 b(Exercise)147 1701 y(2.)446 1824 y(Since)50 b FE(det)17 b FD(A)943 1788 y Ft(tr)1064 1824 y FE(=)57 b(det)17 b FD(A)p FH(,)54 b(w)m(e)d(see)g(that)e(if)g FD(A)h FH(is)f(orthogonal,)k(then)d FE(det)q(\()p FD(A)3435 1788 y Ft(tr)3498 1824 y FD(A)p FE(\))57 b(=)147 1944 y(\(det)17 b FD(A)p FE(\))448 1896 y Fz(2)515 1944 y FE(=)28 b(det)17 b FD(I)36 b FE(=)27 b(1)p FH(.)43 b(Hence)34 b FE(det)17 b FD(A)28 b FE(=)g FA(\006)p FE(1)p FH(,)k(for)g(all)f (orthogonal)g(matrices)g FD(A)p FH(.)446 2067 y(This)46 b(form)m(ula)e(tells)g(us,)50 b(in)45 b(particular,)i(that)f(ev)m(ery)h (orthogonal)d(matrix)g(m)m(ust)i(b)s(e)147 2187 y(in)m(v)m(ertible.)d (But)33 b(if)e FD(A)i FH(is)f(an)g(orthogonal)f(matrix,)g(then)929 2333 y Fv(\012)976 2414 y FD(A)1049 2373 y Fw(\000)p Fz(1)1143 2414 y FD(x;)17 b(A)1315 2373 y Fw(\000)p Fz(1)1410 2414 y FD(y)1462 2333 y Fv(\013)1536 2414 y FE(=)1639 2333 y Fv(\012)1686 2414 y FD(A)1776 2333 y Fv(\000)1822 2414 y FD(A)1895 2373 y Fw(\000)p Fz(1)1989 2414 y FD(x)2044 2333 y Fv(\001)2107 2414 y FD(;)g(A)2241 2333 y Fv(\000)2286 2414 y FD(A)2359 2373 y Fw(\000)p Fz(1)2453 2414 y FD(x)2508 2333 y Fv(\001\013)2629 2414 y FE(=)28 b FA(h)o FD(x;)17 b(y)t FA(i)147 2641 y FH(Th)m(us)32 b(the)f(in)m(v)m(erse)h(of)e(an)h (orthogonal)d(matrix)h(is)i(orthogonal.)40 b(F)-8 b(urthermore,)31 b(the)g(pro)s(duct)f(of)147 2761 y(t)m(w)m(o)e(orthogonal)d(matrices)g (is)i(orthogonal,)f(since)h(if)f FD(A)h FH(and)f FD(B)32 b FH(b)s(oth)27 b(preserv)m(e)i(inner)d(pro)s(ducts,)147 2882 y(then)33 b(so)g(do)s(es)g FD(AB)5 b FH(.)44 b(Th)m(us)34 b(the)f(set)g(of)f(orthogonal)f(matrices)g(forms)h(a)g(group.)446 3004 y(The)h(set)f(of)f(all)e FD(n)20 b FA(\002)h FD(n)32 b FH(real)e(orthogonal)g(matrices)g(is)h(the)h FI(orthogonal)k(group)c FF(O)p FE(\()p FD(n)p FE(\))p FH(,)147 3125 y(and)46 b(is)f(a)g(subgroup)h(of)f FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)88 b(The)46 b(limit)c(of)j(a)g(sequence)j(of)d (orthogonal)f(matices)g(is)147 3245 y(orthogonal,)32 b(b)s(ecause)j(the)f(relation)e FD(A)1626 3209 y Ft(tr)1690 3245 y FD(A)d FE(=)h FD(I)41 b FH(is)33 b(preserv)m(ed)j(under)f (limits.)43 b(Th)m(us)35 b FF(O)p FE(\()p FD(n)p FE(\))f FH(is)f(a)147 3365 y(matrix)e(Lie)h(group.)446 3488 y(The)f(set)g(of)f FD(n)18 b FA(\002)g FD(n)30 b FH(orthogonal)e(matrices)i(with)f (determinan)m(t)h(one)h(is)e(the)i FI(sp)s(ecial)j(or-)147 3608 y(thogonal)k(group)c FF(SO)q FE(\()p FD(n)p FE(\))p FH(.)46 b(Clearly)33 b(this)g(is)g(a)g(subgroup)i(of)e FF(O)p FE(\()p FD(n)p FE(\))p FH(,)h(and)f(hence)i(of)e FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)147 3729 y(Moreo)m(v)m(er,)42 b(b)s(oth)c(orthogonalit)m(y)f(and)h(the)h (prop)s(ert)m(y)g(of)f(ha)m(ving)h(determinan)m(t)f(one)h(are)f(pre-) 147 3849 y(serv)m(ed)45 b(under)g(limits,)d(and)i(so)f FF(SO)p FE(\()p FD(n)p FE(\))h FH(is)f(a)g(matrix)e(Lie)i(group.)75 b(Since)44 b(elemen)m(ts)f(of)g FF(O)p FE(\()p FD(n)p FE(\))147 3970 y FH(already)32 b(ha)m(v)m(e)i(determinan)m(t)e FA(\006)p FE(1)p FH(,)h FF(SO)p FE(\()p FD(n)p FE(\))g FH(is)f(\020half)7 b(\021)31 b(of)h FF(O)p FE(\()p FD(n)p FE(\))p FH(.)446 4092 y(Geometrically)-8 b(,)29 b(elemen)m(ts)j(of)g FF(O)p FE(\()p FD(n)p FE(\))f FH(are)h(either)g(rotations,)f(or)h(com)m (binations)e(of)h(rota-)147 4213 y(tions)h(and)h(re\035ections.)44 b(The)33 b(elemen)m(ts)g(of)f FF(SO)p FE(\()p FD(n)p FE(\))h FH(are)f(just)h(the)g(rotations.)446 4335 y(See)g(also)f (Exercise)i(6.)147 4609 y FI(The)k(unitary)f(and)h(sp)s(ecial)f (unitary)g(groups,)h FF(U)p FE(\()p FD(n)p FE(\))f FI(and)i FF(SU)p FE(\()p FD(n)p FE(\))147 4798 y FH(An)i FD(n)28 b FA(\002)g FD(n)41 b FH(complex)f(matrix)f FD(A)h FH(is)h(said)f(to)g (b)s(e)h FI(unitary)g FH(if)f(the)h(column)e(v)m(ectors)j(of)e FD(A)h FH(are)147 4919 y(orthonormal,)30 b(that)j(is,)f(if)1653 5092 y Ft(n)1603 5121 y Fv(X)1618 5331 y Ft(i)p Fz(=1)p 1763 5136 134 4 v 1763 5216 a FD(A)1836 5231 y Ft(ij)1897 5216 y FD(A)1970 5231 y Ft(ik)2065 5216 y FE(=)27 b FD(\016)2211 5231 y Ft(j)t(k)p eop %%Page: 18 18 18 17 bop 147 48 a FK(18)2825 b FG(Matrix)26 b(Lie)i(Groups)147 298 y FH(Equiv)-5 b(alen)m(tly)d(,)55 b FD(A)d FH(is)e(unitary)h(if)f (it)g(preserv)m(es)k(the)e(inner)f(pro)s(duct,)56 b(namely)-8 b(,)55 b(if)50 b FA(h)p FD(x;)17 b(y)t FA(i)58 b FE(=)147 418 y FA(h)p FD(Ax;)17 b(Ay)t FA(i)26 b FH(for)g(all)f(v)m(ectors)j FD(x;)17 b(y)30 b FH(in)c Fu(C)1497 382 y Ft(n)1550 418 y FH(.)41 b(\(Angled)26 b(brac)m(k)m(ets)j(here)e(denote)h(the)f(inner) f(pro)s(duct)h(on)147 539 y Fu(C)213 502 y Ft(n)266 539 y FH(,)k FA(h)p FD(x;)17 b(y)t FA(i)26 b FE(=)683 464 y Fv(P)788 568 y Ft(i)p 833 484 84 4 v 833 539 a FD(x)888 554 y Ft(i)916 539 y FD(y)964 554 y Ft(i)992 539 y FH(.)43 b(W)-8 b(e)30 b(will)e(adopt)i(the)h(con)m(v)m(en)m(tion)g(of)f (putting)g(the)g(complex)g(conjugate)147 659 y(on)g(the)h(left.\))42 b(Still)28 b(another)i(equiv)-5 b(alen)m(t)30 b(de\034nition)g(is)f (that)i FD(A)f FH(is)g(unitary)g(if)f FD(A)3150 623 y Fw(\003)3190 659 y FD(A)e FE(=)h FD(I)8 b FH(,)31 b(i.e.,)f(if)147 779 y FD(A)220 743 y Fw(\003)287 779 y FE(=)e FD(A)464 743 y Fw(\000)p Fz(1)558 779 y FH(.)44 b(\()p FD(A)740 743 y Fw(\003)812 779 y FH(is)32 b(the)h(adjoin)m(t)e(of)h FD(A)p FH(,)h FE(\()p FD(A)1763 743 y Fw(\003)1803 779 y FE(\))1841 808 y Ft(ij)1929 779 y FE(=)p 2032 699 134 4 v 27 w FD(A)2105 794 y Ft(j)t(i)2166 779 y FH(.\))43 b(See)34 b(Exercise)g(3.)446 922 y(Since)f FE(det)17 b FD(A)926 886 y Fw(\003)993 922 y FE(=)p 1097 841 226 4 v 28 w(det)g FD(A)p FH(,)33 b(w)m(e)g(see)h(that)f(if)e FD(A)i FH(is)f(unitary)-8 b(,)32 b(then)h FE(det)17 b(\()p FD(A)3036 886 y Fw(\003)3076 922 y FD(A)p FE(\))28 b(=)f FA(j)p FE(det)17 b FD(A)p FA(j)3599 874 y Fz(2)3666 922 y FE(=)147 1042 y(det)g FD(I)36 b FE(=)27 b(1)p FH(.)44 b(Hence)33 b FA(j)p FE(det)17 b FD(A)p FA(j)28 b FE(=)f(1)p FH(,)33 b(for)f(all)e(unitary)i(matrices)g FD(A)p FH(.)446 1164 y(This)42 b(in)f(particular)f(sho)m(ws)k(that)d(ev)m(ery)j (unitary)d(matrix)f(is)i(in)m(v)m(ertible.)70 b(The)43 b(same)147 1284 y(argumen)m(t)31 b(as)h(for)e(the)i(orthogonal)e(group) h(sho)m(ws)i(that)e(the)h(set)g(of)e(unitary)h(matrices)g(forms)f(a)147 1405 y(group.)446 1526 y(The)41 b(set)g(of)e(all)f FD(n)28 b FA(\002)f FD(n)41 b FH(unitary)e(matrices)g(is)h(the)g FI(unitary)46 b(group)40 b FF(U)p FE(\()p FD(n)p FE(\))p FH(,)j(and)d(is)f(a)147 1647 y(subgroup)e(of)g FF(GL)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)62 b(The)37 b(limit)c(of)j(unitary)g(matrices)g(is)g(unitary)-8 b(,)37 b(so)g FF(U)p FE(\()p FD(n)p FE(\))g FH(is)f(a)h(matrix)147 1767 y(Lie)30 b(group.)43 b(The)32 b(set)f(of)g(unitary)f(matrices)g (with)g(determinan)m(t)g(one)h(is)g(the)g FI(sp)s(ecial)j(unitary)147 1888 y(group)42 b FF(SU)q FE(\()p FD(n)p FE(\))p FH(.)71 b(It)42 b(is)g(easy)h(to)e(c)m(hec)m(k)j(that)e FF(SU)p FE(\()p FD(n)p FE(\))g FH(is)g(a)f(matrix)f(Lie)i(group.)71 b(Note)42 b(that)f(a)147 2008 y(unitary)34 b(matrix)e(can)i(ha)m(v)m(e) h(determinan)m(t)e FD(e)1813 1972 y Ft(i\022)1910 2008 y FH(for)h(an)m(y)g FD(\022)s FH(,)g(and)g(so)g FF(SU)q FE(\()p FD(n)p FE(\))g FH(is)f(a)h(smaller)d(subset)147 2128 y(of)g FF(U)p FE(\()p FD(n)p FE(\))h FH(than)g FF(SO)p FE(\()p FD(n)p FE(\))f FH(is)g(of)g FF(O)p FE(\()p FD(n)p FE(\))p FH(.)43 b(\(Sp)s(eci\034cally)-8 b(,)31 b FF(SO)p FE(\()p FD(n)p FE(\))g FH(has)h(the)g(same)f(dimension)f(as)i FF(O)p FE(\()p FD(n)p FE(\))p FH(,)147 2249 y(whereas)i FF(SU)q FE(\()p FD(n)p FE(\))e FH(has)h(dimension)e(one)i(less)g(than)g (that)f(of)g FF(U)p FE(\()p FD(n)p FE(\))p FH(.\))446 2370 y(See)h(also)f(Exercise)i(8.)147 2638 y FI(The)k(complex)c (orthogonal)j(groups,)h FF(O)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))43 b FI(and)38 b FF(SO)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))147 2825 y FH(Consider)33 b(the)f(bilinear)e (form)g FE(\()j(\))f FH(on)g Fs(C)1659 2789 y Ft(n)1738 2825 y FH(de\034ned)h(b)m(y)g FE(\()p FD(x;)17 b(y)t FE(\))27 b(=)2565 2750 y Fv(P)2687 2825 y FD(x)2742 2840 y Ft(i)2771 2825 y FD(y)2819 2840 y Ft(i)2847 2825 y FH(.)43 b(This)32 b(form)f(is)g(not)h(an)147 2945 y(inner)i(pro)s (duct,)g(b)s(ecause)i(of)d(the)i(lac)m(k)f(of)f(a)h(complex)f (conjugate)h(in)f(the)i(de\034nition.)46 b(The)35 b(set)147 3066 y(of)30 b(all)e FD(n)18 b FA(\002)g FD(n)31 b FH(complex)f (matrices)f FD(A)i FH(whic)m(h)g(preserv)m(e)h(this)e(form,)g(\(i.e.,)g (suc)m(h)i(that)e FE(\()p FD(Ax;)17 b(Ay)t FE(\))27 b(=)147 3186 y(\()p FD(x;)17 b(y)t FE(\))k FH(for)g(all)f FD(x;)d(y)31 b FA(2)d Fs(C)1011 3150 y Ft(n)1058 3186 y FH(\))21 b(is)h(the)g FI(complex)g(orthogonal)i(group)e FF(O)p FE(\()p FD(n)p FE(;)17 b Fu(C)j 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b(a)h(subgroup)g(of)f FF(GL)o FE(\(3;)17 b Fu(R)5 b FE(\))p FH(.)55 b(Clearly)34 b(the)h(limit)c(of)j(matrices)f(of)h(the)h(form)f(\(2.4\))f(is)147 4681 y(again)e(of)h(that)h(form,)e(and)i(so)f FD(H)40 b FH(is)32 b(a)h(matrix)e(Lie)h(group.)446 4806 y(It)25 b(is)f(not)h(eviden)m(t)g(at)g(the)g(momen)m(t)f(wh)m(y)i(this)e(group) h(should)g(b)s(e)g(called)e(the)i(Heisen)m(b)s(erg)147 4926 y(group.)40 b(W)-8 b(e)23 b(shall)e(see)i(later)e(that)h(the)h (Lie)f(algebra)f(of)h FD(H)29 b FH(giv)m(es)23 b(a)f(realization)e(of)i (the)g(Heisen)m(b)s(erg)147 5046 y(comm)m(utation)39 b(relations)g(of)h(quan)m(tum)h(mec)m(hanics.)68 b(\(See)41 b(esp)s(ecially)f(Chapter)h(5,)i(Exercise)147 5167 y(10.\))446 5292 y(See)33 b(also)f(Exercise)i(10.)p eop %%Page: 21 21 21 20 bop 147 48 a Fx(Examples)28 b(of)f(Matrix)g(Lie)h(Groups)2358 b FG(21)147 298 y FI(The)38 b(groups)g Fu(R)806 262 y Fw(\003)851 298 y FI(,)g Fu(C)986 262 y Fw(\003)1031 298 y FI(,)f FD(S)1165 262 y Fz(1)1205 298 y FI(,)g Fu(R)5 b FI(,)44 b(and)38 b Fu(R)1696 262 y Ft(n)147 482 y FH(Sev)m(eral)30 b(imp)s(ortan)m(t)e(groups)h(whic)m(h)h(are)g(not)f(naturally)f(groups) i(of)f(matrices)f(can)i(\(and)g(will)d(in)147 603 y(these)34 b(notes\))f(b)s(e)g(though)m(t)g(of)f(as)g(suc)m(h.)446 723 y(The)i(group)f Fu(R)990 687 y Fw(\003)1068 723 y FH(of)g(non-zero)g(real)f(n)m(um)m(b)s(ers)i(under)f(m)m(ultiplication) c(is)j(isomorphic)f(to)147 844 y FF(GL)p FE(\(1)p FD(;)17 b Fu(R)t FE(\))p FH(.)57 b(Th)m(us)37 b(w)m(e)f(will)c(regard)j Fu(R)1537 807 y Fw(\003)1618 844 y FH(as)g(a)g(matrix)e(Lie)i(group.)50 b(Similarly)-8 b(,)32 b(the)j(group)g Fu(C)3583 807 y Fw(\003)3663 844 y FH(of)147 964 y(non-zero)k(complex)g(n)m(um)m(b)s (ers)h(under)g(m)m(ultiplication)34 b(is)39 b(isomorphic)e(to)i FF(GL)p FE(\(1;)17 b Fu(C)i FE(\))p FH(,)47 b(and)40 b(the)147 1084 y(group)33 b FD(S)490 1048 y Fz(1)561 1084 y FH(of)f(complex)g(n)m(um)m(b)s(ers)i(with)e(absolute)g(v)-5 b(alue)32 b(one)h(is)f(isomorphic)e(to)j FF(U)p FE(\(1\))p FH(.)446 1205 y(The)38 b(group)f Fu(R)48 b FH(under)38 b(addition)d(is)h(isomorphic)g(to)g FF(GL)p FE(\(1;)17 b Fu(R)t FE(\))2787 1169 y Fz(+)2889 1205 y FH(\()p FE(1)25 b FA(\002)h FE(1)37 b FH(real)f(matrices)147 1325 y(with)22 b(p)s(ositiv)m(e)g(determinan)m(t\))f(via)h(the)h(map)e FD(x)28 b FA(!)g FE([)p FD(e)2084 1289 y Ft(x)2128 1325 y FE(])p FH(.)40 b(The)23 b(group)f Fu(R)2744 1289 y Ft(n)2820 1325 y FH(\(with)g(v)m(ector)h(addition\))147 1446 y(is)39 b(isomorphic)e(to)i(the)g(group)g(of)g(diagonal)d(real)i (matrices)h(with)f(p)s(ositiv)m(e)h(diagonal)d(en)m(tries,)147 1566 y(via)c(the)h(map)1194 1915 y FE(\()p FD(x)1287 1930 y Fz(1)1327 1915 y FD(;)17 b FA(\001)g(\001)g(\001)31 b FD(;)17 b(x)1619 1930 y Ft(n)1666 1915 y FE(\))27 b FA(!)1859 1685 y Fv(0)1859 1860 y(B)1859 1924 y(@)1987 1773 y FD(e)2032 1737 y Ft(x)2072 1746 y Fr(1)2433 1773 y FE(0)2200 1877 y FH(.)2238 1902 y(.)2276 1927 y(.)2025 2056 y FE(0)318 b FD(e)2437 2019 y Ft(x)2477 2027 y Fn(n)2565 1685 y Fv(1)2565 1860 y(C)2565 1924 y(A)2668 1915 y FH(.)147 2314 y FI(The)38 b(Euclidean)e(and)i(P)m(oincare)f(groups)147 2499 y FH(The)30 b(Euclidean)e(group)h FF(E)p FE(\()p FD(n)p FE(\))g FH(is)g(b)m(y)h(de\034nition)d(the)j(group)e(of)h(all)e (one-to-one,)i(on)m(to,)g(distance-)147 2619 y(preserving)e(maps)e(of)g Fu(R)1028 2583 y Ft(n)1107 2619 y FH(to)g(itself,)h(that)f(is,)i(maps)e FD(f)38 b FE(:)28 b Fu(R)2256 2583 y Ft(n)2337 2619 y FA(!)f Fu(R)2530 2583 y Ft(n)2609 2619 y FH(suc)m(h)g(that)e FD(d)17 b FE(\()o FD(f)28 b FE(\()p FD(x)p FE(\))17 b FD(;)g(f)27 b FE(\()o FD(y)t FE(\)\))g(=)147 2740 y FD(d)17 b FE(\()o FD(x;)g(y)t FE(\))33 b FH(for)g(all)e FD(x;)17 b(y)33 b FA(2)c Fu(R)1101 2703 y Ft(n)1154 2740 y FD(:)34 b FH(Here)g FD(d)f FH(is)g(the)h(usual)f(distance)h(on)f Fu(R)2637 2703 y Ft(n)2690 2740 y FD(;)g(d)17 b FE(\()p FD(x;)g(y)t FE(\))28 b(=)h FA(j)p FD(x)22 b FA(\000)h FD(y)t FA(j)15 b FD(:)34 b FH(Note)147 2860 y(that)28 b(w)m(e)i(don't)e(assume)h FC(anything)36 b FH(ab)s(out)28 b(the)h(structure)g(of)f FD(f)39 b FH(b)s(esides)29 b(the)g(ab)s(o)m(v) m(e)g(prop)s(erties.)147 2980 y(In)f(particular,)f FD(f)38 b FH(need)28 b(not)f(b)s(e)h(linear.)40 b(The)28 b(orthogonal)d(group)j FF(O)p FE(\()p FD(n)p FE(\))f FH(is)g(a)g(subgroup)h(of)f FF(E)p FE(\()p FD(n)p FE(\))p FH(,)147 3101 y(and)41 b(is)g(the)g(group)f(of)h(all)e FC(line)-5 b(ar)50 b FH(distance-preserving)42 b(maps)e(of)h Fu(R)2774 3065 y Ft(n)2867 3101 y FH(to)g(itself.)67 b(The)42 b(set)f(of)147 3221 y(translations)29 b(of)h Fu(R)852 3185 y Ft(n)935 3221 y FH(\(i.e.,)h(the)g(set)g(of)f(maps)g(of)g(the)h(form)e FD(T)2394 3236 y Ft(x)2438 3221 y FE(\()p FD(y)t FE(\))e(=)g FD(x)19 b FE(+)e FD(y)t FH(\))30 b(is)g(also)f(a)h(subgroup)147 3342 y(of)i FF(E)p FE(\()p FD(n)p FE(\))p FH(.)147 3556 y FI(Prop)s(osition)k(15)48 b FC(Every)37 b(element)e FD(T)50 b FC(of)36 b FF(E)p FE(\()p FD(n)p FE(\))h FC(c)-5 b(an)36 b(b)-5 b(e)36 b(written)g(uniquely)g(as)g(an)g(ortho)-5 b(gonal)147 3676 y(line)g(ar)34 b(tr)-5 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b(R)q FA(g)g FD(y)31 b FE(=)c FD(R)q(y)e FE(+)d FD(x)147 5084 y FH(and)33 b(that)595 5292 y FA(f)p FD(x)700 5307 y Fz(1)739 5292 y FD(;)17 b(R)857 5307 y Fz(1)897 5292 y FA(gf)p FD(x)1052 5307 y Fz(2)1091 5292 y FD(;)g(R)1209 5307 y Fz(2)1249 5292 y FA(g)p FD(y)30 b FE(=)e FD(R)1555 5307 y Fz(1)1594 5292 y FE(\()p FD(R)1706 5307 y Fz(2)1746 5292 y FD(y)d FE(+)d FD(x)1972 5307 y Fz(2)2012 5292 y FE(\))g(+)g FD(x)2225 5307 y Fz(1)2293 5292 y FE(=)27 b FD(R)2470 5307 y Fz(1)2510 5292 y FD(R)2584 5307 y Fz(2)2624 5292 y FD(y)e FE(+)d(\()p FD(x)2888 5307 y Fz(1)2950 5292 y FE(+)g FD(R)3122 5307 y Fz(1)3162 5292 y FD(x)3217 5307 y Fz(2)3257 5292 y FE(\))p eop %%Page: 22 22 22 21 bop 147 48 a FK(22)2825 b FG(Matrix)26 b(Lie)i(Groups)147 298 y FH(Th)m(us)34 b(the)f(pro)s(duct)g(op)s(eration)e(for)h FF(E)p FE(\()p FD(n)p FE(\))h FH(is)f(the)h(follo)m(wing:)1130 506 y FA(f)p FD(x)1235 521 y Fz(1)1275 506 y FD(;)17 b(R)1393 521 y Fz(1)1432 506 y FA(gf)p FD(x)1587 521 y Fz(2)1627 506 y 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b(is)g(a)g(standard)h(theorem)f(from)147 1455 y(elemen)m(tary)40 b(analysis)g(that)g(a)f(subset)j(of)e Fu(C)1827 1419 y Ft(m)1939 1455 y FH(is)g(compact)g(\(in)f(the)i(usual) e(sense)j(that)e(ev)m(ery)147 1576 y(op)s(en)33 b(co)m(v)m(er)h(has)f (a)f(\034nite)g(sub)s(co)m(v)m(er\))j(if)c(and)i(only)f(if)f(it)h(is)g (closed)g(and)h(b)s(ounded.)446 1698 y(All)h(of)h(our)g(examples)h(of)f (matrix)f(Lie)h(groups)h(except)h FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))41 b FH(and)36 b FF(GL)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))42 b FH(ha)m(v)m(e)147 1819 y(prop)s(ert)m(y)33 b(\(1\).)43 b(Th)m(us)35 b(it)c(is)h(the)h(b)s (oundedness)i(condition)c(\(2\))h(that)g(is)g(most)g(imp)s(ortan)m(t.) 446 1941 y(The)c(prop)s(ert)m(y)f(of)f(compactness)i(has)f(v)m(ery)h (imp)s(ortan)m(t)d(implications.)38 b(F)-8 b(or)26 b(example,)h(if)147 2062 y FD(G)f FH(is)f(compact,)i(then)g(ev)m(ery)g(irreducible)e (unitary)g(represen)m(tation)i(of)e FD(G)h FH(is)f (\034nite-dimensional.)147 2334 y 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FD(S)701 3089 y Fz(1)768 3098 y FA(\030)769 3130 y FE(=)873 3125 y FF(U)p FE(\(1\))p FH(.\))147 3398 y FI(Examples)j(of)i(non-compact)f (groups)147 3587 y FH(All)43 b(of)i(the)g(other)g(examples)g(giv)m(en)f (of)h(matrix)e(Lie)h(groups)h(are)g(non-compact.)80 b FF(GL)o FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))147 3708 y FH(and)39 b FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))45 b FH(violate)37 b(prop)s(ert)m(y)j(\(1\),)g(since)g(a)f(limit) c(of)k(in)m(v)m(ertible)f(matrices)g(ma)m(y)h(b)s(e)g(non-)147 3828 y(in)m(v)m(ertible.)k FF(SL)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))39 b FH(and)32 b FF(SL)17 b FE(\()p FD(n)p FE(;)g Fu(C)j FE(\))38 b FH(violate)31 b(\(2\),)h(except)i(in)e (the)h(trivial)d(case)j FD(n)28 b FE(=)g(1)p FH(,)k(since)1358 4321 y FD(A)1431 4336 y Ft(n)1506 4321 y FE(=)1609 3971 y Fv(0)1609 4147 y(B)1609 4206 y(B)1609 4266 y(B)1609 4326 y(B)1609 4386 y(B)1609 4450 y(@)1738 4059 y FD(n)1893 4140 y Fz(1)p 1889 4156 43 4 v 1889 4213 a Ft(n)2025 4299 y FE(1)2162 4403 y 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Fu(R)2151 5135 y Fw(\003)2237 5171 y FH(and)c Fu(C)2502 5135 y Fw(\003)2547 5171 y FH(.)70 b(It)41 b(is)g(left)f(to)h(the)h(reader)f(to)147 5292 y(pro)m(vide)33 b(examples)f(to)h(sho)m(w)g(that)g(this)f(is)g(the)h (case.)p eop %%Page: 24 24 24 23 bop 147 48 a FK(24)2825 b FG(Matrix)26 b(Lie)i(Groups)147 298 y FI(2.4)112 b(Connectedness)147 469 y(De\034nition)36 b(17)49 b FC(A)36 b(matrix)f(Lie)h(gr)-5 b(oup)35 b FD(G)h FC(is)f(said)g(to)h(b)-5 b(e)35 b FB(c)-6 b(onne)g(cte)g(d)37 b FC(if)f(given)f(any)g(two)h(ma-)147 589 y(tric)-5 b(es)35 b FD(A)h FC(and)f FD(B)40 b FC(in)35 b FD(G)p FC(,)g(ther)-5 b(e)36 b(exists)f(a)g(c)-5 b(ontinuous)35 b(p)-5 b(ath)35 b FD(A)p FE(\()p FD(t)p FE(\))p FC(,)g FD(a)29 b FA(\024)g FD(t)g FA(\024)g FD(b)p FC(,)36 b(lying)f(in)g FD(G)g FC(with)147 710 y FD(A)p FE(\()p FD(a)p FE(\))28 b(=)g FD(A)p FC(,)34 b(and)h FD(A)p FE(\()p FD(b)p FE(\))28 b(=)f FD(B)5 b FC(.)147 992 y FI(Remarks)147 1184 y FH(This)25 b(prop)s(ert)m(y)g(is)g(what)g(is)f(called)f 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FH(Consider)40 b(\034rst)f(the)h(case)g FD(n)f FE(=)g(1)p FH(.)62 b(A)40 b FE(1)26 b FA(\002)h FE(1)39 b FH(in)m(v)m(ertible)f(complex)h(matrix)e FD(A)j FH(is)e(of)h(the)g (form)147 4576 y FD(A)30 b FE(=)g([)p FD(\025)p FE(])k FH(with)f FD(\025)d FA(2)g Fu(C)973 4539 y Fw(\003)1019 4576 y FH(,)k(the)g(set)h(of)e(non-zero)h(complex)f(n)m(um)m(b)s(ers.) 48 b(But)34 b(giv)m(en)g(an)m(y)h(t)m(w)m(o)f(non-)147 4696 y(zero)42 b(complex)f(n)m(um)m(b)s(ers,)k(w)m(e)e(can)f(easily)f (\034nd)h(a)f(con)m(tin)m(uous)h(path)g(whic)m(h)g(connects)h(them)147 4816 y(and)33 b(do)s(es)g(not)f(pass)i(through)e(zero.)446 4940 y(F)-8 b(or)34 b(the)h(case)h FD(n)c FA(\025)g FE(1)p FH(,)j(w)m(e)h(use)g(the)f(Jordan)g(canonical)f(form.)49 b(Ev)m(ery)36 b FD(n)24 b FA(\002)h FD(n)35 b FH(complex)147 5061 y(matrix)c FD(A)i FH(can)g(b)s(e)f(written)h(as)1679 5292 y FD(A)28 b FE(=)f FD(C)7 b(B)e(C)2116 5250 y Fw(\000)p Fz(1)p eop %%Page: 25 25 25 24 bop 147 48 a Fx(Connectedness)2980 b FG(25)147 298 y FH(where)36 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FI(Remarks)147 3893 y FH(Y)-8 b(ou)39 b(should)g(think)g(of)f FD(A)p FE(\()p FD(t)p FE(\))h FH(as)g(a)g(lo)s(op,)g(and)g FD(A)p FE(\()p FD(s;)17 b(t)p FE(\))39 b FH(as)g(a)g(parameterized)f (family)f(of)h(lo)s(ops)147 4014 y(whic)m(h)29 b(shrinks)h FD(A)p FE(\()p FD(t)p FE(\))f FH(to)f(a)h(p)s(oin)m(t.)41 b(Condition)28 b(1\))g(sa)m(ys)i(that)f(for)f(eac)m(h)i(v)-5 b(alue)28 b(of)g(the)i(parameter)147 4134 y FD(s)p FH(,)39 b(w)m(e)f(ha)m(v)m(e)h(a)e(lo)s(op;)h(condition)e(2\))h(sa)m(ys)i(that) e(when)h FD(s)e FE(=)g(0)h FH(the)h(lo)s(op)e(is)h(the)g(sp)s (eci\034ed)i(lo)s(op)147 4254 y FD(A)p FE(\()p FD(t)p FE(\))p FH(;)33 b(and)g(condition)e(3\))h(sa)m(ys)i(that)e(when)i FD(s)27 b FE(=)h(1)k FH(our)h(lo)s(op)e(is)h(a)g(p)s(oin)m(t.)446 4375 y(It)41 b(is)f(customary)g(to)h(sp)s(eak)g(of)f (simple-connectedness)i(only)e(for)f(connected)k(matrix)147 4495 y(Lie)32 b(groups,)h(ev)m(en)h(though)f(the)g(de\034nition)e(mak)m (es)i(sense)h(for)e(disconnected)i(groups.)147 4725 y FI(Prop)s(osition)i(24)48 b FC(The)35 b(gr)-5 b(oup)34 b FF(SU)q FE(\(2\))g FC(is)h(simply)f(c)-5 b(onne)g(cte)g(d.)147 4986 y FI(Pro)s(of)147 5171 y FH(Exercise)23 b(8)f(sho)m(ws)h(that)e FF(SU)q FE(\(2\))g FH(ma)m(y)h(b)s(e)g(though)m(t)g(of)f(\(top)s (ologically\))c(as)22 b(the)g(three-dimensional)147 5292 y(sphere)34 b FD(S)517 5255 y Fz(3)589 5292 y FH(sitting)d(inside)h Fu(R)1236 5255 y Fz(4)1281 5292 y FH(.)44 b(It)32 b(is)g(w)m(ell-kno)m (wn)h(that)f FD(S)2339 5255 y Fz(3)2411 5292 y FH(is)g(simply)f (connected.)45 b Fy(\003)p eop %%Page: 28 28 28 27 bop 147 48 a FK(28)2825 b FG(Matrix)26 b(Lie)i(Groups)446 298 y FH(The)44 b(condition)e(of)h(simple-connectedness)h(is)f (extremely)h(imp)s(ortan)m(t.)74 b(One)43 b(of)g(our)147 418 y(most)25 b(imp)s(ortan)m(t)f(theorems)i(will)d(b)s(e)j(that)g(if)e FD(G)i FH(is)f(simply)f(connected,)29 b(then)d(there)h(is)e(a)g (natural)147 539 y(one-to-one)32 b(corresp)s(ondence)i(b)s(et)m(w)m (een)g(the)e(represen)m(tations)i(of)d FD(G)h FH(and)g(the)h(represen)m (tations)147 659 y(of)f(its)g(Lie)g(algebra.)446 783 y(Without)g(pro)s(of,)g(w)m(e)h(giv)m(e)g(the)g(follo)m(wing)c(table.) 1287 1006 y FI(Group)277 b(Simply)33 b(connected?)1260 1127 y FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))663 b FH(no)1258 1247 y FF(SL)16 b FE(\()p FD(n)p FE(;)h Fu(C)j FE(\))646 b FH(y)m(es)1260 1367 y FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))663 b FH(no)1258 1488 y FF(SL)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))660 b FH(no)1314 1608 y FF(SO)p FE(\()p FD(n)p FE(\))711 b FH(no)1344 1729 y FF(U)p FE(\()p FD(n)p FE(\))741 b FH(no)1317 1849 y FF(SU)p FE(\()p FD(n)p FE(\))700 b FH(y)m(es)1273 1969 y FF(SO)p FE(\(1;)17 b(1\))654 b FH(y)m(es)1094 2090 y FF(SO)p FE(\()p FD(n)p FE(;)17 b(1\))32 b FH(\()p FD(n)c FA(\025)g FE(2)p FH(\))490 b(no)1212 2210 y(Heisen)m(b)s(erg)596 b(y)m(es)147 2479 y FI(2.6)112 b(Homomorphism)n(s)31 b(and)39 b(Isomorphisms)147 2650 y(De\034nition)d(25)49 b FC(L)-5 b(et)32 b FD(G)g FC(and)f FD(H)39 b FC(b)-5 b(e)31 b(matrix)h(Lie)f(gr)-5 b(oups.)44 b(A)32 b(map)f FD(\036)g FC(fr)-5 b(om)31 b FD(G)h FC(to)g FD(H)39 b FC(is)31 b(c)-5 b(al)5 b(le)-5 b(d)31 b(a)147 2771 y FB(Lie)39 b(gr)-6 b(oup)41 b(homomorphism)c FC(if)d(1\))g FD(\036)g FC(is)g(a)h(gr)-5 b(oup)34 b(homomorphism)e (and)i(2\))g FD(\036)g FC(is)g(c)-5 b(ontinu-)147 2891 y(ous.)44 b(If)30 b(in)g(addition,)h FD(\036)g FC(is)f(one-to-one)g (and)g(onto,)i(and)e(the)h(inverse)f(map)g FD(\036)3031 2855 y Fw(\000)p Fz(1)3156 2891 y FC(is)g(c)-5 b(ontinuous,)147 3011 y(then)35 b FD(\036)f FC(is)h(c)-5 b(al)5 b(le)-5 b(d)34 b(a)h FB(Lie)k(gr)-6 b(oup)42 b(isomorphism)p FC(.)147 3295 y FI(Remarks)147 3487 y FH(The)c(condition)e(that)h FD(\036)g FH(b)s(e)h(con)m(tin)m(uous)g(should)f(b)s(e)g(regarded)h(as) g(a)f(tec)m(hnicalit)m(y)-8 b(,)37 b(in)g(that)g(it)147 3608 y(is)i(v)m(ery)h(di\036cult)f(to)f(giv)m(e)h(an)g(example)g(of)g (a)g(group)f(homomorphism)f(b)s(et)m(w)m(een)k(t)m(w)m(o)f(matrix)147 3728 y(Lie)35 b(groups)h(whic)m(h)f(is)g(not)g(con)m(tin)m(uous.)53 b(In)36 b(fact,)g(if)e FD(G)e FE(=)h Fu(R)46 b FH(and)35 b FD(H)40 b FE(=)33 b Fu(C)2978 3692 y Fw(\003)3023 3728 y FH(,)j(then)g(an)m(y)g(group)147 3849 y(homorphism)28 b(from)h FD(G)g FH(to)h FD(H)37 b FH(whic)m(h)31 b(is)e(ev)m(en)i (measurable)f(\(a)f(v)m(ery)i(w)m(eak)h(condition\))c(m)m(ust)i(b)s(e) 147 3969 y(con)m(tin)m(uous.)44 b(\(See)34 b(W.)e(Rudin,)g FC(R)-5 b(e)g(al)35 b(and)f(Complex)g(A)n(nalysis,)d FH(Chap.)44 b(9,)33 b(Ex.)44 b(17.\))446 4093 y(If)34 b FD(G)h FH(and)f FD(H)42 b FH(are)34 b(matrix)f(Lie)h(groups,)h(and)f (there)i(exists)f(a)f(Lie)f(group)i(isomorphism)147 4214 y(from)c FD(G)i FH(to)f FD(H)8 b FH(,)32 b(then)h FD(G)g FH(and)g FD(H)40 b FH(are)32 b(said)g(to)g(b)s(e)h FI(isomorphic)p FH(,)c(and)j(w)m(e)i(write)e FD(G)3291 4186 y FA(\030)3292 4218 y FE(=)3396 4214 y FD(H)8 b FH(.)43 b(Tw)m(o)147 4334 y(matrix)f(Lie)g(groups)i(whic)m(h)g(are)f(isomorphic)e(should)i (b)s(e)h(though)m(t)f(of)g(as)g(b)s(eing)g(essen)m(tially)147 4454 y(the)d(same)g(group.)65 b(\(Note)40 b(that)f(b)m(y)i (de\034nition,)f(the)g(in)m(v)m(erse)h(of)f(Lie)f(group)g(isomorphism)e (is)147 4575 y(con)m(tin)m(uous,)c(and)g(so)g(also)e(a)i(Lie)f(group)g (isomorphism.\))147 4858 y FI(Example:)47 b FF(SU)q FE(\(2\))37 b FI(and)h FF(SO)p FE(\(3\))147 5051 y FH(A)32 b(v)m(ery)h(imp)s(ortan) m(t)c(topic)i(for)g(us)h(will)e(b)s(e)h(the)h(relationship)e(b)s(et)m (w)m(een)j(the)f(groups)g FF(SU)q FE(\(2\))f FH(and)147 5171 y FF(SO)p FE(\(3\))p FH(.)47 b(This)33 b(example)g(is)h(designed)g (to)f(sho)m(w)i(that)e FF(SU)q FE(\(2\))g FH(and)h FF(SO)p FE(\(3\))f FH(are)h(almost)d(\(but)j(not)147 5292 y(quite!\))75 b(isomorphic.)d(Sp)s(eci\034cally)-8 b(,)44 b(there)g(exists)f(a)g(Lie) f(group)h(homomorphism)d FD(\036)j FH(whic)m(h)p eop %%Page: 29 29 29 28 bop 147 48 a Fx(Lie)28 b(Groups)3107 b FG(29)147 298 y FH(maps)29 b FF(SU)q FE(\(2\))g FH(on)m(to)g FF(SO)p FE(\(3\))p FH(,)g(and)h(whic)m(h)g(is)e FC(two)6 b FH(-to-one.)42 b(\(See)30 b(Miller)d(7.1)i(and)g(Br\366c)m(k)m(er,)i(Chap.)147 418 y(I,)i(6.18.\))446 539 y(Consider)f(the)f(space)h FD(V)53 b FH(of)31 b(all)e FE(2)19 b FA(\002)h FE(2)31 b FH(complex)f(matrices)h(whic)m(h)g(are)h(self-adjoin)m(t)d(and)147 660 y(ha)m(v)m(e)g(trace)f(zero.)42 b(This)28 b(is)f(a)g (three-dimensional)e FC(r)-5 b(e)g(al)38 b FH(v)m(ector)28 b(space)h(with)e(the)h(follo)m(wing)d(basis)767 934 y FD(A)840 949 y Fz(1)907 934 y FE(=)1011 794 y Fv(\022)1126 873 y FE(0)83 b(1)1126 993 y(1)g(0)1348 794 y Fv(\023)1438 934 y FE(;)g FD(A)1621 949 y Fz(2)1688 934 y FE(=)1792 794 y Fv(\022)1937 873 y FE(0)122 b FD(i)1906 993 y FA(\000)p FD(i)84 b FE(0)2191 794 y Fv(\023)2281 934 y FE(;)f FD(A)2464 949 y Fz(3)2531 934 y FE(=)2634 794 y Fv(\022)2749 873 y FE(1)122 b(0)2749 993 y(0)83 b FA(\000)p FE(1)3049 794 y Fv(\023)147 1208 y FH(W)-8 b(e)33 b(ma)m(y)f(de\034ne)i(an)f (inner)f(pro)s(duct)h(on)f FD(V)54 b FH(b)m(y)34 b(the)f(form)m(ula) 1488 1469 y FA(h)p FD(A;)17 b(B)5 b FA(i)27 b FE(=)1903 1401 y(1)p 1903 1446 49 4 v 1903 1537 a(2)1962 1469 y(trace\()p FD(AB)5 b FE(\))147 1722 y FH(\(Exercise:)45 b(c)m(hec)m(k)35 b(that)d(this)g(is)g(an)h(inner)f(pro)s(duct.\))446 1843 y(Direct)42 b(computation)f(sho)m(ws)j(that)d FA(f)p FD(A)1964 1858 y Fz(1)2004 1843 y FD(;)17 b(A)2121 1858 y Fz(2)2160 1843 y FD(;)g(A)2277 1858 y Fz(3)2316 1843 y FA(g)42 b FH(is)g(an)g(orthonormal)e(basis)i(for)f FD(V)22 b FH(.)147 1963 y(Ha)m(ving)32 b(c)m(hosen)i(an)f(orthonormal)d (basis)i(for)g FD(V)22 b FH(,)33 b(w)m(e)g(can)g(inden)m(tify)f FD(V)54 b FH(with)32 b Fu(R)3143 1927 y Fz(3)3189 1963 y FH(.)446 2084 y(No)m(w,)g(if)f FD(U)42 b FH(is)31 b(an)g(elemen)m(t)h (of)f FF(SU)p FE(\(2\))p FH(,)h(and)f FD(A)h FH(is)f(an)h(elemen)m(t)f (of)g FD(V)21 b FH(,)32 b(then)g(it)f(is)g(easy)h(to)147 2204 y(see)h(that)f FD(U)10 b(AU)740 2168 y Fw(\000)p Fz(1)868 2204 y FH(is)32 b(in)f FD(V)21 b FH(.)44 b(Th)m(us)33 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FH(\),)k(and)f(that)g(it)f(is)147 3895 y(con)m(tin)m(uous.)44 b(Th)m(us)34 b FD(U)39 b FA(!)27 b FD(\036)1210 3910 y Ft(U)1301 3895 y FH(is)32 b(a)h(Lie)e(group)i(homomorphism)c(of)j FF(SU)q FE(\(2\))g FH(in)m(to)g FF(O)p FE(\(3\))p FH(.)446 4016 y(Recall)38 b(that)i(ev)m(ery)i(elemen)m(t)d(of)h FF(O)p FE(\(3\))f FH(has)i(determinan)m(t)e FA(\006)p FE(1)p FH(.)66 b(Since)40 b FF(SU)p FE(\(2\))g FH(is)f(con-)147 4136 y(nected)34 b(\(Exercise)g(8\),)f(and)g(the)g(map)f FD(U)39 b FA(!)28 b FD(\036)1891 4151 y Ft(U)1983 4136 y FH(is)k(con)m(tin)m(uous,)i FD(\036)2655 4151 y Ft(U)2746 4136 y FH(m)m(ust)f(actually)f(map)g(in)m(to)147 4256 y FF(SO)p FE(\(3\))p FH(.)43 b(Th)m(us)34 b FD(U)39 b FA(!)27 b FD(\036)1005 4271 y Ft(U)1096 4256 y FH(is)32 b(a)h(Lie)f(group)g(homomorphism)e(of)i FF(SU)p FE(\(2\))g FH(in)m(to)g FF(SO)p FE(\(3\))p FH(.)446 4377 y(The)43 b(map)f FD(U)54 b FA(!)44 b FD(\036)1205 4392 y Ft(U)1306 4377 y FH(is)e(not)g(one-to-one,)i(since)f(for)e(an)m(y)i FD(U)55 b FA(2)45 b FF(SU)p FE(\(2\))p FH(,)f FD(\036)3320 4392 y Ft(U)3423 4377 y FE(=)g FD(\036)3601 4392 y Fw(\000)p Ft(U)3715 4377 y FH(.)147 4498 y(\(Observ)m(e)c(that)e(if)e FD(U)49 b FH(is)37 b(in)h FF(SU)p FE(\(2\))p FH(,)h(then)g(so)f(is)f FA(\000)p FD(U)10 b FH(.\))61 b(It)38 b(is)g(p)s(ossible)f(to)g(sho)m (w)i(that)f FD(\036)3493 4513 y Ft(U)3590 4498 y FH(is)f(a)147 4618 y(t)m(w)m(o-to-one)32 b(map)g(of)g FF(SU)q FE(\(2\))g FH(on)m(to)g FF(SO)q FE(\(3\))p FH(.)43 b(\(See)33 b(Miller.\))147 4883 y FI(2.7)112 b(Lie)37 b(Groups)147 5051 y FH(A)22 b(Lie)f(group)h(is)f(something)g(whic)m(h)h(is)g(sim)m(ultaneously)e(a) h(group)h(and)g(a)f(di\033eren)m(tiable)g(manifold)147 5171 y(\(see)44 b(De\034nition)e(26\).)73 b(As)44 b(the)f(terminology)d (suggests,)47 b(ev)m(ery)d(matrix)d(Lie)h(group)h(is)f(a)h(Lie)147 5292 y(group,)28 b(although)d(this)h(requires)h(pro)s(of)e(\(Theorem)i (27\).)40 b(I)27 b(ha)m(v)m(e)h(decided)e(to)g(restrict)h(atten)m(tion) p eop %%Page: 30 30 30 29 bop 147 48 a FK(30)2825 b FG(Matrix)26 b(Lie)i(Groups)147 298 y FH(to)33 b(matrix)e(Lie)i(groups,)g(except)i(in)d(emergencies,)h (for)g(three)h(reasons.)45 b(First,)32 b(this)h(mak)m(es)h(the)147 418 y(course)41 b(accessible)f(to)g(studen)m(ts)h(who)f(are)g(not)g (familiar)35 b(with)40 b(the)g(theory)g(of)f(di\033eren)m(tiable)147 539 y(manifolds.)g(Second,)29 b(this)e(mak)m(es)g(the)g(de\034nition)f (of)g(the)h(Lie)f(algebra)g(and)g(of)h(the)g(exp)s(onen)m(tial)147 659 y(mapping)d(far)h(more)g(comprehensible.)41 b(Third,)27 b(all)d(of)h(the)h(imp)s(ortan)m(t)e(examples)h(of)h(Lie)f(groups)147 779 y(are)33 b(\(or)f(can)h(easily)e(b)s(e)i(represen)m(ted)i(as\))e (matrix)e(Lie)h(groups.)446 900 y(Alas,)j(there)h(is)f(a)g(price)g(to)g (pa)m(y)g(for)g(this)g(simpli\034cation.)47 b(Certain)35 b(imp)s(ortan)m(t)e(topics)147 1020 y(\(notably)-8 b(,)32 b(the)g(univ)m(ersal)g(co)m(v)m(er\))i(are)e(considerably)g (complicated)e(b)m(y)j(restricting)e(to)h(the)h(ma-)147 1140 y(trix)i(case.)52 b(Nev)m(ertheless,)38 b(I)d(feel)g(that)g(the)h (adv)-5 b(an)m(tages)35 b(out)m(w)m(eigh)h(the)f(disadv)-5 b(an)m(tages)36 b(in)e(an)147 1261 y(in)m(tro)s(ductory)e(course)i(suc) m(h)g(as)f(this.)147 1454 y FI(De\034nition)j(26)49 b FC(A)43 b FB(Lie)48 b(gr)-6 b(oup)44 b FC(is)f(a)f(di\033er)-5 b(entiable)41 b(manifold)g FD(G)i FC(which)f(is)g(also)g(a)g(gr)-5 b(oup,)147 1574 y(and)34 b(such)h(that)g(the)g(gr)-5 b(oup)35 b(pr)-5 b(o)g(duct)1691 1763 y FD(G)22 b FA(\002)h FD(G)k FA(!)g FD(G)147 1952 y FC(and)34 b(the)h(inverse)f(map)g FD(g)d FA(!)d FD(g)1304 1916 y Fw(\000)p Fz(1)1432 1952 y FC(ar)-5 b(e)35 b(di\033er)-5 b(entiable.)446 2145 y FH(F)d(or)28 b(the)h(reader)g(who)g(is)f(not)g(familiar)d(with)j(the) h(notion)e(of)h(a)h(di\033eren)m(tiable)e(manifold,)147 2265 y(here)i(is)f(a)f(brief)h(recap.)42 b(\(I)29 b(will)c(consider)k (only)e(manifolds)f(em)m(b)s(edded)j(in)e(some)h Fu(R)3216 2229 y Ft(n)3269 2265 y FH(,)h(whic)m(h)g(is)e(a)147 2386 y(harmless)j(assumption.\))43 b(A)30 b(subset)j FD(M)41 b FH(of)31 b Fs(R)1869 2349 y Ft(n)1946 2386 y FH(is)f(called)g(a)g FD(k)s FI(-dimensional)i(di\033eren)m(tiable)147 2506 y(manifold)25 b FH(if)h(giv)m(en)i(an)m(y)g FD(m)1198 2521 y Fz(0)1266 2506 y FA(2)g FD(M)10 b FH(,)29 b(there)g(exists)f(a)f (smo)s(oth)g(\(non-linear\))e(co)s(ordinate)i(system)147 2626 y FE(\()p FD(x)240 2590 y Fz(1)280 2626 y FD(;)17 b FA(\001)g(\001)g(\001)e FD(x)512 2590 y Ft(n)559 2626 y FE(\))32 b FH(de\034ned)i(in)e(a)g(neigh)m(b)s(orho)s(o)s(d)g FD(U)43 b FH(of)32 b FD(m)2078 2641 y Fz(0)2150 2626 y FH(suc)m(h)i(that)815 2815 y FD(M)f FA(\\)22 b FD(U)39 b FE(=)1238 2735 y Fv(\010)1296 2815 y FD(m)28 b FA(2)h FD(U)1597 2731 y Fv(\014)1597 2790 y(\014)1630 2815 y FD(x)1685 2774 y Ft(k)r Fz(+1)1818 2815 y FE(\()p FD(m)p FE(\))f(=)g FD(c)2153 2830 y Fz(1)2192 2815 y FD(;)17 b FA(\001)g(\001)g(\001)31 b FD(;)17 b(x)2484 2774 y Ft(n)2531 2815 y FE(\()p FD(m)p FE(\))28 b(=)g FD(c)2866 2830 y Ft(n)p Fw(\000)p Ft(k)3016 2735 y Fv(\011)147 3004 y FH(This)51 b(sa)m(ys)i(that)d(lo)s(cally)-8 b(,)53 b(after)d(a)h(suitable)f(c)m(hange)i(of)e(v)-5 b(ariables,)54 b FD(M)62 b FH(lo)s(oks)50 b(lik)m(e)g(the)h FD(k)s FH(-)147 3125 y(dimensional)45 b(h)m(yp)s(erplane)j(in)f Fu(R)1410 3089 y Ft(n)1510 3125 y FH(obtained)g(b)m(y)i(setting)e(all)e(but)j (the)g(\034rst)g FD(k)j FH(co)s(ordinates)147 3245 y(equal)33 b(to)f(constan)m(ts.)446 3365 y(F)-8 b(or)32 b(example,)h FD(S)1097 3329 y Fz(1)1164 3365 y FA(\032)c Fu(R)1336 3329 y Fz(2)1414 3365 y FH(is)j(a)h(one-dimensional)d(di\033eren)m (tiable)i(manifold)e(b)s(ecause)k(in)147 3486 y(the)j(usual)f(p)s(olar) f(co)s(ordinates)h FE(\()p FD(\022)s(;)17 b(r)s FE(\))p FH(,)37 b FD(S)1701 3450 y Fz(1)1776 3486 y FH(is)f(the)h(set)g FD(r)g FE(=)d(1)p FH(.)55 b(Of)36 b(course,)j(p)s(olar)c(co)s (ordinates)147 3606 y(are)j(not)g(globally)d(de\034ned,)41 b(b)s(ecause)f FD(\022)h FH(is)d(unde\034ned)h(at)f(the)h(origin,)e (and)h(b)s(ecause)h FD(\022)j FH(is)37 b(not)147 3727 y(\020single-v)-5 b(alued.\021)45 b(But)34 b(giv)m(en)g(an)m(y)h(p)s (oin)m(t)e FD(m)1842 3742 y Fz(0)1916 3727 y FH(in)g FD(S)2097 3690 y Fz(1)2136 3727 y FH(,)h(w)m(e)h(can)f(de\034ne)h(p)s (olar)e(co)s(ordinates)g(in)g(a)147 3847 y(neigh)m(b)s(orho)s(o)s(d)e FD(U)44 b FH(of)32 b FD(m)1065 3862 y Fz(0)1104 3847 y FH(,)h(and)g(then)g FD(S)1642 3811 y Fz(1)1703 3847 y FA(\\)23 b FD(U)43 b FH(will)30 b(b)s(e)j(the)g(set)g FD(r)d FE(=)e(1)p FH(.)446 3967 y(Note)44 b(that)f(while)g(w)m(e)i (assume)f(that)f(our)h(di\033eren)m(tiable)e(manifolds)f(are)j(em)m(b)s (edded)147 4088 y(in)37 b(some)f Fu(R)580 4052 y Ft(n)670 4088 y FH(\(a)h(harmless)g(assumption\),)g(w)m(e)h(are)f FC(not)47 b FH(sa)m(ying)37 b(that)g(a)f(Lie)h(group)g(has)g(to)g(b)s (e)147 4208 y(em)m(b)s(edden)28 b(in)d Fu(R)775 4172 y Ft(n)818 4149 y Fr(2)863 4208 y FH(,)j(or)e(that)g(the)h(group)f(op)s (eration)g(has)h(to)f(ha)m(v)m(e)i(an)m(ything)e(to)g(do)g(with)h (matrix)147 4328 y(m)m(ultiplication.)38 b(A)31 b(Lie)e(group)h(is)g (simply)f(a)h(subset)i FD(G)e FH(of)g(some)g Fu(R)2655 4292 y Ft(n)2738 4328 y FH(whic)m(h)h(is)e(a)h(di\033eren)m(tiable)147 4449 y(manifold,)38 b(together)h(with)f FC(any)47 b FH(map)38 b(from)g FD(G)26 b FA(\002)h FD(G)39 b FH(in)m(to)f FD(G)g FH(whic)m(h)i(mak)m(es)f FD(G)g FH(in)m(to)f(a)g(group)147 4569 y(\(and)c(suc)m(h)h(that)f(the)g(group)g(op)s(erations)f(are)g (smo)s(oth\).)47 b(It)33 b(is)h(remark)-5 b(able)32 b(that)i(almost)e (\(but)147 4690 y(not)h(quite!\))43 b(ev)m(ery)34 b(Lie)e(group)h(is)f (isomorphic)e(to)i(a)h(matrix)e(Lie)h(group.)446 4810 y(Note)38 b(also)f(that)h(it)f(is)h(far)g(from)e(ob)m(vious)j(that)f(a) f(matrix)g(Lie)g(group)h(m)m(ust)h(b)s(e)f(a)g(Lie)147 4930 y(group,)g(since)f(our)f(de\034nition)g(of)g(a)g(matrix)f(Lie)h (group)h FD(G)f FH(do)s(es)h(not)g(sa)m(y)h(an)m(ything)e(ab)s(out)g FD(G)147 5051 y FH(b)s(eing)g(a)g(manifold.)51 b(It)37 b(is)e(not)h(to)s(o)g(di\036cult)f(to)h(v)m(erify)h(that)f(all)e(of)i (our)g(examples)g(of)g(matrix)147 5171 y(Lie)31 b(groups)h(are)f(Lie)g (groups,)h(but)g(in)e(fact)i(w)m(e)g(ha)m(v)m(e)h(the)f(follo)m(wing)c (result)k(whic)m(h)g(mak)m(es)g(suc)m(h)147 5292 y(v)m(eri\034cations)h (unnecessary:)p eop %%Page: 31 31 31 30 bop 147 48 a Fx(Exercises)3179 b FG(31)147 298 y FI(Theorem)35 b(27)49 b FC(Every)35 b(matrix)f(Lie)h(gr)-5 b(oup)35 b(is)f(a)h(Lie)g(gr)-5 b(oup.)147 553 y FI(Discussion)147 738 y FH(Let)40 b(us)h(consider)f(\034rst)g(the)g(group)g FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))p FH(.)71 b(The)41 b(space)g(of)e(all)f FD(n)27 b FA(\002)h FD(n)40 b FH(real)f(matrices)g(can)147 858 y(b)s(e)i(though)m(t)g(of)g(as)g Fu(R)976 822 y Ft(n)1019 798 y Fr(2)1064 858 y FH(.)68 b(Since)41 b FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))47 b FH(is)40 b(the)i(set)f(of)g(all)e(matrices)h FD(A)h FH(with)f FE(det)17 b FD(A)42 b FA(6)p FE(=)g(0)p FH(,)147 978 y FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))50 b FH(is)44 b(an)h(op)s(en)f(subset)i(of)e Fu(R)1564 942 y Ft(n)1607 919 y 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Fv(\023)3493 5196 y FH(\(2.11\))p eop %%Page: 34 34 34 33 bop 147 48 a FK(34)2825 b FG(Matrix)26 b(Lie)i(Groups)1571 203 y Fv(\022)1686 282 y FA(\000)17 b FE(cosh)h FD(t)135 b FE(sinh)17 b FD(t)1738 403 y FE(sinh)g FD(t)136 b FA(\000)17 b FE(cosh)g FD(t)2472 203 y Fv(\023)3493 343 y FH(\(2.12\))1618 607 y Fv(\022)1733 687 y FE(cosh)g FD(t)89 b FA(\000)17 b FE(sinh)g FD(t)1738 807 y FE(sinh)g FD(t)89 b FA(\000)17 b FE(cosh)g FD(t)2425 607 y Fv(\023)3493 748 y FH(\(2.13\))1577 1012 y Fv(\022)1691 1091 y FA(\000)g FE(cosh)h FD(t)83 b FA(\000)17 b FE(sinh)g FD(t)1744 1212 y FE(sinh)f FD(t)178 b FE(cosh)17 b FD(t)2467 1012 y Fv(\023)3493 1152 y FH(\(2.14\))391 1409 y(\(Since)27 b FE(cosh)17 b FD(t)27 b FH(is)f(alw)m(a)m(ys)h(p)s (ositiv)m(e,)g(there)h(is)e(no)g(o)m(v)m(erlap)h(among)e(the)i(four)g (cases.)42 b(Matri-)391 1529 y(ces)34 b(of)e(the)i(\034rst)f(t)m(w)m(o) h(forms)e(ha)m(v)m(e)i(determinan)m(t)e(one;)i(matrices)e(of)g(the)h (last)f(t)m(w)m(o)i(forms)391 1649 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b(that)g(the)g(quotien)m(t)f(group) h FD(H)r(=)-5 b(Z)7 b FE(\()p FD(H)h FE(\))31 b FH(is)h(ab)s(elian.)218 4197 y(11.)48 b FC(Conne)-5 b(cte)g(dness)35 b(of)57 b FF(SO)p FE(\()p FD(n)p FE(\))p FH(.)49 b(Sho)m(w)35 b(that)f FF(SO)p FE(\()p FD(n)p FE(\))g FH(is)g(connected,)i(follo)m (wing)c(the)i(outline)391 4318 y(b)s(elo)m(w.)391 4477 y(F)-8 b(or)45 b(the)h(case)h FD(n)k FE(=)f(1)p FH(,)f(there)e(is)e (not)h(m)m(uc)m(h)g(to)g(sho)m(w,)k(since)c(a)g FE(1)31 b FA(\002)h FE(1)45 b FH(matrix)f(with)391 4597 y(determinan)m(t)d(one) g(m)m(ust)h(b)s(e)f FE([1])p FH(.)70 b(Assume,)44 b(then,)g(that)d FD(n)i FA(\025)g FE(2)p FH(.)69 b(Let)42 b FD(e)3203 4612 y Fz(1)3284 4597 y FH(denote)g(the)391 4717 y(v)m(ector)1806 5097 y FD(e)1851 5112 y Fz(1)1918 5097 y FE(=)2021 4807 y Fv(0)2021 4983 y(B)2021 5042 y(B)2021 5102 y(B)2021 5166 y(@)2150 4895 y FE(1)2150 5015 y(0)2161 5111 y FH(.)2161 5144 y(.)2161 5177 y(.)2150 5298 y FE(0)2240 4807 y Fv(1)2240 4983 y(C)2240 5042 y(C)2240 5102 y(C)2240 5166 y(A)p eop %%Page: 35 35 35 34 bop 147 48 a Fx(Exercises)3179 b FG(35)391 298 y FH(in)32 b Fu(R)570 262 y Ft(n)623 298 y FH(.)44 b(Giv)m(en)32 b(an)m(y)h(unit)e(v)m(ector)i FD(v)f FA(2)c Fu(R)1896 262 y Ft(n)1949 298 y FH(,)k(sho)m(w)h(that)f(there)h(exists)g(a)f(con) m(tin)m(uous)h(path)391 418 y FD(R)q FE(\()p FD(t)p FE(\))38 b FH(in)g FF(SO)p FE(\()p FD(n)p FE(\))g FH(with)g FD(R)q FE(\(0\))f(=)g FD(I)46 b FH(and)38 b FD(R)q FE(\(1\))p FD(v)j FE(=)c FD(e)2341 433 y Fz(1)2380 418 y FH(.)60 b(\(Th)m(us)40 b(an)m(y)f(unit)e(v)m(ector)i(can)g(b)s(e)391 539 y(\020con)m(tin)m(uously)33 b(rotated\021)f(to)g FD(e)1561 554 y Fz(1)1601 539 y FH(.\))391 700 y(No)m(w)45 b(sho)m(w)g(that)e(an)m(y)i(elemen)m(t)f FD(R)h FH(of)e FF(SO)p FE(\()p FD(n)p FE(\))h FH(can)h(b)s(e)f(connected)h(to)f(an)g (elemen)m(t)f(of)391 821 y FF(SO)p FE(\()p FD(n)23 b FA(\000)f FE(1\))p FH(,)32 b(and)h(pro)s(ceed)g(b)m(y)h(induction.)218 1024 y(12.)48 b FC(The)36 b(p)-5 b(olar)36 b(de)-5 b(c)g(omp)g(osition) 35 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4030 y FE(0)304 b FD(\025)2224 4045 y Ft(n)2312 3659 y Fv(1)2312 3835 y(C)2312 3899 y(A)2416 3890 y FH(.)147 4241 y(Observ)m(e)34 b(that)f FD(e)778 4205 y Ft(D)874 4241 y FH(is)f(the)h(diagonal)d(matrix)g(with)i(eigen)m(v)-5 b(alues)33 b FD(e)2618 4205 y Ft(\025)2659 4214 y Fr(1)2698 4241 y FD(;)17 b FA(\001)g(\001)g(\001)31 b FD(;)17 b(e)2980 4205 y Ft(\025)3021 4213 y Fn(n)3068 4241 y FH(,)32 b(and)h(so)f(in)g (ligh)m(t)147 4361 y(of)g(Prop)s(osition)f(31,)h(w)m(e)i(ha)m(v)m(e) 1270 4700 y FD(e)1315 4659 y Ft(X)1411 4700 y FE(=)27 b FD(C)1608 4470 y Fv(0)1608 4645 y(B)1608 4709 y(@)1736 4558 y FD(e)1781 4522 y Ft(\025)1822 4531 y Fr(1)2184 4558 y FE(0)1950 4662 y FH(.)1988 4687 y(.)2026 4712 y(.)1775 4840 y FE(0)318 b FD(e)2187 4804 y Ft(\025)2228 4812 y Fn(n)2317 4470 y Fv(1)2317 4645 y(C)2317 4709 y(A)2420 4700 y FD(C)2497 4659 y Fw(\000)p Fz(1)2592 4700 y FH(.)147 5051 y(Th)m(us)37 b(if)d(y)m(ou)i(can)g(explicitly)d (diagonalize)g FD(X)8 b FH(,)35 b(y)m(ou)h(can)g(explicitly)d(compute)j FD(e)3179 5015 y Ft(X)3246 5051 y FH(.)52 b(Note)35 b(that)147 5171 y(if)29 b FD(X)38 b FH(is)29 b(real,)h(then)g(although)f FD(C)37 b FH(ma)m(y)30 b(b)s(e)g(complex)f(and)i(the)f FD(\025)2521 5186 y Ft(i)2549 5171 y FH('s)g(ma)m(y)g(b)s(e)g(complex,) g FD(e)3435 5135 y Ft(X)3533 5171 y FH(m)m(ust)147 5292 y(come)i(out)h(to)f(b)s(e)h(real,)e(since)i(eac)m(h)h(term)e(in)g(the)h (series)g(\(3.1\))f(is)g(real.)p eop %%Page: 41 41 41 40 bop 147 48 a Fx(Computing)28 b(the)g(Exp)r(onen)n(tial)f(of)h(a)f (Matrix)2057 b FG(41)446 298 y FH(F)-8 b(or)32 b(example,)g(tak)m(e) 1566 560 y FD(X)k FE(=)1786 420 y Fv(\022)1903 499 y FE(0)84 b FA(\000)p FD(a)1901 620 y(a)124 b FE(0)2206 420 y Fv(\023)2296 560 y FH(.)147 899 y(Then)47 b(the)f(eigen)m(v)m (ectors)h(of)e FD(X)53 b FH(are)1587 759 y Fv(\022)1702 838 y FE(1)1710 959 y FD(i)1793 759 y Fv(\023)1911 899 y FH(and)2114 759 y Fv(\022)2236 838 y FD(i)2229 959 y FE(1)2319 759 y Fv(\023)2392 899 y FH(,)c(with)c(eigen)m(v)-5 b(alues)45 b FA(\000)p FD(ia)i FH(and)e FD(ia)p FH(,)147 1072 y(resp)s(ectiv)m(ely)-8 b(.)45 b(Th)m(us)34 b(the)f(in)m(v)m (ertible)e(matrix)1635 1334 y FD(C)k FE(=)1844 1194 y Fv(\022)1958 1273 y FE(1)91 b FD(i)1966 1394 y(i)g FE(1)2181 1194 y Fv(\023)147 1673 y FH(maps)38 b(the)h(basis)f(v)m(ectors)1164 1533 y Fv(\022)1279 1612 y FE(1)1279 1732 y(0)1369 1533 y Fv(\023)1481 1673 y FH(and)1676 1533 y Fv(\022)1791 1612 y FE(0)1791 1732 y(1)1881 1533 y Fv(\023)1993 1673 y FH(to)g(the)h(eigen)m(v)m(ectors)h(of)d FD(X)8 b FH(,)40 b(and)e(so)h(\(c)m(hec)m(k\))147 1856 y FD(C)224 1820 y Fw(\000)p Fz(1)318 1856 y FD(X)8 b(C)40 b FH(is)32 b(a)g(diagonal)e(matrix)h FD(D)s FH(.)43 b(Th)m(us)34 b FD(X)i FE(=)27 b FD(C)7 b(D)s(C)2262 1820 y Fw(\000)p Fz(1)2356 1856 y FH(:)933 2118 y FD(e)978 2077 y Ft(X)1074 2118 y FE(=)1177 1978 y Fv(\022)1292 2057 y FE(1)91 b FD(i)1300 2178 y(i)g FE(1)1514 1978 y Fv(\023)17 b(\022)1719 2057 y FD(e)1764 2021 y Fw(\000)p Ft(ia)1999 2057 y FE(0)1777 2178 y(0)142 b FD(e)2013 2142 y Ft(ia)2120 1978 y Fv(\023)17 b(\022)2356 2057 y FE(1)p FD(=)p FE(2)113 b FA(\000)p FD(i=)p FE(2)2325 2178 y FA(\000)p FD(i=)p FE(2)h(1)p FD(=)p FE(2)2866 1978 y Fv(\023)1074 2392 y FE(=)1177 2252 y Fv(\022)1292 2331 y FE(cos)17 b FD(a)83 b FA(\000)17 b FE(sin)g FD(a)1297 2452 y FE(sin)g FD(a)130 b FE(cos)17 b FD(a)1897 2252 y Fv(\023)1987 2392 y FH(.)147 2667 y(Note)33 b(that)f(explicitly)f(if)g FD(X)40 b FH(\(and)33 b(hence)h FD(a)p FH(\))f(is)f(real,)f(then)j FD(e)2430 2631 y Ft(X)2530 2667 y FH(is)e(real.)147 2926 y FI(Case)38 b(2:)50 b FD(X)45 b FI(is)37 b(nilp)s(oten)m(t)147 3110 y FH(An)f FD(n)24 b FA(\002)g FD(n)36 b FH(matrix)d FD(X)43 b FH(is)35 b(said)g(to)g(b)s(e)g FI(nilp)s(oten)m(t)f FH(if)g FD(X)2252 3074 y Ft(m)2350 3110 y FE(=)e(0)j FH(for)g(some)g(p)s(ositiv)m(e)g(in)m(teger)g FD(m)p FH(.)147 3231 y(Of)g(course,)h(if)e FD(X)797 3195 y Ft(m)895 3231 y FE(=)e(0)p FH(,)k(then)f FD(X)1428 3195 y Ft(l)1486 3231 y FE(=)c(0)k 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FE(+)2198 4915 y(1)p 2198 4960 49 4 v 2198 5051 a(2)2257 4982 y FD(t)2292 4946 y Fz(2)2332 4982 y FD(ac)1690 5135 y FE(0)102 b(1)279 b FD(tc)1690 5255 y FE(0)102 b(0)294 b(1)2466 4866 y Fv(1)2466 5041 y(C)2466 5105 y(A)2570 5096 y FH(.)p eop %%Page: 42 42 42 41 bop 147 48 a FK(42)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FI(Case)38 b(3:)50 b FD(X)45 b FI(arbitrary)147 482 y FH(A)30 b(general)g(matrix)e FD(X)38 b FH(ma)m(y)30 b(b)s(e)g(neither)h(nilp)s(oten)m(t)d(nor)i (diagonalizable.)39 b(Ho)m(w)m(ev)m(er,)33 b(it)c(follo)m(ws)147 603 y(from)46 b(the)i(Jordan)f(canonical)f(form)g(that)h FD(X)55 b FH(can)47 b(b)s(e)h(written)f(\(Exercise)h(2\))f(in)g(the)g (form)147 723 y FD(X)e FE(=)37 b FD(S)31 b FE(+)26 b FD(N)48 b FH(with)38 b FD(S)44 b FH(diagonalizable,)36 b FD(N)48 b FH(nilp)s(oten)m(t,)38 b(and)g FD(S)6 b(N)47 b FE(=)37 b FD(N)10 b(S)c FH(.)60 b(\(See)39 b(Exercise)g(2.\))147 844 y(Then,)34 b(since)f FD(N)43 b FH(and)33 b FD(S)38 b FH(comm)m(ute,)1546 1035 y FD(e)1591 993 y Ft(X)1686 1035 y FE(=)28 b FD(e)1835 993 y Ft(S)t Fz(+)p Ft(N)2031 1035 y FE(=)g FD(e)2180 993 y Ft(S)2231 1035 y FD(e)2276 993 y Ft(N)147 1226 y FH(and)33 b FD(e)382 1189 y Ft(S)465 1226 y FH(and)g FD(e)700 1189 y Ft(N)800 1226 y FH(can)g(b)s(e)f (computed)h(as)g(ab)s(o)m(v)m(e.)446 1346 y(F)-8 b(or)32 b(example,)g(tak)m(e)1605 1590 y FD(X)k FE(=)1825 1449 y Fv(\022)1940 1529 y FD(a)88 b(b)1941 1649 y FE(0)c FD(a)2167 1449 y Fv(\023)2257 1590 y FH(.)147 1833 y(Then)1340 2057 y FD(X)35 b FE(=)1560 1916 y Fv(\022)1675 1996 y FD(a)84 b FE(0)1676 2116 y(0)g FD(a)1902 1916 y Fv(\023)1997 2057 y FE(+)2095 1916 y Fv(\022)2210 1996 y FE(0)j FD(b)2210 2116 y FE(0)c(0)2432 1916 y Fv(\023)2522 2057 y FH(.)147 2306 y(The)31 b(t)m(w)m(o)f(terms)g(clearly)f(comm)m(ute)g(\(since)h (the)h(\034rst)f(one)g(is)f(a)h(m)m(ultiple)d(of)j(the)g(iden)m(tit)m (y\),)g(and)147 2426 y(so)1015 2650 y FD(e)1060 2609 y Ft(X)1155 2650 y FE(=)1258 2510 y Fv(\022)1373 2589 y 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4849 y Fz(log)13 b Ft(z)1961 4890 y FE(=)27 b FD(z)t FC(.)147 5081 y(F)-7 b(or)34 b(al)5 b(l)35 b FD(u)f FC(with)h FA(j)o FD(u)p FA(j)27 b FD(<)h FE(log)16 b(2)p FC(,)35 b FA(j)o FD(e)1341 5045 y Ft(u)1409 5081 y FA(\000)22 b FE(1)p FA(j)28 b FD(<)f FE(1)35 b FC(and)1720 5272 y FE(log)17 b FD(e)1908 5231 y Ft(u)1980 5272 y FE(=)28 b FD(u)p FC(.)p eop %%Page: 43 43 43 42 bop 147 48 a Fx(The)28 b(Matrix)f(Logarithm)2683 b FG(43)147 298 y FI(Pro)s(of)147 482 y FH(The)34 b(usual)e(logarithm)d (for)j(real,)g(p)s(ositiv)m(e)g(n)m(um)m(b)s(ers)h(satis\034es)1004 650 y FD(d)p 976 694 107 4 v 976 785 a(dx)1109 717 y FE(log\(1)22 b FA(\000)g FD(x)p FE(\))28 b(=)1728 650 y FA(\000)p FE(1)p 1678 694 226 4 v 1678 785 a(1)22 b FA(\000)h FD(x)1942 717 y FE(=)k FA(\000)2139 636 y Fv(\000)2185 717 y FE(1)22 b(+)g FD(x)g FE(+)g FD(x)2584 676 y Fz(2)2646 717 y FE(+)g FA(\001)17 b(\001)g(\001)2877 636 y Fv(\001)147 947 y FH(for)32 b FA(j)p FD(x)p FA(j)c FD(<)f FE(1)p FH(.)43 b(In)m(tegrating)32 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FE(=)1403 1738 y Fw(1)1366 1768 y Fv(X)1362 1977 y Ft(m)p Fz(=1)1514 1863 y FE(\()p FA(\000)p FE(1\))1716 1821 y Ft(m)p Fz(+1)1883 1795 y FE(\()p FD(z)27 b FA(\000)c FE(1\))2180 1759 y Ft(m)p 1883 1840 363 4 v 2022 1931 a FD(m)2256 1863 y FH(.)446 2134 y(This)49 b(series)f(has)h(radius)f(of)g(con)m(v)m (ergence)j(one,)h(and)d(de\034nes)h(a)e(complex)f(analytic)147 2255 y(function)37 b(on)g(the)g(set)h FA(fj)o FD(z)27 b FA(\000)c FE(1)p FA(j)k FD(<)h FE(1)p FA(g)o FH(,)38 b(whic)m(h)g(coincides)f(with)f(the)i(usual)f(logarithm)c(for)k(real) 147 2375 y FD(z)44 b FH(in)39 b(the)h(in)m(terv)-5 b(al)38 b FE(\(0)p FD(;)17 b FE(2\))p FH(.)63 b(No)m(w,)42 b FE(exp)q(\(log)16 b FD(z)t FE(\))40 b(=)g FD(z)k FH(for)39 b FD(z)44 b FA(2)39 b FE(\(0)p FD(;)17 b FE(2\))p FH(,)41 b(and)e(b)m(y)i(analyticit)m(y)c(this)147 2496 y(iden)m(tit)m(y)c(con)m (tin)m(ues)g(to)f(hold)g(on)g(the)h(whole)g(set)g FA(fj)o FD(z)27 b FA(\000)c FE(1)p FA(j)k FD(<)h FE(1)p FA(g)o FH(.)446 2616 y(On)33 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FD(A)g FC(is)f(r)-5 b(e)g(al.)446 4691 y(F)e(or)34 b(al)5 b(l)34 b FD(A)h FC(with)g FA(k)p FD(A)22 b FA(\000)h FD(I)8 b FA(k)27 b FD(<)h FE(1)p FC(,)1725 4881 y FD(e)1770 4840 y Fz(log)13 b Ft(A)1958 4881 y FE(=)27 b FD(A)p FC(.)147 5072 y(F)-7 b(or)34 b(al)5 b(l)35 b FD(X)42 b FC(with)35 b FA(k)p FD(X)8 b FA(k)27 b FD(<)g FE(log)17 b(2)p FC(,)1380 4987 y Fv(\015)1380 5047 y(\015)1435 5072 y FD(e)1480 5036 y Ft(X)1570 5072 y FA(\000)22 b FE(1)1718 4987 y Fv(\015)1718 5047 y(\015)1801 5072 y FD(<)28 b FE(1)34 b FC(and)1692 5272 y FE(log)17 b FD(e)1880 5231 y Ft(X)1975 5272 y FE(=)27 b FD(X)8 b FC(.)p eop %%Page: 44 44 44 43 bop 147 48 a FK(44)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FI(Pro)s(of)147 482 y FH(It)f(is)g(easy)h(to)f(see)i(that)e(the)g(series)h(\(3.6\))f(con)m (v)m(erges)i(absolutely)d(whenev)m(er)k FA(k)p FD(A)22 b FA(\000)h FD(I)8 b FA(k)27 b FD(<)h FE(1)p FH(.)41 b(The)147 603 y(pro)s(of)c(of)g(con)m(tin)m(uit)m(y)g(is)h(essen)m 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3473 y Fz(log)13 b Ft(z)2267 3481 y Fn(n)2355 3139 y Fv(1)2355 3314 y(C)2355 3378 y(A)2459 3369 y FD(C)2536 3328 y Fw(\000)p Fz(1)2658 3369 y FE(=)28 b FD(A)p FH(.)446 3727 y FC(Case)34 b(2:)50 b FD(A)35 b FC(is)g(not)g(diagonalizable.)446 3848 y FH(If)43 b FD(A)g FH(is)f(not)g(diagonalizable,)g(then,)k(using)c(the)h (Jordan)g(canonical)e(form,)j(it)e(is)g(not)147 3968 y(di\036cult)35 b(to)h(construct)h(a)f(sequence)i FD(A)1628 3983 y Ft(m)1731 3968 y FH(of)e(diagonalizable)c(matrices)j(with)h FD(A)3176 3983 y Ft(m)3276 3968 y FA(!)d FD(A)p FH(.)54 b(\(See)147 4088 y(Exercise)34 b(4.\))44 b(If)33 b FA(k)o FD(A)23 b FA(\000)f FD(I)8 b FA(k)28 b FD(<)g FE(1)p FH(,)k(then)i FA(k)o FD(A)1717 4103 y Ft(m)1806 4088 y FA(\000)23 b FD(I)8 b FA(k)28 b FD(<)f FE(1)33 b FH(for)f(all)f (su\036cien)m(tly)i(large)f FD(m)p FH(.)44 b(By)33 b(Case)147 4209 y(1,)g FE(exp)q(\(log)16 b FD(A)658 4224 y Ft(m)725 4209 y FE(\))27 b(=)h FD(A)967 4224 y Ft(m)1034 4209 y FH(,)k(and)h(so)g(b)m(y)g(the)g(con)m(tin)m(uit)m(y)g(of)f FE(exp)h FH(and)g FE(log)p FH(,)f FE(exp)r(\(log)16 b FD(A)p FE(\))28 b(=)f FD(A)p FH(.)446 4329 y(Th)m(us)e(w)m(e)g(ha)m(v)m (e)g(sho)m(wn)h(that)d FE(exp)q(\(log)17 b FD(A)p FE(\))28 b(=)f FD(A)d FH(for)f(all)f FD(A)i FH(with)f FA(k)p FD(A)g FA(\000)f FD(I)8 b FA(k)27 b FD(<)h FE(1)p FH(.)40 b(No)m(w,)27 b(the)147 4450 y(same)h(argumen)m(t)g(as)g(in)g(the)g(complex)g(case)h (sho)m(ws)g(that)f(if)f FA(k)p FD(X)8 b FA(k)27 b FD(<)h FE(log)16 b(2)p FH(,)29 b(then)3139 4365 y Fv(\015)3139 4425 y(\015)3194 4450 y FD(e)3239 4413 y Ft(X)3329 4450 y FA(\000)23 b FD(I)3480 4365 y Fv(\015)3480 4425 y(\015)3563 4450 y FD(<)k FE(1)p FH(.)147 4570 y(But)39 b(then)g(the)g(same)f(t)m (w)m(o-case)i(argumen)m(t)e(as)h(ab)s(o)m(v)m(e)g(sho)m(ws)h(that)e FE(log\(exp)18 b FD(X)8 b FE(\))37 b(=)h FD(X)46 b FH(for)38 b(all)147 4690 y(suc)m(h)c FD(X)8 b FH(.)43 b Fy(\003)147 4915 y FI(Prop)s(osition)36 b(35)48 b FC(Ther)-5 b(e)39 b(exists)g(a)g(c)-5 b(onstant)39 b FD(c)g FC(such)g(that)h(for)f(al)5 b(l)39 b FD(n)26 b FA(\002)g FD(n)39 b FC(matric)-5 b(es)39 b FD(B)44 b FC(with)147 5036 y FA(k)p FD(B)5 b FA(k)28 b FD(<)467 4997 y Fz(1)p 467 5013 36 4 v 467 5070 a(2)1340 5267 y FA(k)p FE(log\()p FD(I)i FE(+)22 b FD(B)5 b FE(\))22 b FA(\000)h FD(B)5 b FA(k)28 b(\024)g FD(c)17 b FA(k)o FD(B)5 b FA(k)2463 5218 y Fz(2)2519 5267 y FC(.)p eop %%Page: 45 45 45 44 bop 147 48 a Fx(F)-7 b(urther)28 b(Prop)r(erties)e(of)i(the)g (Matrix)f(Exp)r(onen)n(tial)1858 b FG(45)147 298 y FI(Pro)s(of)147 482 y FH(Note)33 b(that)787 706 y FE(log\()p FD(I)d FE(+)22 b FD(B)5 b FE(\))22 b FA(\000)h FD(B)33 b FE(=)1612 582 y Fw(1)1576 611 y Fv(X)1571 821 y Ft(m)p Fz(=2)1724 706 y FE(\()p FA(\000)p FE(1\))1926 665 y Ft(m)2003 639 y FD(B)2082 603 y Ft(m)p 2003 683 146 4 v 2033 774 a FD(m)2186 706 y FE(=)28 b FD(B)2369 665 y Fz(2)2466 582 y Fw(1)2429 611 y Fv(X)2425 821 y Ft(m)p Fz(=2)2577 706 y FE(\()p FA(\000)p FE(1\))2779 665 y Ft(m)2856 639 y FD(B)2935 603 y Ft(m)p Fw(\000)p Fz(2)p 2856 683 236 4 v 2931 774 a FD(m)147 960 y FH(so)33 b(that)1170 1183 y FA(k)p FE(log\()p FD(I)d FE(+)22 b FD(B)5 b FE(\))22 b FA(\000)g FD(B)5 b FA(k)28 b(\024)g(k)p FD(B)5 b FA(k)2234 1135 y Fz(2)2331 1059 y Fw(1)2294 1089 y Fv(X)2290 1298 y Ft(m)p Fz(=2)2469 1029 y Fv(\000)2525 1070 y Fz(1)p 2525 1087 36 4 v 2525 1144 a(2)2570 1029 y Fv(\001)2616 1051 y Ft(m)p 2469 1160 214 4 v 2533 1252 a FD(m)2692 1183 y FH(.)147 1437 y(This)33 b(is)f(what)h(w)m(e)g(w)m(an)m(t.)45 b Fy(\003)147 1611 y FI(Prop)s(osition)36 b(36)48 b FC(L)-5 b(et)40 b FD(X)48 b FC(b)-5 b(e)39 b(any)g FD(n)26 b FA(\002)g FD(n)40 b FC(c)-5 b(omplex)38 b(matrix,)j(and)e(let)g FD(C)2942 1626 y Ft(m)3048 1611 y FC(b)-5 b(e)39 b(a)h(se)-5 b(quenc)g(e)38 b(of)147 1732 y(matric)-5 b(es)34 b(such)h(that)g FA(k)p FD(C)1080 1747 y Ft(m)1146 1732 y FA(k)28 b(\024)1339 1692 y Fz(const)p Ft(:)p 1339 1709 181 4 v 1381 1766 a(m)1443 1747 y Fr(2)1530 1732 y FC(.)44 b(Then)1378 1973 y FE(lim)1344 2033 y Ft(m)p Fw(!1)1564 1833 y Fv(\024)1616 1973 y FD(I)30 b FE(+)1797 1906 y FD(X)p 1797 1950 89 4 v 1799 2042 a(m)1918 1973 y FE(+)22 b FD(C)2086 1988 y Ft(m)2153 1833 y Fv(\025)2205 1855 y Ft(m)2300 1973 y FE(=)27 b FD(e)2448 1932 y Ft(X)2516 1973 y FC(.)147 2277 y FI(Pro)s(of)147 2462 y FH(The)k(expression)g(inside)e(the)h (brac)m(k)m(ets)i(is)e(clearly)e(tending)i(to)f FD(I)38 b FH(as)30 b FD(m)e FA(!)f(1)p FH(,)j(and)g(so)g(is)g(in)f(the)147 2582 y(domain)i(of)h(the)h(logarithm)c(for)j(all)f(su\036cien)m(tly)i (large)e FD(m)p FH(.)44 b(No)m(w)1154 2807 y FE(log)1297 2666 y Fv(\022)1370 2807 y FD(I)30 b FE(+)1551 2739 y FD(X)p 1551 2784 V 1553 2875 a(m)1672 2807 y FE(+)22 b FD(C)1840 2822 y Ft(m)1906 2666 y Fv(\023)2007 2807 y FE(=)2121 2739 y FD(X)p 2121 2784 V 2123 2875 a(m)2242 2807 y FE(+)g FD(C)2410 2822 y Ft(m)2498 2807 y FE(+)g FD(E)2668 2822 y Ft(m)147 3063 y FH(where)30 b FD(E)497 3078 y Ft(m)592 3063 y FH(is)e(an)h(error)f(term)g(whic)m(h,)i(b)m(y)f (Prop)s(osition)e(35)h(satis\034es)h FA(k)p FD(E)2839 3078 y Ft(m)2906 3063 y FA(k)e(\024)i FD(c)3148 2978 y Fv(\015)3148 3038 y(\015)3212 3024 y Ft(X)p 3212 3040 64 4 v 3212 3097 a(m)3308 3063 y FE(+)22 b FD(C)3476 3078 y Ft(m)3542 2978 y Fv(\015)3542 3038 y(\015)3597 3005 y Fz(2)3665 3063 y FA(\024)157 3144 y Fz(const)p Ft(:)p 157 3161 181 4 v 199 3218 a(m)261 3199 y Fr(2)348 3184 y FH(.)43 b(But)33 b(then)1121 3423 y FD(I)d FE(+)1302 3355 y FD(X)p 1302 3400 89 4 v 1304 3491 a(m)1423 3423 y FE(+)22 b FD(C)1591 3438 y Ft(m)1685 3423 y FE(=)27 b(exp)1954 3282 y Fv(\022)2037 3355 y FD(X)p 2037 3400 V 2039 3491 a(m)2158 3423 y FE(+)22 b FD(C)2326 3438 y Ft(m)2415 3423 y FE(+)g FD(E)2585 3438 y Ft(m)2651 3282 y Fv(\023)2741 3423 y FH(,)147 3647 y(and)33 b(so)995 3713 y Fv(\024)1048 3854 y FD(I)d FE(+)1229 3787 y FD(X)p 1229 3831 V 1231 3922 a(m)1349 3854 y FE(+)22 b FD(C)1517 3869 y Ft(m)1584 3713 y Fv(\025)1636 3736 y Ft(m)1731 3854 y FE(=)27 b(exp)18 b(\()p FD(X)30 b FE(+)22 b FD(mC)2402 3869 y Ft(m)2491 3854 y FE(+)g FD(mE)2746 3869 y Ft(m)2813 3854 y FE(\))16 b FH(.)147 4091 y(Since)25 b(b)s(oth)f FD(C)686 4106 y Ft(m)777 4091 y FH(and)h FD(E)1031 4106 y Ft(m)1122 4091 y FH(are)f(of)h(order)1667 4052 y Fz(1)p 1636 4068 98 4 v 1636 4126 a Ft(m)1698 4107 y Fr(2)1743 4091 y FH(,)h(w)m(e)g(obtain)d(the)i(desired)g(result)g(b)m(y)g (letting)e FD(m)28 b FA(!)f(1)147 4212 y FH(and)33 b(using)f(the)h(con) m(tin)m(uit)m(y)g(of)f(the)h(exp)s(onen)m(tial.)42 b Fy(\003)147 4467 y FI(3.4)112 b(F)-9 b(urther)38 b(Prop)s(erties)e(of)h (the)h(Matrix)e(Exp)s(onen)m(tial)147 4634 y FH(In)c(this)e(section)i (w)m(e)g(giv)m(e)f(three)h(additional)c(results)j(in)m(v)m(olving)f (the)i(exp)s(onen)m(tial)e(of)h(a)g(matrix,)147 4755 y(whic)m(h)i(will)d(b)s(e)j(imp)s(ortan)m(t)e(in)h(our)g(study)i(of)e (Lie)g(algebras.)147 4929 y FI(Theorem)j(37)j(\(Lie)e(Pro)s(duct)h(F)-9 b(orm)m(ula\))45 b FC(L)-5 b(et)43 b FD(X)51 b FC(and)42 b FD(Y)64 b FC(b)-5 b(e)43 b FD(n)28 b FA(\002)g FD(n)43 b FC(c)-5 b(omplex)42 b(matric)-5 b(es.)147 5049 y(Then)1417 5222 y FD(e)1462 5181 y Ft(X)5 b Fz(+)p Ft(Y)1669 5222 y FE(=)61 b(lim)1772 5282 y Ft(m)p Fw(!1)1993 5111 y Fv(\020)2052 5222 y FD(e)2107 5154 y Fn(X)p 2107 5166 55 3 v 2107 5207 a(m)2176 5222 y FD(e)2234 5154 y Fn(Y)p 2231 5166 V 2231 5207 a(m)2300 5111 y Fv(\021)2359 5134 y Ft(m)2442 5222 y FC(.)p eop %%Page: 46 46 46 45 bop 147 48 a FK(46)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FI(Remark)147 494 y FH(This)34 b(theorem)f(has)h(a)g(big)e(brother,)i(called)f(the)h(T)-8 b(rotter)34 b(pro)s(duct)f(form)m(ula,)f(whic)m(h)j(giv)m(es)f(the)147 615 y(same)46 b(result)g(in)f(the)h(case)h(where)g FD(X)54 b FH(and)46 b FD(Y)67 b FH(are)46 b(suitable)f(un)m(b)s(ounded)j(op)s (erators)d(on)h(an)147 735 y(in\034nite-dimensional)29 b(Hilb)s(ert)i(space.)44 b(The)34 b(T)-8 b(rotter)32 b(form)m(ula)f(is)h(describ)s(ed,)i(for)e(example,)g(in)147 855 y(M.)h(Reed)g(and)g(B.)g(Simon,)e FC(Metho)-5 b(ds)34 b(of)h(Mo)-5 b(dern)34 b(Mathematic)-5 b(al)35 b(Physics)p FH(,)d(V)-8 b(ol.)42 b(I,)33 b(VI)s(I)s(I.8.)147 1151 y FI(Pro)s(of)147 1347 y FH(Using)f(the)h(p)s(o)m(w)m(er)h(series)f (for)f(the)h(exp)s(onen)m(tial)f(and)g(m)m(ultiplying,)e(w)m(e)j(get) 1361 1627 y FD(e)1416 1559 y Fn(X)p 1416 1571 55 3 v 1416 1612 a(m)1485 1627 y FD(e)1543 1559 y Fn(Y)p 1540 1571 V 1540 1612 a(m)1636 1627 y FE(=)28 b FD(I)i FE(+)1921 1560 y FD(X)p 1921 1604 89 4 v 1923 1696 a(m)2042 1627 y FE(+)2153 1560 y FD(Y)p 2149 1604 86 4 v 2149 1696 a(m)2267 1627 y FE(+)22 b FD(C)2435 1642 y Ft(m)2501 1627 y FH(,)147 1919 y(where)36 b(\(c)m(hec)m(k!\))53 b FA(k)o FD(C)934 1934 y Ft(m)1001 1919 y FA(k)31 b(\024)1201 1880 y Ft(const:)p 1201 1896 186 4 v 1245 1953 a(m)1307 1934 y Fr(2)1396 1919 y FH(.)51 b(Since)35 b FD(e)1786 1855 y Fn(X)p 1786 1867 55 3 v 1786 1908 a(m)1854 1919 y FD(e)1912 1855 y Fn(Y)p 1910 1867 V 1910 1908 a(m)2010 1919 y FA(!)c FD(I)43 b FH(as)35 b FD(m)d FA(!)f(1)p FH(,)k FD(e)2814 1855 y Fn(X)p 2814 1867 V 2814 1908 a(m)2883 1919 y FD(e)2941 1855 y Fn(Y)p 2938 1867 V 2938 1908 a(m)3041 1919 y FH(is)f(in)h(the)g(domain)147 2039 y(of)d(the)h(logarithm)d(for)i(all)e(su\036cien)m(tly)j(large)f FD(m)p FH(.)43 b(But)1091 2328 y FE(log)1233 2218 y Fv(\020)1293 2328 y FD(e)1348 2260 y Fn(X)p 1348 2272 V 1348 2313 a(m)1417 2328 y FD(e)1475 2260 y Fn(Y)p 1472 2272 V 1472 2313 a(m)1540 2218 y Fv(\021)1628 2328 y FE(=)27 b(log)1874 2188 y Fv(\022)1947 2328 y FD(I)j FE(+)2128 2261 y FD(X)p 2128 2305 89 4 v 2130 2397 a(m)2249 2328 y FE(+)2360 2261 y FD(Y)p 2357 2305 86 4 v 2357 2397 a(m)2474 2328 y FE(+)22 b FD(C)2642 2343 y Ft(m)2709 2188 y Fv(\023)1628 2592 y FE(=)1741 2524 y FD(X)p 1741 2569 89 4 v 1743 2660 a(m)1862 2592 y FE(+)1973 2524 y FD(Y)p 1970 2569 86 4 v 1970 2660 a(m)2087 2592 y FE(+)g FD(C)2255 2607 y Ft(m)2344 2592 y FE(+)g FD(E)2514 2607 y Ft(m)147 2886 y FH(where)43 b(b)m(y)g(Prop)s(osition)d(35)h FA(k)p FD(C)1376 2901 y Ft(m)1442 2886 y FA(k)i(\024)h FD(const:)1928 2802 y Fv(\015)1928 2862 y(\015)1993 2847 y Ft(X)p 1993 2864 64 4 v 1993 2921 a(m)2089 2886 y FE(+)2200 2847 y Ft(Y)p 2197 2864 63 4 v 2197 2921 a(m)2291 2886 y FE(+)22 b FD(C)2459 2901 y Ft(m)2526 2802 y Fv(\015)2526 2862 y(\015)2581 2828 y Fz(2)2664 2886 y FA(\024)2795 2847 y Ft(const:)p 2795 2864 186 4 v 2839 2921 a(m)2901 2902 y Fr(2)2990 2886 y FH(.)71 b(Exp)s(onen)m(tiating)147 3007 y(the)33 b(logarithm)c(giv)m(es)1174 3296 y FD(e)1229 3228 y Fn(X)p 1229 3240 55 3 v 1229 3281 a(m)1298 3296 y FD(e)1356 3228 y Fn(Y)p 1353 3240 V 1353 3281 a(m)1450 3296 y FE(=)e(exp)1719 3155 y Fv(\022)1802 3228 y FD(X)p 1802 3273 89 4 v 1804 3364 a(m)1923 3296 y FE(+)2034 3228 y FD(Y)p 2031 3273 86 4 v 2031 3364 a(m)2148 3296 y FE(+)22 b FD(C)2316 3311 y Ft(m)2405 3296 y FE(+)g FD(E)2575 3311 y Ft(m)2642 3155 y Fv(\023)147 3584 y FH(and)1033 3711 y Fv(\020)1093 3822 y FD(e)1148 3753 y Fn(X)p 1148 3765 55 3 v 1148 3807 a(m)1216 3822 y FD(e)1274 3753 y Fn(Y)p 1272 3765 V 1272 3807 a(m)1340 3711 y Fv(\021)1400 3733 y Ft(m)1494 3822 y FE(=)28 b(exp)17 b(\()p FD(X)30 b FE(+)22 b FD(Y)43 b FE(+)22 b FD(mC)2363 3837 y Ft(m)2452 3822 y FE(+)g FD(mE)2707 3837 y Ft(m)2774 3822 y FE(\))17 b FH(.)147 4094 y(Since)37 b(b)s(oth)g FD(C)711 4109 y Ft(m)815 4094 y FH(and)g FD(E)1081 4109 y Ft(m)1184 4094 y FH(are)h(of)e(order)1767 4055 y Fz(1)p 1736 4071 98 4 v 1736 4128 a Ft(m)1798 4109 y Fr(2)1843 4094 y FH(,)i(w)m(e)h(ha)m(v)m(e)f(\(using)f(the)g(con)m(tin)m(uit)m(y)g(of)g (the)g(exp)s(o-)147 4214 y(nen)m(tial\))1322 4481 y FE(lim)1288 4541 y Ft(m)p Fw(!1)1508 4371 y Fv(\020)1568 4481 y FD(e)1623 4413 y Fn(X)p 1623 4425 55 3 v 1623 4466 a(m)1691 4481 y FD(e)1749 4413 y Fn(Y)p 1747 4425 V 1747 4466 a(m)1815 4371 y Fv(\021)1875 4393 y Ft(m)1969 4481 y FE(=)28 b(exp)17 b(\()p FD(X)30 b FE(+)22 b FD(Y)f FE(\))147 4745 y FH(whic)m(h)33 b(is)f(the)h(Lie)f(pro)s(duct)h(form)m(ula.)41 b Fy(\003)147 5019 y FI(Theorem)35 b(38)49 b FC(L)-5 b(et)35 b FD(X)43 b FC(b)-5 b(e)34 b(an)h FD(n)22 b FA(\002)h FD(n)35 b FC(r)-5 b(e)g(al)34 b(or)h(c)-5 b(omplex)34 b(matrix.)44 b(Then)1526 5257 y FE(det)1678 5176 y Fv(\000)1724 5257 y FD(e)1769 5216 y Ft(X)1836 5176 y Fv(\001)1910 5257 y FE(=)27 b FD(e)2058 5216 y Fz(trace)q(\()p Ft(X)5 b Fz(\))2334 5257 y FC(.)p eop %%Page: 47 47 47 46 bop 147 48 a Fx(F)-7 b(urther)28 b(Prop)r(erties)e(of)i(the)g (Matrix)f(Exp)r(onen)n(tial)1858 b FG(47)147 298 y FI(Pro)s(of)147 482 y FH(There)34 b(are)e(three)i(cases,)g(as)e(in)g(Section)h(3.2.)446 603 y FC(Case)k(1:)55 b FD(A)35 b FC(is)i(diagonalizable)p FH(.)50 b(Supp)s(ose)36 b(there)g(is)f(a)g(complex)g(in)m(v)m(ertible)f (matrix)g FD(C)147 723 y FH(suc)m(h)g(that)1311 1060 y FD(X)i FE(=)27 b FD(C)1625 829 y Fv(0)1625 1005 y(B)1625 1069 y(@)1754 918 y FD(\025)1811 933 y Fz(1)2158 918 y FE(0)1938 1021 y FH(.)1977 1046 y(.)2015 1072 y(.)1777 1200 y FE(0)304 b FD(\025)2187 1215 y Ft(n)2276 829 y Fv(1)2276 1005 y(C)2276 1069 y(A)2380 1060 y FD(C)2457 1018 y Fw(\000)p Fz(1)2551 1060 y FH(.)147 1415 y(Then)1270 1751 y FD(e)1315 1710 y Ft(X)1411 1751 y FE(=)27 b FD(C)1608 1521 y Fv(0)1608 1696 y(B)1608 1760 y(@)1736 1609 y FD(e)1781 1573 y Ft(\025)1822 1582 y Fr(1)2184 1609 y FE(0)1950 1713 y FH(.)1988 1738 y(.)2026 1763 y(.)1775 1892 y FE(0)318 b FD(e)2187 1855 y Ft(\025)2228 1863 y Fn(n)2317 1521 y Fv(1)2317 1696 y(C)2317 1760 y(A)2420 1751 y FD(C)2497 1710 y Fw(\000)p Fz(1)2592 1751 y FH(.)147 2124 y(Th)m(us)36 b FE(trace\()p FD(X)8 b FE(\))30 b(=)908 2049 y Fv(P)1030 2124 y FD(\025)1087 2139 y Ft(i)1115 2124 y FH(,)35 b(and)f FE(det\()p FD(e)1586 2088 y Ft(X)1654 2124 y FE(\))c(=)1828 2049 y Fv(Q)1939 2124 y FD(e)1984 2088 y Ft(\025)2025 2098 y Fn(i)2086 2124 y FE(=)g FD(e)2237 2038 y Fl(P)2324 2088 y Ft(\025)2365 2098 y Fn(i)2395 2124 y FH(.)48 b(\(Recall)33 b(that)h FE(trace\()p FD(C)7 b(D)s(C)3504 2088 y Fw(\000)p Fz(1)3598 2124 y FE(\))30 b(=)147 2244 y(trace)q(\()p FD(D)s FE(\))p FH(.\))446 2364 y FC(Case)39 b(2:)59 b FD(X)48 b FC(is)39 b(nilp)-5 b(otent)p FH(.)59 b(If)37 b FD(X)46 b FH(is)37 b(nilp)s(oten)m(t,)h(then)g(it)f(cannot)h(ha)m(v)m (e)h(an)m(y)f(non-zero)147 2485 y(eigen)m(v)-5 b(alues)42 b(\(c)m(hec)m(k!\),)k(and)c(so)g(all)e(the)i(ro)s(ots)g(of)f(the)h(c)m (haracteristic)g(p)s(olynomial)c(m)m(ust)k(b)s(e)147 2605 y(zero.)h(Th)m(us)31 b(the)f(Jordan)f(canonical)f(form)g(of)h FD(X)37 b FH(will)27 b(b)s(e)i(strictly)g(upp)s(er)h(triangular.)40 b(That)29 b(is,)147 2726 y FD(X)40 b FH(can)33 b(b)s(e)g(written)f(as) 1362 3062 y FD(X)k FE(=)27 b FD(C)1676 2832 y Fv(0)1676 3007 y(B)1676 3071 y(@)1804 2920 y FE(0)281 b FA(\003)1942 3024 y FH(.)1980 3049 y(.)2018 3074 y(.)1804 3202 y FE(0)g(0)2225 2832 y Fv(1)2225 3007 y(C)2225 3071 y(A)2329 3062 y FD(C)2406 3021 y Fw(\000)p Fz(1)2500 3062 y FH(.)147 3430 y(In)30 b(that)e(case)i(\(it)e(is)h(easy)h(to)f(see\))h FD(e)1467 3394 y Ft(X)1563 3430 y FH(will)d(b)s(e)i(upp)s(er)h(triangular,)d (with)i(ones)h(on)f(the)g(diagonal:)1350 3791 y FD(e)1395 3750 y Ft(X)1490 3791 y FE(=)f 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5236 y Ft(X)549 5272 y FC(is)g(an)h(element)f(of)g(the)h(identity)g(c)-5 b(omp)g(onent)34 b(of)g FD(G:)p eop %%Page: 54 54 54 53 bop 147 48 a FK(54)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FI(Pro)s(of)147 490 y FH(By)33 b(de\034nition)f(of)g(the)h(Lie)f(algebra,)f FD(e)1590 454 y Ft(tX)1716 490 y FH(lies)h(in)f FD(G)i FH(for)f(all)f(real)g FD(t)p FH(.)44 b(But)33 b(as)g FD(t)f FH(v)-5 b(aries)33 b(from)e FE(0)h FH(to)147 610 y FE(1)p FH(,)h FD(e)301 574 y Ft(tX)426 610 y FH(is)f(a)g(con)m(tin)m (uous)i(path)e(connecting)h(the)g(iden)m(tit)m(y)f(to)g FD(e)2501 574 y Ft(X)2569 610 y FH(.)43 b Fy(\003)147 867 y FI(Prop)s(osition)36 b(43)48 b FC(L)-5 b(et)49 b FD(G)e FC(b)-5 b(e)48 b(a)f(matrix)h(Lie)g(gr)-5 b(oup,)50 b(with)e(Lie)g(algebr)-5 b(a)46 b Fk(g)p FC(.)84 b(L)-5 b(et)48 b FD(X)56 b FC(b)-5 b(e)47 b(an)147 987 y(element)34 b(of)h Fk(g)p FC(,)g(and)f FD(A)h FC(an)f(element)g(of)h FD(G)p FC(.)45 b(Then)34 b FD(AX)8 b(A)2291 951 y Fw(\000)p 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4298 y Fv(i)1488 4408 y FC(,)34 b(for)h(al)5 b(l)34 b FD(X)r(;)17 b(Y)49 b FA(2)28 b Fk(g)p FC(.)263 4672 y(3.)400 4646 y Fv(e)391 4672 y FD(\036)p FE(\()p FD(X)8 b FE(\))27 b(=)777 4633 y Ft(d)p 765 4649 62 4 v 765 4706 a(dt)836 4587 y Fv(\014)836 4647 y(\014)870 4711 y Ft(t)p Fz(=0)1006 4672 y FD(\036)p FE(\()p FD(e)1147 4636 y Ft(tX)1240 4672 y FE(\))p FC(,)34 b(for)h(al)5 b(l)34 b FD(X)i FA(2)28 b Fk(g)p FC(.)446 4914 y(If)38 b FD(G)p FC(,)h FD(H)8 b FC(,)39 b(and)f FD(K)46 b FC(ar)-5 b(e)38 b(matrix)h(Lie)f(gr)-5 b(oups)38 b(and)g FD(\036)d FE(:)f FD(H)42 b FA(!)35 b FD(K)45 b FC(and)38 b FD( )h FE(:)34 b FD(G)h FA(!)f FD(H)46 b FC(ar)-5 b(e)147 5034 y(Lie)35 b(gr)-5 b(oup)35 b(homomorphisms,)d (then)1661 5249 y Fu(])1645 5272 y FD(\036)22 b FA(\016)g FD( )32 b FE(=)2005 5246 y Fv(e)1995 5272 y FD(\036)22 b FA(\016)2164 5246 y Fv(e)2147 5272 y FD( )t FC(.)p eop %%Page: 56 56 56 55 bop 147 48 a FK(56)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FI(Remarks)147 483 y FH(In)34 b(practice,)f(giv)m(en)g(a)g(Lie)f(group)h(homomorphism)d FD(\036)p FH(,)j(the)h(w)m(a)m(y)g(one)g(go)s(es)f(ab)s(out)f (computing)157 577 y Fv(e)147 603 y FD(\036)i FH(is)g(b)m(y)h(using)e (Prop)s(ert)m(y)i(3.)48 b(Of)34 b(course,)h(since)1985 577 y Fv(e)1976 603 y FD(\036)f FH(is)f(\(real\))g(linear,)g(it)g (su\036ces)j(to)e(compute)3694 577 y Fv(e)3684 603 y FD(\036)147 735 y FH(on)h(a)g(basis)f(for)h Fk(g)p FH(.)50 b(In)35 b(the)h(language)d(of)i(di\033eren)m(tiable)e(manifolds,)g (Prop)s(ert)m(y)j(3)e(sa)m(ys)j(that)3694 709 y Fv(e)3684 735 y FD(\036)147 855 y FH(is)31 b(the)g(deriv)-5 b(ativ)m(e)31 b(\(or)f(di\033eren)m(tial\))f(of)i FD(\036)f FH(at)h(the)g(iden)m(tit) m(y)-8 b(,)31 b(whic)m(h)h(is)e(the)i(standard)f(de\034nition)147 987 y(of)268 960 y Fv(e)258 987 y FD(\036)p FH(.)43 b(\(See)34 b(also)d(Exercise)j(19.\))446 1107 y(A)45 b(linear)f(map)g(with)h(prop) s(ery)g(\(2\))g(is)g(called)f(a)g FI(Lie)52 b(algebra)g(homomorphism)n FH(.)147 1228 y(\(See)35 b(Section)e(3.8.\))47 b(This)34 b(theorem)f(sa)m(ys)i(that)f(ev)m(ery)i(Lie)d(group)g(homomorphism)e (giv)m(es)j(rise)147 1348 y(to)42 b(a)g(Lie)g(algebra)f(homomorphism.) 71 b(W)-8 b(e)42 b(will)f(see)i(ev)m(en)m(tually)g(that)f(the)h(con)m (v)m(erse)i(is)d(true)147 1469 y FC(under)33 b(c)-5 b(ertain)33 b(cir)-5 b(cumstanc)g(es)p FH(.)41 b(Sp)s(eci\034cally)-8 b(,)30 b(supp)s(ose)i(that)e FD(G)h FH(and)f FD(H)38 b FH(are)31 b(Lie)f(groups,)h(and)157 1563 y Fv(e)147 1589 y FD(\036)j FE(:)h Fk(g)f FA(!)g Fk(h)i FH(is)g(a)g(Lie)g(algebra) f(homomorphism.)52 b(If)37 b FD(G)f FH(is)g FC(c)-5 b(onne)g(cte)g(d)37 b(and)h(simply)g(c)-5 b(onne)g(cte)g(d)p FH(,)147 1709 y(then)37 b(there)f(exists)h(a)e(unique)i(Lie)e(group)g(homomorphism)e FD(\036)g FE(:)h FD(G)f FA(!)g FD(H)43 b FH(suc)m(h)37 b(that)f FD(\036)f FH(and)3694 1683 y Fv(e)3684 1709 y FD(\036)147 1830 y FH(are)e(related)f(as)h(in)e(Theorem)i(46.)147 2091 y FI(Pro)s(of)k(of)h(the)f(Theorem)147 2277 y FH(The)47 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522 y FE(=)27 b FD(e)2553 478 y Ft(t)2578 486 y FE(\()2624 461 y Fl(e)2617 478 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\)+)2838 461 y Fl(e)2831 478 y Ft(\036)q Fz(\()p Ft(Y)16 b Fz(\))2985 486 y FE(\))3027 522 y FH(.)147 773 y(Di\033eren)m(tiating)31 b(this)h(result)h(at)f FD(t)c FE(=)f(0)33 b FH(giv)m(es)g(the)f(desired)i(result.)446 1016 y FC(Step)h(4)14 b FH(:)806 990 y Fv(e)797 1016 y FD(\036)i FE(\()p FD(AX)8 b(A)1144 980 y Fw(\000)p Fz(1)1238 1016 y FE(\))28 b(=)f FD(\036)p FE(\()p FD(A)p FE(\))1623 990 y Fv(e)1614 1016 y FD(\036)p FE(\()p FD(X)8 b FE(\))p FD(\036)p FE(\()p FD(A)p FE(\))2044 980 y Fw(\000)p Fz(1)2137 1016 y FH(.)446 1138 y(By)33 b(Steps)h(1)e(and)h(2,)780 1362 y FE(exp)17 b FD(t)989 1336 y Fv(e)980 1362 y FD(\036)p FE(\()p FD(AX)8 b(A)1311 1321 y Fw(\000)p Fz(1)1406 1362 y FE(\))27 b(=)h(exp)1750 1336 y Fv(e)1740 1362 y FD(\036)p FE(\()p FD(tAX)8 b(A)2106 1321 y Fw(\000)p Fz(1)2201 1362 y FE(\))27 b(=)h FD(\036)2445 1282 y Fv(\000)2490 1362 y FE(exp)18 b FD(tAX)8 b(A)2926 1321 y Fw(\000)p Fz(1)3020 1282 y Fv(\001)3082 1362 y FH(.)147 1587 y(Using)32 b(a)h(prop)s(ert)m(y)g(of)f(the)h(exp)s(onen)m(tial)f(and)g(Step)i(1,)e (this)g(b)s(ecomes)820 1811 y FE(exp)17 b FD(t)1029 1785 y Fv(e)1020 1811 y FD(\036)p FE(\()p FD(AX)8 b(A)1351 1770 y Fw(\000)p Fz(1)1445 1811 y FE(\))28 b(=)g FD(\036)1690 1730 y Fv(\000)1735 1811 y FD(Ae)1853 1770 y Ft(tX)1946 1811 y FD(A)2019 1770 y Fw(\000)p Fz(1)2113 1730 y Fv(\001)2187 1811 y FE(=)f FD(\036)p FE(\()p FD(A)p FE(\))p FD(\036)p FE(\()p FD(e)2638 1770 y Ft(tX)2731 1811 y FE(\))p FD(\036)p FE(\()p FD(A)p FE(\))2976 1770 y Fw(\000)p Fz(1)1511 1987 y FE(=)h FD(\036)p FE(\()p FD(A)p FE(\))p FD(e)1867 1946 y Ft(t)1899 1928 y Fl(e)1892 1946 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))2056 1987 y FD(\036)p FE(\()p FD(A)p FE(\))2263 1946 y Fw(\000)p Fz(1)2357 1987 y FH(.)147 2211 y(Di\033eren)m(tiating) 31 b(this)h(at)g FD(t)c FE(=)g(0)k FH(giv)m(es)h(the)g(desired)g (result.)446 2455 y FC(Step)i(5)14 b FH(:)806 2429 y Fv(e)797 2455 y FD(\036)o FE(\([)p FD(X)r(;)j(Y)22 b FE(]\))27 b(=)1321 2344 y Fv(h)1377 2429 y(e)1368 2455 y FD(\036)p FE(\()p FD(X)8 b FE(\))p FD(;)1643 2429 y Fv(e)1635 2455 y FD(\036)o FE(\()p FD(Y)21 b FE(\))1846 2344 y Fv(i)1893 2455 y FH(.)446 2598 y(Recall)31 b(from)g(the)i(pro)s (of)f(of)g(Theorem)h(44)f(that)1400 2822 y FE([)p FD(X)r(;)17 b(Y)22 b FE(])28 b(=)1824 2783 y Ft(d)p 1811 2800 62 4 v 1811 2857 a(dt)1883 2738 y Fv(\014)1883 2798 y(\014)1916 2861 y Ft(t)p Fz(=0)2052 2822 y FD(e)2097 2781 y Ft(tX)2191 2822 y FD(Y)21 b(e)2314 2781 y Fw(\000)p Ft(tX)2462 2822 y FH(.)147 3047 y(Hence)782 3245 y Fv(e)772 3271 y FD(\036)c FE(\([)p FD(X)r(;)g(Y)k FE(]\))28 b(=)1323 3245 y Fv(e)1313 3271 y FD(\036)1388 3190 y Fv(\000)1466 3232 y Ft(d)p 1453 3248 V 1453 3305 a(dt)1525 3186 y Fv(\014)1525 3246 y(\014)1558 3310 y Ft(t)p Fz(=0)1695 3271 y FD(e)1740 3230 y Ft(tX)1833 3271 y FD(Y)21 b(e)1956 3230 y Fw(\000)p Ft(tX)2104 3190 y Fv(\001)2177 3271 y FE(=)2313 3232 y Ft(d)p 2301 3248 V 2301 3305 a(dt)2372 3186 y Fv(\014)2372 3246 y(\014)2406 3310 y Ft(t)p Fz(=0)2551 3245 y Fv(e)2542 3271 y FD(\036)2616 3190 y Fv(\000)2662 3271 y FD(e)2707 3230 y Ft(tX)2800 3271 y FD(Y)g(e)2923 3230 y Fw(\000)p Ft(tX)3071 3190 y Fv(\001)147 3495 y FH(where)32 b(w)m(e)f(ha)m(v)m(e)g (used)h(the)e(fact)g(that)g(a)g(deriv)-5 b(ativ)m(e)30 b(comm)m(utes)f(with)h(a)g(linear)f(tranformation.)446 3617 y(But)k(then)g(b)m(y)g(Step)h(4,)1148 3815 y Fv(e)1138 3842 y FD(\036)17 b FE(\([)p FD(X)r(;)g(Y)k FE(]\))28 b(=)1712 3802 y Ft(d)p 1699 3819 V 1699 3876 a(dt)1771 3757 y Fv(\014)1771 3817 y(\014)1804 3881 y Ft(t)p Fz(=0)1941 3842 y FD(\036)p FE(\()p FD(e)2082 3800 y Ft(tX)2174 3842 y FE(\))2221 3815 y Fv(e)2212 3842 y FD(\036)p FE(\()p FD(Y)21 b FE(\))p FD(\036)p FE(\()p FD(e)2565 3800 y Fw(\000)p Ft(tX)2713 3842 y FE(\))1576 4020 y(=)1712 3981 y Ft(d)p 1699 3997 V 1699 4055 a(dt)1771 3936 y Fv(\014)1771 3995 y(\014)1804 4059 y Ft(t)p Fz(=0)1941 4020 y FD(e)1986 3979 y Ft(t)2018 3962 y Fl(e)2011 3979 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))2185 3994 y Fv(e)2176 4020 y FD(\036)o FE(\()p FD(Y)22 b FE(\))p FD(e)2433 3979 y Fw(\000)p Ft(t)2520 3962 y Fl(e)2513 3979 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))1576 4207 y FE(=)1679 4097 y Fv(h)1736 4181 y(e)1726 4207 y FD(\036)p FE(\()p FD(X)j FE(\))p FD(;)2002 4181 y Fv(e)1993 4207 y FD(\036)o FE(\()p FD(Y)21 b FE(\))2204 4097 y Fv(i)2268 4207 y FH(.)446 4555 y FC(Step)35 b(6)14 b FH(:)806 4528 y Fv(e)797 4555 y FD(\036)o FE(\()p FD(X)8 b FE(\))28 b(=)1183 4516 y Ft(d)p 1170 4532 V 1170 4589 a(dt)1242 4470 y Fv(\014)1242 4530 y(\014)1275 4594 y Ft(t)p Fz(=0)1411 4555 y FD(\036)p FE(\()p FD(e)1552 4519 y Ft(tX)1645 4555 y FE(\))p FH(.)446 4702 y(This)33 b(follo)m(ws)e(from)g(\(3.15\))h(and)h(our)f(de\034nition)f(of)2418 4675 y Fv(e)2409 4702 y FD(\036)o FH(.)446 4945 y FC(Step)k(7)14 b FH(:)806 4919 y Fv(e)797 4945 y FD(\036)34 b FC(is)h(the)g(unique)f (r)-5 b(e)g(al-line)g(ar)34 b(map)g(such)h(that)44 b FD(\036)p FE(\()p FD(e)2715 4909 y Ft(X)2783 4945 y FE(\))27 b(=)h FD(e)3004 4892 y Fl(e)2997 4909 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))3161 4945 y FH(.)446 5067 y(Supp)s(ose)34 b(that)e FD( )k FH(is)c(another)h(suc)m(h)h(map.)43 b(Then)1436 5292 y FD(e)1481 5250 y Ft(t )r Fz(\()p Ft(X)5 b Fz(\))1705 5292 y FE(=)28 b FD(e)1854 5250 y Ft( )r Fz(\()p Ft(tX)5 b Fz(\))2078 5292 y FE(=)27 b FD(\036)p FE(\()p FD(e)2322 5250 y Ft(tX)2415 5292 y FE(\))p eop %%Page: 58 58 58 57 bop 147 48 a FK(58)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FH(so)33 b(that)1483 531 y FD( )t FE(\()p FD(X)8 b FE(\))27 b(=)1879 491 y Ft(d)p 1866 508 62 4 v 1866 565 a(dt)1938 446 y Fv(\014)1938 506 y(\014)1971 570 y Ft(t)p Fz(=0)2107 531 y FD(\036)p FE(\()p FD(e)2248 490 y Ft(tX)2341 531 y FE(\))p FH(.)147 789 y(Th)m(us)34 b(b)m(y)g(Step)f(6,)f FD( )37 b FH(coincides)32 b(with)1604 762 y Fv(e)1594 789 y FD(\036)p FH(.)446 1038 y FC(Step)j(8:)808 1015 y Fu(])793 1038 y FD(\036)22 b FA(\016)g FD( )31 b FE(=)1152 1012 y Fv(e)1143 1038 y FD(\036)22 b FA(\016)1311 1012 y Fv(e)1295 1038 y FD( )t FH(.)446 1163 y(F)-8 b(or)32 b(an)m(y)h FD(X)i FA(2)29 b Fk(g)p FH(,)876 1417 y FD(\036)22 b FA(\016)g FD( )1111 1337 y Fv(\000)1157 1417 y FD(e)1202 1376 y Ft(tX)1295 1337 y Fv(\001)1368 1417 y FE(=)28 b FD(\036)1547 1337 y Fv(\000)1592 1417 y FD( )1676 1337 y Fv(\000)1721 1417 y FD(e)1766 1376 y Ft(tX)1859 1337 y Fv(\001\001)1978 1417 y FE(=)g FD(\036)2157 1307 y Fv(\020)2216 1417 y FD(e)2261 1376 y Ft(t)2299 1359 y Fl(e)2286 1376 y Ft( )s Fz(\()p Ft(X)5 b Fz(\))2457 1307 y Fv(\021)2544 1417 y FE(=)28 b FD(e)2693 1376 y Ft(t)2725 1359 y Fl(e)2718 1376 y Ft(\036)q Fz(\()2800 1359 y Fl(e)2788 1376 y Ft( )r Fz(\()p Ft(X)5 b Fz(\)\))2986 1417 y FH(.)147 1708 y(Th)m(us)410 1685 y Fu(])394 1708 y FD(\036)22 b FA(\016)g FD( )t FE(\()p FD(X)8 b FE(\))27 b(=)918 1682 y Fv(e)909 1708 y FD(\036)22 b FA(\016)1077 1682 y Fv(e)1061 1708 y FD( )t FE(\()p FD(X)8 b FE(\))p FH(.)43 b Fy(\003)147 1971 y FI(De\034nition)36 b(47)i(\(The)f(Adjoin)m(t)f(Mapping\))49 b FC(L)-5 b(et)48 b FD(G)f FC(b)-5 b(e)47 b(a)g(matrix)g(Lie)g(gr)-5 b(oup,)50 b(with)d(Lie)147 2091 y(algebr)-5 b(a)34 b Fk(g)p FC(.)45 b(Then)34 b(for)h(e)-5 b(ach)34 b FD(A)28 b FA(2)g FD(G)p FC(,)34 b(de\034ne)g(a)h(line)-5 b(ar)34 b(map)g FE(Ad)p FD(A)28 b FE(:)g Fk(g)g FA(!)f Fk(g)35 b FC(by)g(the)g(formula)1517 2324 y FE(Ad)p FD(A)p FE(\()p FD(X)8 b FE(\))28 b(=)f FD(AX)8 b(A)2248 2283 y Fw(\000)p Fz(1)2343 2324 y FC(.)147 2557 y(We)35 b(wil)5 b(l)34 b(let)h FE(Ad)h FC(denote)e(the)h(map)f FD(A)28 b FA(!)f FE(Ad)p FD(A)p FC(.)147 2815 y FI(Prop)s(osition)36 b(48)48 b FC(L)-5 b(et)44 b FD(G)g FC(b)-5 b(e)43 b(a)g(matrix)h(Lie)f(gr)-5 b(oup,)45 b(with)f(Lie)f(algebr)-5 b(a)43 b Fk(g)p FC(.)71 b(Then)42 b(for)i(e)-5 b(ach)147 2936 y FD(A)50 b FA(2)g FD(G)p FC(,)f FE(Ad)q FD(A)e FC(is)f(an)g(invertible)g(line)-5 b(ar)46 b(tr)-5 b(anformation)46 b(of)g Fk(g)h FC(with)g(inverse)e FE(Ad)q FD(A)3414 2900 y Fw(\000)p Fz(1)3508 2936 y FC(,)50 b(and)147 3056 y FE(Ad)28 b(:)g FD(G)g FA(!)f FF(GL)p FE(\()p Fk(g)p FE(\))35 b FC(is)f(a)h(gr)-5 b(oup)34 b(homomorphism.)147 3342 y FI(Pro)s(of)147 3536 y FH(Easy)-8 b(.)42 b(Note)25 b(that)f(Prop)s(osition)f(43)h(guaran)m(tees)h(that)f FE(Ad)q FD(A)p FE(\()p FD(X)8 b FE(\))24 b FH(is)g(actually)f(in)h Fk(g)h FH(for)f(all)e FD(X)36 b FA(2)28 b Fk(g)p FH(.)147 3656 y Fy(\003)446 3781 y FH(Since)34 b Fk(g)g FH(is)f(a)h(real)f(v)m (ector)i(space)g(with)e(some)h(dimension)e FD(k)s FH(,)i FF(GL)p FE(\()p Fk(g)p FE(\))g FH(is)f(essen)m(tially)h(the)147 3901 y(same)40 b(as)f FF(GL)p FE(\()p FD(k)s FE(;)17 b Fu(R)5 b FE(\))p FH(.)70 b(Th)m(us)42 b(w)m(e)e(will)d(regard)j FF(GL)p FE(\()p Fk(g)p FE(\))f FH(as)h(a)f(matrix)g(Lie)f(group.)65 b(It)40 b(is)f(easy)h(to)147 4022 y(sho)m(w)35 b(that)f FE(Ad)d(:)g FD(G)f FA(!)g FF(GL)p FE(\()p Fk(g)p FE(\))k FH(is)g(con)m(tin)m(uous,)h(and)g(so)f(is)g(a)g(Lie)f(group)h (homomorphism.)46 b(By)147 4156 y(Theorem)35 b(46,)f(there)i(is)e(an)g 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b(Lie)h(algebr)-5 b(a)33 b(map.)44 b(Then)34 b(for)h(al)5 b(l)35 b FD(X)r(;)17 b(Y)49 b FA(2)28 b Fk(g)1563 5240 y Fv(f)1549 5267 y FE(Ad)p FD(X)8 b FE(\()p FD(Y)21 b FE(\))28 b(=)g([)p FD(X)r(;)17 b(Y)k FE(])p FC(.)p eop %%Page: 59 59 59 58 bop 147 48 a Fx(The)28 b(Exp)r(onen)n(tial)f(Mapping)2556 b FG(59)147 298 y FI(Pro)s(of)147 486 y FH(Recall)31 b(that)h(b)m(y)i(Theorem)f(46,)1371 460 y Fv(f)1357 486 y FE(Ad)g FH(can)f(b)s(e)h(computed)g(as)g(follo)m(ws:)1470 685 y Fv(f)1456 711 y FE(Ad)p FD(X)j FE(=)1836 672 y Ft(d)p 1823 688 62 4 v 1823 745 a(dt)1895 626 y Fv(\014)1895 686 y(\014)1928 750 y Ft(t)p Fz(=0)2065 711 y FE(Ad\()p FD(e)2275 670 y Ft(tX)2368 711 y FE(\))p FH(.)147 936 y(Th)m(us)928 1135 y Fv(f)915 1161 y FE(Ad)p FD(X)8 b FE(\()p FD(Y)21 b FE(\))28 b(=)1449 1122 y Ft(d)p 1436 1138 V 1436 1196 a(dt)1508 1077 y Fv(\014)1508 1136 y(\014)1541 1200 y Ft(t)p Fz(=0)1677 1161 y FE(Ad)q(\()p FD(e)1888 1120 y Ft(tX)1981 1161 y FE(\)\()p FD(Y)21 b FE(\))27 b(=)2337 1122 y Ft(d)p 2324 1138 V 2324 1196 a(dt)2396 1077 y Fv(\014)2396 1136 y(\014)2429 1200 y Ft(t)p Fz(=0)2565 1161 y FD(e)2610 1120 y Ft(tX)2703 1161 y FD(Y)22 b(e)2827 1120 y Fw(\000)p Ft(tX)1313 1318 y FE(=)27 b([)p FD(X)r(;)17 b(Y)22 b FE(])147 1543 y FH(whic)m(h)33 b(is)f(what)h(w)m(e)h(w)m(an)m (ted)g(to)e(pro)m(v)m(e.)45 b(See)33 b(also)f(Exercise)h(13.)43 b Fy(\003)147 1811 y FI(3.7)112 b(The)38 b(Exp)s(onen)m(tial)e(Mapping) 147 1981 y(De\034nition)g(50)49 b FC(If)34 b FD(G)h FC(is)g(a)f(matrix) h(Lie)f(gr)-5 b(oup)35 b(with)g(Lie)f(algebr)-5 b(a)34 b Fk(g)p FC(,)h(then)g(the)g FB(exp)-6 b(onential)147 2101 y(mapping)36 b FC(for)e FD(G)h FC(is)g(the)g(map)1673 2326 y FE(exp)29 b(:)e Fk(g)h FA(!)g FD(G)p FC(.)446 2566 y FH(In)33 b(general)e(the)i(exp)s(onen)m(tial)e(mapping)f(is)i (neither)g(one-to-one)f(nor)h(on)m(to.)44 b(Nev)m(erthe-)147 2687 y(less,)30 b(it)e(pro)m(vides)h(an)g(crucial)f(mec)m(hanism)g(for) g(passing)h(information)c(b)s(et)m(w)m(een)31 b(the)e(group)g(and)147 2807 y(the)42 b(Lie)f(algebra.)70 b(The)43 b(follo)m(wing)c(result)j (sa)m(ys)h(that)e(the)i(exp)s(onen)m(tial)e(mapping)f(is)j FC(lo)-5 b(c)g(al)5 b(ly)147 2928 y FH(one-to-one)32 b(and)h(on)m(to,)f(a)g(result)h(that)f(will)e(b)s(e)j(essen)m(tial)g (later.)147 3170 y FI(Prop)s(osition)j(51)48 b FC(L)-5 b(et)34 b FD(G)g FC(b)-5 b(e)33 b(a)h(matrix)f(Lie)g(gr)-5 b(oup)34 b(with)f(Lie)g(algebr)-5 b(a)33 b Fk(g)p FC(.)45 b(Then)32 b(ther)-5 b(e)34 b(exist)f(a)147 3290 y(neighb)-5 b(orho)g(o)g(d)29 b FD(U)41 b FC(of)30 b(zer)-5 b(o)30 b(in)g Fk(g)g FC(and)g(a)g(neighb)-5 b(orho)g(o)g(d)29 b FD(V)52 b FC(of)30 b FD(I)38 b FC(in)30 b FD(G)g FC(such)g(that)h (the)f(exp)-5 b(onential)147 3410 y(mapping)34 b(takes)g FD(U)46 b FC(home)-5 b(omorphic)g(al)5 b(ly)32 b(onto)j FD(V)21 b FC(.)147 3681 y FI(Pro)s(of)147 3869 y FH(W)-8 b(e)30 b(follo)m(w)d(the)j(pro)s(of)e(of)h(Theorem)g(I.3.11)g(in)f (Br\366c)m(k)m(er)j(and)e(tom)f(Diec)m(k.)44 b(In)29 b(view)h(of)f(what)g(w)m(e)147 3989 y(ha)m(v)m(e)g(pro)m(v)m(ed)g(ab)s (out)e(the)g(matrix)f(logarithm,)f(w)m(e)k(kno)m(w)f(this)f(result)h (for)e(the)i(case)h(of)d FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)147 4110 y(T)-8 b(o)39 b(pro)m(v)m(e)g(the)g (general)f(case,)j(w)m(e)f(consider)e(a)h(matrix)e(Lie)g(group)i FD(G)e(<)h FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(,)46 b(with)38 b(Lie)147 4230 y(algebra)32 b Fk(g)p FH(.)147 4472 y FI(Lemma)h(52)49 b FC(Supp)-5 b(ose)34 b FD(g)1123 4487 y Ft(n)1205 4472 y FC(ar)-5 b(e)35 b(elements)g(of)g FD(G)p FC(,)g(and)g(that)g FD(g)2466 4487 y Ft(n)2541 4472 y FA(!)29 b FD(I)8 b FC(.)45 b(L)-5 b(et)36 b FD(Y)3022 4487 y Ft(n)3097 4472 y FE(=)28 b(log)17 b FD(g)3391 4487 y Ft(n)3438 4472 y FC(,)35 b(which)147 4592 y(is)30 b(de\034ne)-5 b(d)30 b(for)g(al)5 b(l)30 b(su\036ciently)g(lar)-5 b(ge)30 b FD(n)p FC(.)43 b(Supp)-5 b(ose)30 b FD(Y)2131 4607 y Ft(n)2178 4592 y FD(=)17 b FA(k)o FD(Y)2350 4607 y Ft(n)2396 4592 y FA(k)28 b(!)f FD(Y)49 b FA(2)28 b FF(gl)16 b 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b(\()p FD(m)3415 5307 y Ft(n)3462 5292 y FD(Y)3519 5307 y Ft(n)3566 5292 y FE(\))62 b(=)p eop %%Page: 60 60 60 59 bop 147 48 a FK(60)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FE(exp)18 b([\()p FD(m)463 313 y Ft(n)527 298 y FA(k)p FD(Y)634 313 y Ft(n)680 298 y FA(k)p FE(\))f(\()o FD(Y)879 313 y Ft(n)926 298 y FD(=)g FA(k)o FD(Y)1098 313 y Ft(n)1145 298 y FA(k)p FE(\)])28 b FA(!)h FE(exp)17 b(\()p FD(tY)22 b FE(\))p FH(.)45 b(But)33 b FE(exp)18 b(\()o FD(m)2326 313 y Ft(n)2374 298 y FD(Y)2431 313 y Ft(n)2477 298 y FE(\))29 b(=)f(exp)18 b(\()p FD(Y)2909 313 y Ft(n)2956 298 y FE(\))2994 249 y Ft(m)3056 257 y Fn(n)3131 298 y FE(=)29 b(\()p FD(g)3321 313 y Ft(n)3367 298 y FE(\))3405 249 y Ft(m)3467 257 y Fn(n)3543 298 y FA(2)g FD(G)p FH(,)147 418 y(and)k FD(G)f FH(is)g(closed,)h(so)g FE(exp)18 b(\()o FD(tY)k FE(\))28 b FA(2)g FD(G)p FH(.)43 b Fy(\003)446 539 y FH(W)-8 b(e)41 b(think)f(of)g FF(gl)17 b FE(\()p FD(n)p FE(;)g Fu(C)j FE(\))46 b FH(as)41 b 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584 4240 77 4 v 34 w FD(U)10 b FH(.)48 b(\(The)35 b(existence)g(of)f(the)g(matrix)f(logarithm)d(implies)i (that)i(the)g(exp)s(onen)m(tial)f(is)147 4441 y(one-to-one)h(near)g (zero.\))49 b(Let)35 b FE(log)f FH(denote)h(the)g(in)m(v)m(erse)h(map,) e(de\034ned)h(on)g FE(exp)3165 4360 y Fv(\000)p 3211 4361 V 81 x FD(U)3288 4360 y Fv(\001)3333 4441 y FH(.)49 b(Since)p 3665 4361 V 34 w FD(U)147 4572 y FH(is)29 b(compact,)g(and)h FE(exp)g FH(is)e(one-to-one)h(and)g(con)m(tin)m(uous)h(on)p 2392 4492 V 29 w FD(U)10 b FH(,)30 b(log)e(will)f(b)s(e)i(con)m(tin)m (uous.)43 b(\(This)147 4692 y(is)35 b(a)g(standard)g(top)s(ological)d (result.\))51 b(So)35 b(tak)m(e)h FD(V)57 b FH(to)35 b(b)s(e)g(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h FD(I)43 b FH(con)m(tained)35 b(in)147 4813 y FE(exp)313 4732 y Fv(\000)p 358 4732 V 358 4813 a FD(U)435 4732 y Fv(\001)481 4813 y FH(,)f(and)f(let)g FD(U)950 4776 y Fw(0)1003 4813 y FE(=)c(exp)1257 4776 y Fw(\000)p Fz(1)1368 4813 y FE(\()p FD(V)21 b FE(\))i FA(\\)g FD(U)10 b FH(.)47 b(Then)34 b FD(U)2115 4776 y Fw(0)2173 4813 y FH(is)f(op)s(en)g(and)h(the)g(exp)s (onen)m(tial)e(tak)m(es)j FD(U)3718 4776 y Fw(0)147 4933 y FH(homeomorphically)29 b(on)m(to)k FD(V)21 b FH(.)44 b Fy(\003)147 5152 y FI(De\034nition)36 b(53)49 b FC(If)39 b FD(U)51 b FC(and)39 b FD(V)62 b FC(ar)-5 b(e)40 b(as)f(in)h(Pr)-5 b(op)g(osition)39 b(51,)i(then)e(the)h(inverse)f(map)g FE(exp)3584 5116 y Fw(\000)p Fz(1)3715 5152 y FE(:)147 5272 y FD(V)49 b FA(!)28 b Fk(g)35 b FC(is)f(c)-5 b(al)5 b(le)-5 b(d)34 b(the)h(lo)-5 b(garithm)34 b(for)h FD(G)p FC(.)p eop %%Page: 61 61 61 60 bop 147 48 a Fx(Lie)28 b(Algebras)3054 b FG(61)147 298 y FI(Corollary)36 b(54)49 b FC(If)34 b FD(G)g FC(is)h(a)f(c)-5 b(onne)g(cte)g(d)34 b(matrix)g(Lie)h(gr)-5 b(oup,)34 b(then)g(every)h(element)f FD(A)h FC(of)f FD(G)h FC(c)-5 b(an)147 418 y(b)g(e)35 b(written)g(in)f(the)h(form)1551 643 y FD(A)28 b FE(=)f FD(e)1800 602 y Ft(X)1858 611 y Fr(1)1897 643 y FD(e)1942 602 y Ft(X)2000 611 y Fr(2)2056 643 y FA(\001)17 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b(brac)m(k)m(et)i (generates)g(four)f(terms,)g(for)f(a)h(total)e(of)i(t)m(w)m(elv)m(e.)44 b(Eac)m(h)32 b(of)f(the)g(six)g(orderings)f(of)147 2289 y FA(f)p FD(X)r(;)17 b(Y)5 b(;)17 b(Z)7 b FA(g)32 b FH(o)s(ccurs)h(t)m (wice,)g(once)g(with)g(a)f(plus)g(sign)g(and)h(once)g(with)f(a)h(min)m (us)f(sign.)42 b Fy(\003)147 2513 y FI(De\034nition)36 b(57)49 b FC(A)34 b FB(sub)-6 b(algebr)g(a)34 b FC(of)f(a)g(r)-5 b(e)g(al)33 b(or)g(c)-5 b(omplex)33 b(Lie)g(algebr)-5 b(a)32 b Fk(g)i FC(is)f(a)g(subsp)-5 b(ac)g(e)32 b Fk(h)i FC(of)f Fk(g)147 2633 y FC(such)c(that)h FE([)p FD(H)665 2648 y Fz(1)704 2633 y FD(;)17 b(H)829 2648 y Fz(2)868 2633 y FE(])28 b FA(2)g Fk(h)h FC(for)g(al)5 b(l)29 b FD(H)1463 2648 y Fz(1)1502 2633 y FD(;)17 b(H)1627 2648 y Fz(2)1693 2633 y FA(2)28 b Fk(h)p FC(.)43 b(If)29 b Fk(g)g FC(is)g(a)g(c)-5 b(omplex)28 b(Lie)h(algebr)-5 b(a,)29 b(and)f Fk(h)i FC(is)e(a)h(r)-5 b(e)g(al)147 2753 y(subsp)g(ac)g(e)35 b(of)h Fk(g)g FC(which)f(is)h(close)-5 b(d)35 b(under)h(br)-5 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FC(.)147 3614 y FI(Remarks)147 3799 y FH(A)33 b(subalgebra)e(of)h(a)g(Lie)g(algebra)f(is)h(again)e(a)i(Lie) g(algebra.)42 b(A)33 b(real)e(subalgebra)h(of)g(a)g(complex)147 3919 y(Lie)f(algebra)f(is)h(a)f(real)h(Lie)f(algebra.)42 b(The)32 b(in)m(v)m(erse)h(of)e(a)g(Lie)f(algebra)g(isomorphism)f(is)i (again)e(a)147 4040 y(Lie)j(algebra)f(isomorphism.)147 4263 y FI(Prop)s(osition)36 b(58)48 b FC(The)35 b(Lie)f(algebr)-5 b(a)34 b Fk(g)h FC(of)g(a)f(matrix)h(Lie)g(gr)-5 b(oup)34 b FD(G)h FC(is)g(a)f(r)-5 b(e)g(al)35 b(Lie)g(algebr)-5 b(a.)147 4522 y FI(Pro)s(of)147 4707 y FH(By)29 b(Theorem)f(44,)g Fk(g)g FH(is)g(a)f(real)g(subalgebra)h(of)f FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))34 b FH(complex)27 b(matrices,)h(and)g (is)f(th)m(us)i(a)f(real)147 4827 y(Lie)k(algebra.)42 b Fy(\003)147 5051 y FI(Theorem)35 b(59)j(\(Ado\))47 b FC(Every)h(\034nite-dimensional)d(r)-5 b(e)g(al)47 b(Lie)h(algebr)-5 b(a)46 b(is)i(isomorphic)e(to)h(a)147 5171 y(sub)-5 b(algebr)g(a)33 b(of)h FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))p FC(.)50 b(Every)35 b(\034nite-dimensional)c(c)-5 b(omplex)33 b(Lie)h(algebr)-5 b(a)33 b(is)h(isomorphic)f(to)147 5292 y(a)i(\(c)-5 b(omplex\))33 b(sub)-5 b(algebr)g(a)34 b(of)h FF(gl)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FC(.)p eop %%Page: 63 63 63 62 bop 147 48 a Fx(Lie)28 b(Algebras)3054 b FG(63)446 298 y FH(This)33 b(remark)-5 b(able)32 b(theorem)h(is)g(pro)m(v)m(ed)h (in)f(V)-8 b(aradara)5 b(jan.)44 b(The)35 b(pro)s(of)d(is)h(w)m(ell)f (b)s(ey)m(ond)147 418 y(the)27 b(scop)s(e)g(of)e(this)h(course)h (\(whic)m(h)g(is)f(after)g(all)e(a)i(course)h(on)f(Lie)g FC(gr)-5 b(oups)8 b FH(\),)27 b(and)f(requires)h(a)f(deep)147 539 y(understanding)40 b(of)e(the)i(structure)h(of)d(complex)h(Lie)f (algebras.)63 b(The)40 b(theorem)f(tells)f(us)i(that)147 659 y(ev)m(ery)h(Lie)d(algebra)g(is)g(\(isomorphic)f(to\))i(a)g(Lie)f (algebra)g(of)g(matrices.)62 b(\(This)39 b(is)f(in)g(con)m(trast)147 779 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FD(X)8 b FE(\()p FD(Y)21 b FE(\))36 b FH(is)g(just)g FE([)p FD(X)r(;)17 b(Y)k FE(])p FH(,)37 b(it)e(migh)m(t)g(seem)h(fo)s (olish)e(to)h(in)m(tro)s(duce)h(the)h(additional)c(\020)p FE(ad)o FH(\021)147 2382 y(notation.)43 b(Ho)m(w)m(ev)m(er,)35 b(thinking)c(of)i FE([)p FD(X)r(;)17 b(Y)k FE(])33 b FH(as)g(a)f(linear)f(map)h(in)g FD(Y)54 b FH(for)32 b(eac)m(h)i (\034xed)g FD(X)8 b FH(,)32 b(giv)m(es)h(a)147 2502 y(somewhat)d (di\033eren)m(t)g(p)s(ersp)s(ectiv)m(e.)44 b(In)29 b(an)m(y)i(case,)g (the)f(\020)p FE(ad)p FH(\021)f(notation)f(is)h(extremely)h(useful)g (in)147 2623 y(some)j(situations.)42 b(F)-8 b(or)32 b(example,)g (instead)g(of)g(writing)1543 2844 y FE([)p FD(X)r(;)17 b FE([)p FD(X)r(;)g FE([)p FD(X)r(;)g FE([)p FD(X)r(;)g(Y)22 b FE(]]]])147 3065 y FH(w)m(e)34 b(can)f(no)m(w)g(write)1692 3286 y FE(\(ad)p FD(X)8 b FE(\))1960 3238 y Fz(4)2016 3286 y FE(\()p FD(Y)21 b FE(\))p FH(.)147 3507 y(This)33 b(kind)f(of)g(notation)g(will)e(b)s(e)i(essen)m(tial)h(in)f(Section)g (4.1.)147 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b(the)f(asso)s(ciated)h(Lie)f (algebra)f(homomorphism)2199 1836 y Fv(f)2185 1862 y FE(Ad)d(:)f Fk(g)h FA(!)g FF(gl)o FE(\()p Fk(g)p FE(\))33 b FH(is)f(giv)m(en)h(b)m(y)1564 2068 y Fv(f)1551 2094 y FE(Ad)p FD(X)8 b FE(\()p FD(Y)21 b FE(\))28 b(=)f([)p FD(X)r(;)17 b(Y)k FE(])p FH(.)147 2313 y(In)33 b(our)f(new)i(notation,) d(w)m(e)j(ma)m(y)e(sa)m(y)1778 2505 y Fv(f)1764 2531 y FE(Ad)c(=)f(ad)147 2749 y FH(By)33 b(the)g(de\034ning)g(prop)s(ert)m (y)g(of)1360 2723 y Fv(f)1346 2749 y FE(Ad)p FH(,)g(w)m(e)g(ha)m(v)m(e) h(the)f(follo)m(wing)d(iden)m(tit)m(y:)43 b(F)-8 b(or)32 b(all)e FD(X)36 b FA(2)28 b Fk(g)p FH(,)1614 2968 y FE(Ad\()p FD(e)1824 2927 y Ft(X)1892 2968 y FE(\))f(=)h FD(e)2106 2927 y Fz(ad)q Ft(X)2248 2968 y FH(.)1218 b(\(3.17\))147 3186 y(Note)37 b(that)f(b)s(oth)h(sides)g(of)f(\(3.17\))g(are)g(linear) f(op)s(erators)i(on)f(the)h(Lie)f(algebra)f Fk(g)p FH(.)56 b(This)37 b(is)f(an)147 3306 y(imp)s(ortan)m(t)c(relation,)g(whic)m(h)i 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x(\000)p FE(1)h FH(di\033erence)g(b)s(et)m(w)m(een)i(the)e(ph)m(ysics)h (de\034nition)d(of)h(the)h(Lie)f(algebra)f(and)i(our)f(o)m(wn.\))p eop %%Page: 65 65 65 64 bop 147 48 a Fx(The)28 b(Complexi\034cation)e(of)i(a)f(Real)g (Lie)h(Algebra)1924 b FG(65)446 298 y FH(The)34 b(structure)f(constan)m (ts)h(satisfy)f(the)g(follo)m(wing)c(t)m(w)m(o)34 b(conditions,)2183 523 y FD(c)2225 538 y Ft(ij)t(k)2346 523 y FE(+)22 b FD(c)2486 538 y Ft(j)t(ik)2613 523 y FE(=)28 b(0)1124 602 y Fv(X)1165 811 y Ft(m)1268 697 y FE(\()p FD(c)1348 712 y Ft(ij)t(m)1471 697 y FD(c)1513 712 y Ft(mk)r(l)1662 697 y FE(+)22 b FD(c)1802 712 y Ft(j)t(k)r(m)1939 697 y FD(c)1981 712 y Ft(mil)2116 697 y FE(+)g FD(c)2256 712 y Ft(k)r(im)2385 697 y FD(c)2427 712 y Ft(mj)t(l)2548 697 y FE(\))27 b(=)h(0)147 1004 y FH(for)39 b(all)e FD(i;)17 b(j;)g(k)s(;)g(l)r FH(.)63 b(The)40 b(\034rst)g(of)f(these)h (conditions)e(comes)i(from)e(the)h(sk)m(ew-symmetry)i(of)e(the)147 1124 y(brac)m(k)m(et,)45 b(and)40 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b(and)d(that)h(it)e(satis\034es)i(the)g (Jacobi)f(iden)m(tit)m(y)-8 b(.)79 b(It)44 b(is)h(clear)e(that)i (\(3.18\))e(is)h FC(r)-5 b(e)g(al)147 5292 y FH(bilinear,)37 b(and)g(sk)m(ew-symmetric.)60 b(The)38 b(sk)m(ew-symmetry)h(means)f (that)f(if)g(\(3.18\))f(is)i(complex)p eop %%Page: 66 66 66 65 bop 147 48 a FK(66)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)147 298 y FH(linear)j(in)g(the)h(\034rst)h (factor,)e(it)g(is)h(also)f(complex)g(linear)g(in)g(the)h(second)i (factor.)42 b(Th)m(us)34 b(w)m(e)f(need)147 418 y(only)f(sho)m(w)i (that)940 657 y FE([)p FD(i)p FE(\()p FD(X)1119 672 y Fz(1)1181 657 y FE(+)22 b FD(iX)1393 672 y Fz(2)1433 657 y FE(\))p FD(;)17 b(Y)1572 672 y Fz(1)1632 657 y FE(+)22 b FD(iY)1820 672 y Fz(2)1860 657 y FE(])27 b(=)h FD(i)17 b FE([)p FD(X)2176 672 y Fz(1)2238 657 y FE(+)22 b FD(iX)2450 672 y Fz(2)2489 657 y FD(;)17 b(Y)2590 672 y Fz(1)2651 657 y FE(+)22 b FD(iY)2839 672 y Fz(2)2878 657 y FE(])17 b FH(.)544 b(\(3.19\))147 895 y(W)-8 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1626 y Fz(2)2180 1611 y FD(;)g(Y)2281 1626 y Fz(1)2320 1611 y FE(])22 b(+)g([)p FD(X)2575 1626 y Fz(1)2615 1611 y FD(;)17 b(Y)2716 1626 y Fz(2)2754 1611 y FE(])q(\))p FA(g)926 1757 y FE(=)28 b(\()p FA(\000)17 b FE([)p FD(X)1270 1772 y Fz(2)1310 1757 y FD(;)g(Y)1411 1772 y Fz(1)1449 1757 y FE(])23 b FA(\000)f FE([)p FD(X)1706 1772 y Fz(1)1746 1757 y FD(;)17 b(Y)1847 1772 y Fz(2)1885 1757 y FE(]\))23 b(+)f FD(i)17 b FE(\()o([)p FD(X)2266 1772 y Fz(1)2306 1757 y FD(;)g(Y)2407 1772 y Fz(1)2446 1757 y FE(])22 b FA(\000)g FE([)q FD(X)2703 1772 y Fz(2)2742 1757 y FD(;)17 b(Y)2843 1772 y Fz(2)2882 1757 y FE(]\))f FH(,)147 1995 y(and)33 b(indeed)g(these)g(are)g(equal.)446 2122 y(It)k(remains)f(to)h(c)m(hec) m(k)i(the)e(Jacobi)f(iden)m(tit)m(y)-8 b(.)57 b(Of)36 b(course,)j(the)f(Jacobi)e(iden)m(tit)m(y)g(holds)147 2242 y(if)e FD(X)r(;)17 b(Y)5 b(;)36 b FH(and)f FD(Z)42 b FH(are)35 b(in)g Fk(g)p FH(.)52 b(But)35 b(no)m(w)h(observ)m(e)h (that)e(the)h(expression)g(on)f(the)h(left)e(side)i(of)f(the)147 2363 y(Jacobi)27 b(iden)m(tit)m(y)g(is)g(\(complex!\))41 b(linear)25 b(in)i FD(X)35 b FH(for)27 b(\034xed)h FD(Y)49 b FH(and)27 b FD(Z)7 b FH(.)42 b(It)27 b(follo)m(ws)f(that)h(the)h (Jacobi)147 2483 y(iden)m(tit)m(y)35 b(holds)g(if)f FD(X)43 b FH(is)35 b(in)g Fk(g)1252 2498 y Fi(C)1304 2483 y FH(,)h(and)g FD(Y)5 b(;)17 b(Z)41 b FH(in)35 b Fk(g)p FH(.)52 b(The)36 b(same)f(argumen)m(t)g(then)h(sho)m(ws)h(that)e(w)m(e)147 2603 y(can)c(extend)i(to)d FD(Y)52 b FH(in)30 b Fk(g)1028 2618 y Fi(C)1080 2603 y FH(,)i(and)f(then)g(to)g FD(Z)37 b FH(in)30 b Fk(g)1931 2618 y Fi(C)1984 2603 y FH(.)43 b(Th)m(us)32 b(the)f(Jacobi)f(iden)m(tit)m(y)h(holds)f(in)g Fk(g)3542 2618 y Fi(C)3595 2603 y FH(.)43 b Fy(\003)147 2882 y FI(Prop)s(osition)36 b(64)48 b FC(The)34 b(Lie)g(algebr)-5 b(as)34 b FF(gl)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FC(,)41 b FF(sl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FC(,)40 b FF(so)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FC(,)40 b(and)34 b FF(sp)q FE(\()p FD(n)p FE(;)17 b Fu(C)i FE(\))41 b FC(ar)-5 b(e)34 b(c)-5 b(om-)147 3002 y(plex)34 b(Lie)g(algebr)-5 b(as,)33 b(as)h(is)g(the)g(Lie)g(algebr)-5 b(a)33 b(of)h(the)g(c)-5 b(omplex)33 b(Heisenb)-5 b(er)g(g)33 b(gr)-5 b(oup.)45 b(In)33 b(addition,)147 3122 y(we)i(have)f(the)h(fol) 5 b(lowing)33 b(isomorphisms)g(of)h(c)-5 b(omplex)34 b(Lie)g(algebr)-5 b(as)1444 3353 y FF(gl)17 b FE(\()o FD(n)p FE(;)g Fu(R)5 b FE(\))1782 3382 y Fi(C)1917 3325 y FA(\030)1918 3357 y FE(=)2101 3353 y FF(gl)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))1521 3473 y FF(u)p FE(\()p FD(n)p FE(\))1705 3488 y Fi(C)1917 3445 y FA(\030)1918 3477 y FE(=)2101 3473 y FF(gl)o FE(\()p FD(n)p FE(;)d Fu(C)j FE(\))1450 3593 y FF(sl)d FE(\()p FD(n)p FE(;)g Fu(R)5 b FE(\))1777 3622 y Fi(C)1917 3566 y FA(\030)1918 3597 y FE(=)2106 3593 y FF(sl)q FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))1503 3714 y FF(so)p FE(\()p FD(n)p FE(\))1723 3729 y Fi(C)1917 3686 y FA(\030)1918 3718 y FE(=)2094 3714 y FF(so)p FE(\()p FD(n)p FE(;)d Fu(C)j FE(\))1445 3834 y FF(sp)p FE(\()p FD(n)p FE(;)d Fu(R)5 b FE(\))1776 3849 y Fi(C)1917 3806 y FA(\030)1918 3838 y FE(=)2093 3834 y FF(sp)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))1502 3954 y FF(sp)q FE(\()p FD(n)p FE(\))1724 3969 y Fi(C)1917 3927 y FA(\030)1918 3959 y FE(=)2078 3954 y FF(sp)p FE(\()p FD(n)p FE(;)d Fu(C)j FE(\))p FC(.)147 4245 y FI(Pro)s(of)147 4443 y FH(F)-8 b(rom)38 b(the)j(computations)d(in)h(the)i(previous)f (section)g(w)m(e)g(see)h(easily)e(that)h(the)g(sp)s(eci\034ed)h(Lie)147 4563 y(algebras)30 b(are)h(in)f(fact)g(complex)g(subalgebras)h(of)f FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(,)37 b(and)31 b(hence)h(are)e(complex)h(Lie)f(alge-)147 4683 y(bras.)446 4810 y(No)m(w,)37 b FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))41 b FH(is)36 b(the)g(space)h(of)e(all)e FD(n)25 b FA(\002)g FD(n)36 b FH(complex)f(matrices,)h(whereas)h FF(gl)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))41 b FH(is)147 4930 y(the)31 b(space)g(of)f(all)f FD(n)17 b FA(\002)i FD(n)30 b FH(real)g(matrices.)42 b(Clearly)-8 b(,)29 b(then,)j(ev)m(ery)g FD(X)k FA(2)28 b FF(gl)16 b FE(\()p FD(n)p FE(;)h Fu(C)j FE(\))36 b FH(can)31 b(b)s(e)f(written) 147 5051 y(uniquely)43 b(in)e(the)i(form)e FD(X)1175 5066 y Fz(1)1244 5051 y FE(+)29 b FD(iX)1463 5066 y Fz(2)1502 5051 y FH(,)45 b(with)d FD(X)1887 5066 y Fz(1)1927 5051 y FD(;)17 b(X)2052 5066 y Fz(2)2135 5051 y FA(2)45 b FF(gl)17 b FE(\()p FD(n)p FE(;)g Fu(R)5 b FE(\))h FH(.)73 b(This)42 b(giv)m(es)h(us)g(a)g(complex)147 5171 y(v)m(ector)f(space)h (isomorphism)38 b(of)j FF(gl)17 b FE(\()o FD(n)p FE(;)g Fu(R)5 b FE(\))1754 5200 y Fi(C)1848 5171 y FH(with)40 b FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(,)49 b(and)42 b(it)e(is)h(a)g(trivialit)m(y)d(to)j(c)m(hec)m(k)147 5292 y(that)33 b(this)f(is)g(a)g(Lie)g(algebra)f(isomorphism.)p eop %%Page: 67 67 67 66 bop 147 48 a Fx(Exercises)3179 b FG(67)446 298 y FH(On)40 b(the)f(other)h(hand,)i FF(u)p FE(\()p FD(n)p FE(\))d 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FD(X)47 b FH(in)39 b FF(gl)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))45 b FH(can)40 b(b)s(e)f(written)147 1447 y(uniquely)33 b(as)f FD(X)743 1462 y Fz(1)805 1447 y FE(+)22 b FD(iX)1017 1462 y Fz(2)1057 1447 y FH(,)32 b(with)g FD(X)1419 1462 y Fz(1)1491 1447 y FH(and)h FD(X)1762 1462 y Fz(2)1834 1447 y FH(in)f FF(u)p FE(\()p FD(n)p FE(\))p FH(.)43 b(It)33 b(follo)m(ws)e(that)h FF(u)p FE(\()p FD(n)p FE(\))3023 1462 y Fi(C)3103 1419 y FA(\030)3104 1451 y FE(=)3208 1447 y FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)446 1574 y(The)44 b(v)m(eri\034cation)d(of)i(the) f(remaining)f(isomorphisms)f(is)i(similar,)g(and)h(is)f(left)f(as)i(an) 147 1694 y(exercise)34 b(to)e(the)h(reader.)44 b Fy(\003)147 1992 y FI(Remarks)147 2189 y FH(Note)29 b(that)g FF(u)p FE(\()p FD(n)p FE(\))771 2204 y Fi(C)851 2161 y FA(\030)852 2193 y FE(=)956 2189 y FF(gl)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))1294 2218 y Fi(C)1375 2161 y FA(\030)1376 2193 y FE(=)1480 2189 y FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)48 b(Ho)m(w)m(ev)m(er,)32 b FF(u)p FE(\()p FD(n)p FE(\))d FH(is)g FC(not)38 b FH(isomorphic)27 b(to)i FF(gl)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))h FH(,)147 2309 y(except)39 b(when)f FD(n)d FE(=)g(1)p FH(.)57 b(The)38 b(real)e(Lie)g(algebras)h FF(u)p FE(\()p FD(n)p FE(\))g FH(and)g FF(gl)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))43 b FH(are)37 b(called)f FI(real)41 b(forms)147 2430 y FH(of)c(the)g(complex)f(Lie)h(algebra)e FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)62 b(A)37 b(giv)m(en)g(complex)g(Lie)f(algebra)f(ma)m(y)i(ha)m(v)m(e)h(sev)m (eral)147 2550 y(non-isomorphic)30 b(real)i(forms.)43 b(See)33 b(Exercise)h(11.)446 2677 y(Ph)m(ysicists)k(do)e(not)h(alw)m (a)m(ys)g(clearly)e(distinguish)h(b)s(et)m(w)m(een)i(a)f(matrix)e(Lie)h (group)g(and)147 2797 y(its)g(\(real\))g(Lie)f(algebra,)i(or)f(b)s(et)m (w)m(een)j(a)d(real)f(Lie)h(algebra)g(and)g(its)g(complexi\034cation.) 53 b(Th)m(us,)147 2917 y(for)36 b(example,)h(some)f(references)i(in)e (the)h(ph)m(ysics)h(literature)c(to)i Fp(SU)p FE(\(2\))h FH(actually)e(refer)i(to)f(the)147 3038 y(complexi\034ed)c(Lie)g (algebra,)f FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(.)147 3322 y FI(3.10)112 b(Exercises)266 3496 y FH(1.)49 b FC(The)37 b(pr)-5 b(o)g(duct)37 b(rule)7 b(.)51 b FH(Recall)34 b(that)h(a)g(matrix-v)-5 b(alued)32 b(function)j FD(A)p FE(\()p FD(t)p FE(\))g FH(is)g(smo)s(oth)f(if)g(eac)m(h)391 3616 y FD(A)464 3631 y Ft(ij)525 3616 y FE(\()p FD(t)p FE(\))e FH(is)g(smo)s(oth.)43 b(The)33 b(deriv)-5 b(ativ)m(e)32 b(of)g(suc)m(h)i(a)f(function)f(is)g(de\034ned)i(as)1723 3774 y Fv(\022)1807 3847 y FD(dA)p 1807 3892 124 4 v 1826 3983 a(dt)1940 3774 y Fv(\023)2014 4013 y Ft(ij)2102 3915 y FE(=)2215 3847 y FD(dA)2339 3862 y Ft(ij)p 2215 3892 185 4 v 2265 3983 a FD(dt)391 4221 y FH(or)e(equiv)-5 b(alen)m(tly)d(,)1445 4442 y FD(d)p 1428 4486 86 4 v 1428 4578 a(dt)1523 4509 y(A)p FE(\()p FD(t)p FE(\))28 b(=)33 b(lim)1839 4572 y Ft(h)p Fw(!)p Fz(0)2012 4442 y FD(A)p FE(\()p FD(t)22 b FE(+)g FD(h)p FE(\))h FA(\000)f FD(A)p FE(\()p FD(t)p FE(\))p 2012 4486 667 4 v 2317 4578 a FD(h)2688 4509 y FH(.)391 4840 y(Let)30 b FD(A)p FE(\()p FD(t)p FE(\))f FH(and)h FD(B)5 b FE(\()p FD(t)p FE(\))29 b FH(b)s(e)h(t)m(w)m(o)g(suc)m(h)h(functions.)42 b(Pro)m(v)m(e)31 b(that)e FD(A)p FE(\()p FD(t)p FE(\))p FD(B)5 b FE(\()p FD(t)p FE(\))30 b FH(is)f(again)f(smo)s(oth,)391 4960 y(and)33 b(that)1345 5156 y FD(d)p 1328 5200 86 4 v 1328 5292 a(dt)1440 5223 y FE([)p FD(A)p FE(\()p FD(t)p FE(\))p FD(B)5 b FE(\()p FD(t)p FE(\)])29 b(=)2010 5156 y FD(dA)p 2010 5200 124 4 v 2029 5292 a(dt)2144 5223 y(B)5 b FE(\()p FD(t)p FE(\))22 b(+)g FD(A)p FE(\()p FD(t)p FE(\))2648 5156 y FD(dB)p 2648 5200 130 4 v 2670 5292 a(dt)2788 5223 y FH(.)p eop %%Page: 68 68 68 67 bop 147 48 a FK(68)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)266 298 y FH(2.)49 b(Using)25 b(the)g(Jordan)g(canonical)f(form,)h(sho)m(w)h(that)f(ev)m(ery)i FD(n)7 b FA(\002)g FD(n)25 b FH(matrix)f FD(A)h FH(can)g(b)s(e)g (written)391 418 y(as)38 b FD(A)f FE(=)g FD(S)31 b FE(+)26 b FD(N)10 b FH(,)40 b(with)d FD(S)44 b FH(diagonalizable)34 b(\(o)m(v)m(er)39 b Fu(C)20 b FH(\),)45 b FD(N)j FH(nilp)s(oten)m(t,)38 b(and)g FD(S)6 b(N)47 b FE(=)37 b FD(N)10 b(S)c FH(.)391 539 y(Recall)29 b(that)h(the)h(Jordan)g(canonical)e(form)g(is)h(blo)s (c)m(k)h(diagonal,)d(with)j(eac)m(h)g(blo)s(c)m(k)f(of)h(the)391 659 y(form)1719 752 y Fv(0)1719 928 y(B)1719 992 y(@)1848 841 y FD(\025)284 b FA(\003)1993 944 y FH(.)2031 969 y(.)2069 995 y(.)1852 1123 y FE(0)g FD(\025)2283 752 y Fv(1)2283 928 y(C)2283 992 y(A)2387 983 y FH(.)266 1364 y(3.)49 b(Let)30 b FD(X)38 b FH(and)30 b FD(Y)51 b FH(b)s(e)30 b FD(n)17 b FA(\002)h FD(n)30 b FH(matrices.)42 b(Sho)m(w)30 b(that)g(there)h(exists)g(a)e(constan)m(t)i FD(C)37 b FH(suc)m(h)31 b(that)1423 1513 y Fv(\015)1423 1572 y(\015)1478 1597 y FD(e)1523 1556 y Fz(\()p Ft(X)5 b Fz(+)p Ft(Y)17 b Fz(\))p Ft(=m)1877 1597 y FA(\000)23 b FD(e)2022 1556 y Ft(X)o(=m)2181 1597 y FD(e)2226 1556 y Ft(Y)8 b(=m)2377 1513 y Fv(\015)2377 1572 y(\015)2460 1597 y FA(\024)2599 1530 y FD(C)p 2575 1574 125 4 v 2575 1666 a(m)2660 1637 y Fz(2)391 1824 y FH(for)32 b(all)f(in)m(tegers)h FD(m)c FA(\025)h FE(1)p FH(.)266 2022 y(4.)49 b(Using)36 b(the)g(Jordan)g(canonical)e(form,)i(sho)m(w)h(that)e(ev)m(ery)j FD(n)25 b FA(\002)g FD(n)36 b FH(complex)f(matrix)f FD(A)i FH(is)391 2142 y(the)d(limit)c(of)j(a)g(sequence)k(of)c(diagonalizable) d(matrices.)391 2302 y FC(Hint)9 b FH(:)74 b(If)47 b(the)h(c)m (haracteristic)f(p)s(olynomial)d(of)j FD(A)g FH(has)h FD(n)g FH(distinct)e(ro)s(ots,)51 b(then)d FD(A)f FH(is)391 2422 y(diagonalizable.)266 2620 y(5.)i(Giv)m(e)36 b(an)g(example)g(of)f (a)h(matrix)f(Lie)h(group)g FD(G)g FH(and)g(a)g(matrix)f FD(X)44 b FH(suc)m(h)37 b(that)f FD(e)3436 2584 y Ft(X)3538 2620 y FA(2)e FD(G)p FH(,)391 2740 y(but)f FD(X)47 b(=)-61 b FA(2)28 b Fk(g)p FH(.)266 2939 y(6.)49 b(Sho)m(w)33 b(that)g(t)m(w)m(o)g(isomorphic)e(matrix)g(Lie)g(groups)i(ha)m(v)m(e)h (isomorphic)d(Lie)h(algebras.)266 3137 y(7.)49 b FC(The)38 b(Lie)h(algebr)-5 b(a)45 b Fp(so)p FE(\(3;)17 b(1\))p FH(.)55 b(W)-8 b(rite)36 b(out)h(explicitly)e(the)i(general)f(form)f (of)i(a)f FE(4)25 b FA(\002)g FE(4)37 b FH(real)391 3257 y(matrix)31 b(in)h Fp(so)p FE(\(3;)17 b(1\))p FH(.)266 3455 y(8.)49 b(V)-8 b(erify)32 b(directly)f(that)h(Prop)s(osition)e(43) i(and)g(Theorem)g(44)g(hold)f(for)h(the)g(Lie)g(algebra)e(of)391 3576 y FF(SU)q FE(\()p FD(n)p FE(\))p FH(.)266 3774 y(9.)49 b FC(The)40 b(Lie)g(algebr)-5 b(a)47 b FF(su)p FE(\(2\))p FH(.)61 b(Sho)m(w)39 b(that)f(the)h(follo)m(wing)d(matrices)h(form)g(a) h(basis)h(for)f(the)391 3894 y(real)32 b(Lie)g(algebra)f FF(su)p FE(\(2\))p FH(:)813 4154 y FD(E)885 4169 y Fz(1)952 4154 y FE(=)1066 4115 y Fz(1)p 1066 4131 36 4 v 1066 4188 a(2)1128 4013 y Fv(\022)1250 4093 y FD(i)122 b FE(0)1242 4213 y(0)83 b FA(\000)p FD(i)1526 4013 y Fv(\023)1683 4154 y FD(E)1755 4169 y Fz(2)1822 4154 y FE(=)1936 4115 y Fz(1)p 1936 4131 V 1936 4188 a(2)1997 4013 y Fv(\022)2151 4093 y FE(0)122 b(1)2112 4213 y FA(\000)p FE(1)84 b(0)2412 4013 y Fv(\023)2568 4154 y FD(E)2640 4169 y Fz(3)2708 4154 y FE(=)2821 4115 y Fz(1)p 2821 4131 V 2821 4188 a(2)2883 4013 y Fv(\022)2998 4093 y FE(0)90 b FD(i)3006 4213 y(i)h FE(0)3220 4013 y Fv(\023)3335 4154 y FH(.)391 4453 y(Compute)29 b FE([)p FD(E)909 4468 y Fz(1)949 4453 y FD(;)17 b(E)1065 4468 y Fz(2)1104 4453 y FE(])p FH(,)30 b FE([)p FD(E)1287 4468 y Fz(2)1327 4453 y FD(;)17 b(E)1443 4468 y Fz(3)1482 4453 y FE(])p FH(,)30 b(and)f FE([)p FD(E)1851 4468 y Fz(3)1891 4453 y FD(;)17 b(E)2007 4468 y Fz(1)2046 4453 y FE(])p FH(.)43 b(Sho)m(w)29 b(that)g(there)h(is)f (an)f(in)m(v)m(ertible)h(linear)391 4573 y(map)36 b FD(\036)e FE(:)g FF(su)q FE(\(2\))g FA(!)g Fu(R)1212 4537 y Fz(3)1293 4573 y FH(suc)m(h)k(that)f FD(\036)p FE(\([)o FD(X)r(;)17 b(Y)22 b FE(]\))34 b(=)g FD(\036)p FE(\()p FD(X)8 b FE(\))24 b FA(\002)i FD(\036)p FE(\()p FD(Y)21 b FE(\))36 b FH(for)g(all)e FD(X)r(;)17 b(Y)56 b FA(2)37 b FF(su)p FE(\(2\))p FD(;)391 4693 y FH(where)d FA(\002)f FH(denotes)h(the)f(cross-pro)s(duct)g(on)f Fu(R)2110 4657 y Fz(3)2156 4693 y FH(.)218 4892 y(10.)48 b FC(The)41 b(Lie)g(algebr)-5 b(as)49 b FF(su)p FE(\(2\))40 b FC(and)51 b FF(so)p FE(\(3\))p FH(.)65 b(Sho)m(w)40 b(that)g(the)g(real)f(Lie)g(algebras)g FF(su)q FE(\(2\))g FH(and)391 5012 y FF(so)p FE(\(3\))32 b FH(are)h(isomorphic.)391 5171 y FC(Note)7 b FH(:)43 b(Nev)m(ertheless,)33 b(the)d(corresp)s (onding)g FC(gr)-5 b(oups)38 b FF(SU)p FE(\(2\))30 b FH(and)g FF(SO)p FE(\(3\))g FH(are)f(not)h(isomor-)391 5292 y(phic.)43 b(\(Although)32 b FF(SO)p FE(\(3\))g FH(is)g(isomorphic)f(to)h FF(SU)q FE(\(2\))p FD(=)17 b FA(f)o FD(I)8 b(;)17 b FA(\000)p FD(I)8 b FA(g)o FH(.\))p eop %%Page: 69 69 69 68 bop 147 48 a Fx(Exercises)3179 b FG(69)218 298 y FH(11.)48 b FC(The)38 b(Lie)h(algebr)-5 b(as)46 b FF(su)q FE(\(2\))38 b FC(and)49 b FF(sl)p FE(\(2;)17 b Fu(R)5 b FE(\))p FH(.)62 b(Sho)m(w)38 b(that)f FF(su)p FE(\(2\))i FH(and)e FF(sl)p FE(\(2;)17 b Fu(R)5 b FE(\))42 b FH(are)37 b(not)g(iso-)391 418 y(morphic)31 b(Lie)h(algebras,)g(ev)m(en)i(though) f FF(su)p FE(\(2\))2106 433 y Fi(C)2186 390 y FA(\030)2186 422 y FE(=)2291 418 y FF(sl)p FE(\(2;)17 b Fu(R)5 b FE(\))2586 433 y Fi(C)2644 418 y FH(.)391 590 y FC(Hint)k FH(:)44 b(Using)33 b(Exercise)g(9,)g(sho)m(w)g(that)g FF(su)p FE(\(2\))f FH(has)h(no)g(t)m(w)m(o-dimensional)d(subalgebras.)218 812 y(12.)48 b(Let)31 b FD(G)f FH(b)s(e)h(a)f(matrix)g(Lie)f(group,)i (and)g Fk(g)g FH(its)f(Lie)g(algebra.)41 b(F)-8 b(or)30 b(eac)m(h)i FD(A)27 b FA(2)i FD(G)p FH(,)i(sho)m(w)g(that)391 932 y FE(Ad)p FD(A)i FH(is)f(a)g(Lie)g(algebra)g(automorphism)e(of)i Fk(g)p FH(.)218 1155 y(13.)48 b(Ad)33 b FC(and)42 b FH(ad.)h(Let)33 b FD(X)40 b FH(and)33 b FD(Y)54 b FH(b)s(e)33 b(matrices.)42 b(Sho)m(w)34 b(b)m(y)f(induction)f(that)1266 1457 y FE(\(ad)p FD(X)8 b FE(\))1534 1408 y Ft(n)1597 1457 y FE(\()p FD(Y)21 b FE(\))28 b(=)1933 1332 y Ft(n)1882 1362 y Fv(X)1890 1574 y Ft(k)r Fz(=0)2043 1316 y Fv(\022)2116 1389 y FD(n)2118 1525 y(k)2175 1316 y Fv(\023)2248 1457 y FD(X)2337 1415 y Ft(k)2379 1457 y FD(Y)22 b FE(\()p FA(\000)p FD(X)8 b FE(\))2700 1415 y Ft(n)p Fw(\000)p Ft(k)2840 1457 y FH(.)391 1772 y(No)m(w)33 b(sho)m(w)h(b)m(y)f(direct)g(computation)e (that)1375 2003 y FD(e)1420 1962 y Fz(ad)q Ft(X)1562 2003 y FE(\()p FD(Y)21 b FE(\))28 b(=)f(Ad)q(\()p FD(e)2058 1962 y Ft(X)2125 2003 y FE(\))p FD(Y)49 b FE(=)28 b FD(e)2418 1962 y Ft(X)2485 2003 y FD(Y)22 b(e)2609 1962 y Fw(\000)p Ft(X)2731 2003 y FH(.)391 2235 y(Y)-8 b(ou)36 b(ma)m(y)f(assume)i(that) e(it)g(is)g(legal)f(to)h(m)m(ultiply)f(p)s(o)m(w)m(er)i(series)h (term-b)m(y-term.)52 b(\(This)391 2355 y(result)33 b(w)m(as)g(obtained) f(indirectly)f(in)h(Equation)g(3.17.\))391 2527 y FC(Hint)9 b FH(:)54 b(Recall)36 b(that)h(P)m(ascal's)h(T)-8 b(riangle)36 b(giv)m(es)i(a)f(relationship)e(b)s(et)m(w)m(een)40 b(things)d(of)g (the)391 2647 y(form)621 2566 y Fv(\000)667 2603 y Ft(n)p Fz(+1)714 2681 y Ft(k)800 2566 y Fv(\001)878 2647 y FH(and)c(things)f (of)g(the)h(form)1870 2566 y Fv(\000)1916 2603 y Ft(n)1918 2681 y(k)1959 2566 y Fv(\001)2004 2647 y FH(.)218 2870 y(14.)48 b FC(The)39 b(c)-5 b(omplexi\034c)g(ation)38 b(of)h(a)h(r)-5 b(e)g(al)39 b(Lie)h(algebr)-5 b(a)p FH(.)58 b(Let)38 b Fk(g)g FH(b)s(e)g(a)g(real)f(Lie)g(algebra,)h Fk(g)3548 2885 y Fi(C)3639 2870 y FH(its)391 2990 y (complexi\034cation,)47 b(and)e Fk(h)g FH(an)h(arbitrary)e(complex)h (Lie)g(algebra.)81 b(Sho)m(w)46 b(that)f(ev)m(ery)391 3110 y(real)j(Lie)f(algebra)h(homomorphism)d(of)j Fk(g)h FH(in)m(to)e Fk(h)i FH(extends)h(uniquely)e(to)g(a)h(complex)391 3231 y(Lie)36 b(algebra)e(homomorphism)g(of)h Fk(g)1760 3246 y Fi(C)1849 3231 y FH(in)m(to)g Fk(h)p FH(.)54 b(\(This)36 b(is)g(the)g FI(univ)m(ersal)41 b(prop)s(ert)m(y)c FH(of)391 3351 y(the)43 b(complexi\034cation)e(of)i(a)f(real)h(Lie)f(algebra.)74 b(This)43 b(prop)s(ert)m(y)g(can)h(b)s(e)f(used)h(as)f(an)391 3471 y(alternativ)m(e)32 b(de\034nition)f(of)h(the)h (complexi\034cation.\))218 3694 y(15.)48 b FC(The)31 b(exp)-5 b(onential)29 b(mapping)h(for)42 b FF(SL)16 b FE(\(2;)h Fu(R)5 b FE(\))g FH(.)43 b(Sho)m(w)29 b(that)f(the)h(image) e(of)h(the)h(exp)s(onen)m(tial)391 3814 y(mapping)40 b(for)h FF(SL)16 b FE(\(2;)h Fu(R)5 b FE(\))47 b FH(consists)42 b(of)f(precisely)h(those)g(matrices)f FD(A)i FA(2)g FF(SL)17 b FE(\(2;)g Fu(R)t FE(\))48 b FH(suc)m(h)391 3935 y(that)36 b FE(trace)17 b(\()p FD(A)p FE(\))33 b FD(>)h FA(\000)p FE(2)p FD(;)i FH(together)g(with)f(the)i(matrix)d FA(\000)p FD(I)44 b FH(\(whic)m(h)37 b(has)f(trace)g FA(\000)p FE(2)p FH(\).)54 b(Y)-8 b(ou)391 4055 y(will)31 b(need)i(to)g(consider) g(the)g(p)s(ossibilities)d(for)i(the)i(eigen)m(v)-5 b(alues)32 b(of)h(a)f(matrix)f(in)h(the)i(Lie)391 4175 y(algebra)i FF(sl)17 b FE(\(2;)g Fu(R)t FE(\))43 b FH(and)37 b(in)f(the)i(group)e FF(SL)17 b FE(\(2;)g Fu(R)t FE(\))6 b FH(.)57 b(In)37 b(the)h(Lie)e(algebra,)h(sho)m(w)h(that)f(the)391 4296 y(eigen)m(v)-5 b(alues)30 b(are)f(of)g(the)h(form)e FE(\()p FD(\025;)17 b FA(\000)p FD(\025)p FE(\))30 b FH(or)f FE(\()p FD(i\025;)17 b FA(\000)p FD(i\025)p FE(\))30 b FH(with)f FD(\025)g FH(real.)42 b(In)30 b(the)g(group,)g(sho)m(w)391 4416 y(that)44 b(the)h(eigen)m(v)-5 b(alues)44 b(are)g(of)g(the)h(form) e FE(\()p FD(\013)q(;)17 b FE(1)p FD(=a)p FE(\))43 b FH(or)h FE(\()p FA(\000)p FD(a;)17 b FA(\000)p FE(1)p FD(=a)p FE(\))45 b FH(with)f FD(a)h FH(real)e(and)391 4537 y(p)s(ositiv)m(e,)i(or)e(else)g(of)g(the)g(form)1658 4456 y Fv(\000)1704 4537 y FD(e)1749 4500 y Ft(i\022)1812 4537 y FD(;)17 b(e)1901 4500 y Fw(\000)p Ft(i\022)2019 4456 y Fv(\001)2081 4537 y FD(;)44 b FH(with)e FD(\022)47 b FH(real.)74 b(The)44 b(case)g(of)e(a)h(rep)s(eated)391 4657 y(eigen)m(v)-5 b(alue)40 b(\()p FE(\(0)p FD(;)17 b FE(0\))39 b FH(in)g(the)i(Lie)f(algebra)f(and)h FE(\(1)p FD(;)17 b FE(1\))39 b FH(or)h FE(\()p FA(\000)p FE(1)p FD(;)17 b FA(\000)p FE(1\))40 b FH(in)g(the)g(group\))g(will)391 4777 y(ha)m(v)m(e)34 b(to)e(b)s(e)h(treated)g(separately)-8 b(.)391 4949 y(Sho)m(w)33 b(that)g(the)g(image)d(of)j(the)f(exp)s(onen) m(tial)g(mapping)f(is)h(not)h(dense)h(in)e FF(SL)16 b FE(\(2;)h Fu(R)t FE(\))6 b FH(.)218 5171 y(16.)48 b(Using)29 b(Exercise)j(4,)e(sho)m(w)h(that)e(the)h(exp)s(onen)m(tial)f(mapping)g (for)g FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)i FE(\))36 b FH(maps)30 b(on)m(to)f(a)391 5292 y(dense)34 b(subset)g(of)e FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)p eop %%Page: 70 70 70 69 bop 147 48 a FK(70)1932 b FG(Lie)27 b(Algebras)g(and)g(the)h(Exp) r(onen)n(tial)g(Mapping)218 298 y FH(17.)48 b FC(The)35 b(exp)-5 b(onential)35 b(mapping)f(for)i(the)g(Heisenb)-5 b(er)g(g)35 b(gr)-5 b(oup)p FH(.)47 b(Sho)m(w)34 b(that)g(the)g(exp)s (onen)m(tial)391 418 y(mapping)28 b(from)h(the)h(Lie)f(algebra)g(of)g (the)i(Heisen)m(b)s(erg)f(group)g(to)f(the)i(Heisen)m(b)s(erg)f(group) 391 539 y(is)i(one-to-one)g(and)h(on)m(to.)218 742 y(18.)48 b FC(The)38 b(exp)-5 b(onential)38 b(mapping)g(for)49 b FF(U)p FE(\()p FD(n)p FE(\))p FH(.)57 b(Sho)m(w)38 b(that)f(the)h(exp)s(onen)m(tial)e(mapping)f(from)391 862 y FF(u)p FE(\()p FD(n)p FE(\))k FH(to)g FF(U)p FE(\()p FD(n)p FE(\))g FH(is)g(on)m(to,)i(but)e(not)g(one-to-one.)62 b(\(Note)39 b(that)g(this)g(sho)m(ws)h(that)f FF(U)p FE(\()p FD(n)p FE(\))g FH(is)391 983 y(connected.\))391 1145 y FC(Hint)9 b FH(:)44 b(Ev)m(ery)35 b(unitary)d(matrix)f(has)i(an) f(orthonormal)e(basis)j(of)f(eigen)m(v)m(ectors.)218 1348 y(19.)48 b(Let)29 b FD(G)g FH(b)s(e)g(a)f(matrix)f(Lie)i(group,)g (and)g Fk(g)g FH(its)f(Lie)g(algebra.)41 b(Let)29 b FD(A)p FE(\()p FD(t)p FE(\))g FH(b)s(e)g(a)g(smo)s(oth)e(curv)m(e)391 1468 y(lying)k(in)h FD(G)p FH(,)h(with)f FD(A)p FE(\(0\))27 b(=)h FD(I)8 b FH(.)43 b(Let)33 b FD(X)i FE(=)1982 1429 y Ft(d)p 1970 1445 62 4 v 1970 1503 a(dt)2041 1384 y Fv(\014)2041 1443 y(\014)2075 1507 y Ft(t)p Fz(=0)2211 1468 y FD(A)p FE(\()p FD(t)p FE(\))p FH(.)44 b(Sho)m(w)33 b(that)f FD(X)k FA(2)28 b Fk(g)p FH(.)391 1630 y FC(Hint)9 b FH(:)44 b(Use)34 b(Prop)s(osition)d(36.)391 1792 y FC(Note)7 b FH(:)54 b(This)37 b(sho)m(ws)i(that)d(the)i(Lie)f(algebra)e Fk(g)j FH(coincides)f(with)f(what)i(w)m(ould)f(b)s(e)g(called)391 1913 y(the)26 b FI(tangen)m(t)j(space)g(at)g(the)g(iden)m(tit)m(y)24 b FH(in)h(the)g(language)f(of)h(di\033eren)m(tiable)e(manifolds.)218 2116 y(20.)48 b(Consider)37 b(the)h(space)g FF(gl)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))43 b FH(of)36 b(all)f FD(n)26 b FA(\002)f FD(n)37 b FH(complex)g(matrices.)55 b(As)38 b(usual,)f(for)g FD(X)43 b FA(2)391 2236 y FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(,)44 b(de\034ne)38 b FE(ad)p FD(X)k FE(:)35 b FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))41 b FA(!)35 b FF(gl)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))43 b FH(b)m(y)38 b FE(ad)p FD(X)8 b FE(\()p FD(Y)21 b FE(\))35 b(=)f([)p FD(X)r(;)17 b(Y)22 b FE(])p FH(.)56 b(Supp)s(ose)38 b(that)391 2357 y FD(X)50 b FH(is)41 b(a)h(diagonalizable)d(matrix.)70 b(Sho)m(w,)45 b(then,)g(that)d FE(ad)p FD(X)50 b FH(is)41 b(diagonalizable)e(as)j(an) 391 2477 y(op)s(erator)32 b(on)g FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)391 2639 y FC(Hint)9 b FH(:)44 b(Consider)33 b(\034rst)g(the)g(case)h(where)f FD(X)41 b FH(is)32 b(actually)f(diagonal.)391 2801 y FC(Note)7 b FH(:)73 b(The)48 b(problem)e(of)h(diagonalizing)c FE(ad)p FD(X)55 b FH(is)46 b(an)h(imp)s(ortan)m(t)e(one)i(that)g(w)m(e) h(will)391 2921 y(encoun)m(ter)34 b(again)d(in)h(Chapter)h(6,)g(when)g (w)m(e)h(consider)f(semisimple)d(Lie)i(algebras.)p eop %%Page: 71 71 71 70 bop 1688 1047 a FL(Chapter)37 b(4)518 1222 y(THE)h (BAKER-CAMPBELL-HA)m(USDORFF)f(F)m(ORMULA)147 1491 y FI(4.1)112 b(The)38 b(Bak)m(er-Campb)s(ell-Hausdor\033)e(F)-9 b(orm)m(ula)34 b(for)k(the)f(Heisen)m(b)s(erg)g(Group)190 1659 y FH(A)43 b(crucial)f(result)h(of)g(Chapter)g(5)g(will)e(b)s(e)i (the)g(follo)m(wing:)62 b(Let)43 b FD(G)g FH(and)g FD(H)51 b FH(b)s(e)43 b(matrix)f(Lie)147 1779 y(groups,)51 b(with)46 b(Lie)g(algebras)g Fk(g)g FH(and)h Fk(h)p FH(,)j(and)d(supp)s(ose)h (that)e FD(G)h FH(is)f(connected)i(and)e(simply)147 1900 y(connected.)51 b(Then)36 b(if)1004 1873 y Fv(e)995 1900 y FD(\036)31 b FE(:)g Fk(g)g FA(!)g Fk(h)j FH(is)h(a)f(Lie)g(algebra)f (homomorphism,)g(there)i(exists)g(a)g(unique)147 2020 y(Lie)h(group)g(homomorphism)d FD(\036)g FE(:)h FD(G)g FA(!)f FD(H)44 b FH(suc)m(h)38 b(that)d FD(\036)h FH(and)2543 1994 y Fv(e)2533 2020 y FD(\036)g FH(are)g(related)g(as)g(in)g(Theorem) 147 2141 y(46.)84 b(This)46 b(result)g(is)g(extremely)g(imp)s(ortan)m (t)f(b)s(ecause)i(it)e(implies)f(that)i(if)f FD(G)h FH(is)f(connected) 147 2261 y(and)f(simply)f(connected,)48 b(then)d(there)g(is)e(a)h (natural)f(one-to-one)g(corresp)s(ondence)j(b)s(et)m(w)m(een)147 2381 y(the)32 b(represen)m(tations)h(of)e FD(G)h FH(and)f(the)i (represen)m(tations)f(of)f(its)h(Lie)f(algebra)f Fk(g)i FH(\(as)g(explained)f(in)147 2502 y(Chapter)h(5\).)43 b(In)32 b(practice,)g(it)e(is)h(m)m(uc)m(h)h(easier)f(to)g(determine)g (the)h(represen)m(tations)h(of)e(the)h(Lie)147 2622 y(algebra)g(than)g (to)g(determine)g(directly)g(the)h(represen)m(tations)h(of)e(the)h (corresp)s(onding)f(group.)446 2742 y(This)45 b(result)g(\(relating)e (Lie)h(algebra)f(homomorphisms)g(and)i(Lie)f(group)h(homomor-)147 2863 y(phisms\))i(is)g(deep.)89 b(The)48 b(\020mo)s(dern\021)e(pro)s (of)h(\(e.g.,)k(V)-8 b(aradara)5 b(jan,)50 b(Theorem)e(2.7.5\))e(mak)m (es)147 2983 y(use)38 b(of)e(the)i(F)-8 b(rob)s(enius)36 b(theorem,)i(whic)m(h)f(is)f(b)s(oth)h(hard)g(to)f(understand)i(and)f (hard)g(to)g(pro)m(v)m(e)147 3104 y(\(V)-8 b(aradara)5 b(jan,)27 b(Section)f(1.3\).)41 b(Our)26 b(pro)s(of)f(will)f(instead)i (use)h(the)g(Bak)m(er-Campb)s(ell-Hausdor\033)147 3224 y(form)m(ula,)37 b(whic)m(h)g(is)g(more)f(easily)g(stated)i(and)f(more) f(easily)h(motiv)-5 b(ated)35 b(than)i(the)h(F)-8 b(rob)s(enius)147 3344 y(theorem,)33 b(but)f(still)f(deep.)446 3465 y(The)g(idea)e(is)h (the)g(follo)m(wing.)40 b(The)30 b(desired)h(group)e(homomorphism)e FD(\036)h FE(:)g FD(G)f FA(!)h FD(H)37 b FH(m)m(ust)147 3585 y(satisfy)1622 3781 y FD(\036)1697 3701 y Fv(\000)1742 3781 y FD(e)1787 3740 y Ft(X)1854 3701 y Fv(\001)1928 3781 y FE(=)27 b FD(e)2083 3723 y Fl(e)2076 3740 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))2241 3781 y FH(.)1274 b(\(4.1\))147 3964 y(W)-8 b(e)44 b(w)m(ould)f(lik)m(e,)j(then,)g(to)e FC(de\034ne)49 b FD(\036)43 b FH(b)m(y)i(this)e(relation.)74 b(This)43 b(approac)m(h)h(has)g(t)m(w)m(o)g(serious)147 4084 y(di\036culties.)49 b(First,)35 b(a)f(giv)m(en)h(elemen)m(t)g(of)f FD(G)h FH(ma)m(y)g(not)f(b)s(e)h(expressible)h(as)f FD(e)3039 4048 y Ft(X)3107 4084 y FH(,)g(and)g(ev)m(en)h(if)e(it)147 4205 y(is,)42 b(the)e FD(X)47 b FH(ma)m(y)40 b(not)g(b)s(e)g(unique.)66 b(Second,)43 b(it)38 b(is)i(v)m(ery)h(far)e(from)g(clear)g(wh)m(y)j (the)e FD(\036)f FH(in)h(\(4.1\))147 4325 y(\(ev)m(en)34 b(to)e(the)h(exten)m(t)h(it)e(is)g(w)m(ell-de\034ned\))h(should)f(b)s (e)h(a)f(group)g(homomorphism.)446 4445 y(It)e(is)g(the)h(second)h (issue)e(whic)m(h)h(the)g(Bak)m(er-Campb)s(ell-Hausdor\033)d(form)m (ula)h(addresses.)147 4566 y(\(The)46 b(\034rst)f(issue)g(will)d(b)s(e) j(addressed)h(in)e(the)h(next)g(c)m(hapter;)52 b(it)43 b(is)h(there)h(that)g(the)g(simple)147 4686 y(connectedness)32 b(of)c FD(G)g FH(comes)h(in)m(to)f(pla)m(y)-8 b(.\))41 b(Sp)s(eci\034cally)-8 b(,)28 b(\(one)h(form)e(of)7 b(\))28 b(the)h(Bak)m(er-Campb)s(ell-)147 4806 y(Hausdor\033)k(form)m(ula)e(sa) m(ys)j(that)e(if)f FD(X)41 b FH(and)32 b FD(Y)54 b FH(are)33 b(su\036cien)m(tly)g(small,)d(then)426 4989 y FE(log\()p FD(e)635 4948 y Ft(X)702 4989 y FD(e)747 4948 y Ft(Y)808 4989 y FE(\))e(=)g FD(X)h FE(+)22 b FD(Y)44 b FE(+)1395 4950 y Fz(1)p 1395 4966 36 4 v 1395 5023 a(2)1440 4989 y FE([)p FD(X)r(;)17 b(Y)22 b FE(])g(+)1847 4950 y Fz(1)p 1830 4966 71 4 v 1830 5023 a(12)1910 4989 y FE([)p FD(X)r(;)17 b FE([)p FD(X)r(;)g(Y)22 b FE(]])g FA(\000)2500 4950 y Fz(1)p 2483 4966 V 2483 5023 a(12)2563 4989 y FE([)p FD(Y)5 b(;)17 b FE([)p FD(X)r(;)g(Y)22 b FE(]])g(+)g FA(\001)17 b(\001)g(\001)e FH(.)279 b(\(4.2\))147 5171 y(It)46 b(is)g(not)g(supp)s(osed)i(to)d(b)s(e)i(eviden)m(t)g(at)e(the)i (momen)m(t)e(what)h(\020)p FA(\001)17 b(\001)g(\001)e FH(\021)46 b(refers)h(to.)84 b(The)47 b(only)147 5292 y(imp)s(ortan)m(t)c(p)s(oin)m(t)h(is)g(that)h(all)d(of)j(the)g(terms)f (in)g(\(4.2\))h(are)f(giv)m(en)h(in)f(terms)h(of)f FD(X)53 b FH(and)45 b FD(Y)21 b FH(,)p eop %%Page: 72 72 72 71 bop 147 48 a FK(72)2042 b FG(The)28 b(Bak)n(er-Campb)r (ell-Hausdor\033)c(F)-7 b(orm)n(ula)147 298 y FH(brac)m(k)m(ets)37 b(of)e FD(X)42 b FH(and)35 b FD(Y)22 b FH(,)35 b(brac)m(k)m(ets)i(of)e (brac)m(k)m(ets)i(in)m(v)m(olving)c FD(X)43 b FH(and)35 b FD(Y)21 b FH(,)36 b(etc.)51 b(Then)36 b(b)s(ecause)3694 271 y Fv(e)3684 298 y FD(\036)147 418 y FH(is)c(a)h(Lie)e(algebra)h (homomorphism,)460 626 y Fv(e)450 652 y FD(\036)525 571 y Fv(\000)570 652 y FE(log)713 571 y Fv(\000)759 652 y FD(e)804 611 y Ft(X)871 652 y FD(e)916 611 y Ft(Y)977 571 y Fv(\001\001)1096 652 y FE(=)1209 626 y Fv(e)1200 652 y FD(\036)o FE(\()p FD(X)8 b FE(\))22 b(+)1552 626 y Fv(e)1542 652 y FD(\036)p FE(\()p FD(Y)f FE(\))h(+)1884 613 y Fz(1)p 1884 629 36 4 v 1884 687 a(2)1930 652 y FE([)1966 626 y Fv(e)1957 652 y FD(\036)o FE(\()p FD(X)8 b FE(\))p FD(;)2232 626 y Fv(e)2223 652 y FD(\036)o FE(\()p FD(Y)22 b FE(\)])1091 820 y(+)1216 781 y Fz(1)p 1199 797 71 4 v 1199 854 a(12)1279 820 y FE([)1315 793 y Fv(e)1306 820 y FD(\036)p FE(\()p FD(X)8 b FE(\))p FD(;)17 b FE([)1609 793 y Fv(e)1600 820 y FD(\036)o FE(\()p FD(X)8 b FE(\))p FD(;)1875 793 y Fv(e)1866 820 y FD(\036)o FE(\()p FD(Y)22 b FE(\)]])g FA(\000)2281 781 y Fz(1)p 2263 797 V 2263 854 a(12)2344 820 y FE([)2380 793 y Fv(e)2371 820 y FD(\036)p FE(\()p FD(Y)f FE(\))p FD(;)c FE([)2663 793 y Fv(e)2654 820 y FD(\036)o FE(\()p FD(X)8 b FE(\))p FD(;)2929 793 y Fv(e)2920 820 y FD(\036)p FE(\()p FD(Y)21 b FE(\)]])h(+)g FA(\001)17 b(\001)g(\001)1096 1004 y FE(=)28 b(log)1342 893 y Fv(\020)1402 1004 y FD(e)1454 945 y Fl(e)1447 963 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))1611 1004 y FD(e)1663 945 y Fl(e)1656 963 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))1814 893 y Fv(\021)3542 1004 y FH(\(4.3\))446 1270 y(The)34 b(relation)c(\(4.3\))i(is)g(extremely)h(signi\034can)m(t.)43 b(F)-8 b(or)31 b(of)h(course)1584 1515 y FD(e)1629 1474 y Ft(X)1696 1515 y FD(e)1741 1474 y Ft(Y)1830 1515 y FE(=)27 b FD(e)1978 1474 y Fz(log)r(\()p Ft(e)2130 1451 y Fn(X)2188 1474 y Ft(e)2221 1451 y Fn(Y)2274 1474 y Fz(\))147 1749 y FH(and)33 b(so)g(b)m(y)g(\(4.1\),)1439 2003 y FD(\036)1514 1922 y Fv(\000)1559 2003 y FD(e)1604 1962 y Ft(X)1672 2003 y FD(e)1717 1962 y Ft(Y)1777 1922 y Fv(\001)1851 2003 y FE(=)27 b FD(e)2006 1944 y Fl(e)1999 1962 y Ft(\036)q Fz(\(log)q(\()p Ft(e)2220 1938 y Fn(X)2278 1962 y Ft(e)2311 1938 y Fn(Y)2364 1962 y Fz(\)\))2423 2003 y FH(.)147 2237 y(Th)m(us)34 b(\(4.3\))e(tells)g(us)h(that)808 2514 y FD(\036)883 2433 y Fv(\000)928 2514 y FD(e)973 2473 y Ft(X)1040 2514 y FD(e)1085 2473 y Ft(Y)1146 2433 y Fv(\001)1219 2514 y FE(=)28 b FD(e)1368 2460 y Fz(log)1459 2387 y Fl(\020)1501 2460 y Ft(e)1541 2423 y Fh(e)1534 2436 y Fn(\036)1571 2437 y Fr(\()p Fn(X)t Fr(\))1677 2460 y Ft(e)1717 2423 y Fh(e)1710 2436 y Fn(\036)1747 2437 y Fr(\()p Fn(Y)13 b Fr(\))1848 2387 y Fl(\021)1922 2514 y FE(=)27 b FD(e)2077 2455 y Fl(e)2070 2473 y Ft(\036)q Fz(\()p Ft(X)5 b Fz(\))2235 2514 y FD(e)2287 2455 y Fl(e)2280 2473 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))2465 2514 y FE(=)28 b FD(\036)p FE(\()p FD(e)2710 2473 y Ft(X)2777 2514 y FE(\))p FD(\036)p FE(\()p FD(e)2956 2473 y Ft(Y)3017 2514 y FE(\))p FH(.)147 2748 y(Th)m(us,)j(the)e(Bak)m(er-Campb)s (ell-Hausdor\033)d(form)m(ula)g(sho)m(ws)k(that)e(on)g(elemen)m(ts)h (of)f(the)g(form)f FD(e)3647 2712 y Ft(X)3715 2748 y FH(,)147 2868 y(with)32 b FD(X)41 b FH(small,)30 b FD(\036)i FH(is)g(a)h(group)f(homomorphism.)40 b(\(See)34 b(Corollary)c(68)j(b)s (elo)m(w.\))446 2993 y(The)h(Bak)m(er-Campb)s(ell-Hausdor\033)d(form)m (ula)f(sho)m(ws)35 b(that)d(all)f(the)i(information)d(ab)s(out)147 3114 y(the)35 b(group)g(pro)s(duct,)g(at)f(least)h(near)g(the)g(iden)m (tit)m(y)-8 b(,)35 b(is)f(\020enco)s(ded\021)h(in)f(the)h(Lie)f (algebra.)48 b(Th)m(us)147 3234 y(if)253 3208 y Fv(e)244 3234 y FD(\036)40 b FH(is)f(a)h(Lie)f(algebra)g(homomorphism)e(\(whic)m (h)k(b)m(y)g(de\034nition)e(preserv)m(es)j(the)f(Lie)e(algebra)147 3355 y(structure\),)45 b(and)d(if)e(w)m(e)i(de\034ne)h FD(\036)e FH(near)h(the)g(iden)m(tit)m(y)f(b)m(y)h(\(4.1\),)h(then)f(w) m(e)h(can)f(exp)s(ect)g FD(\036)f FH(to)147 3475 y(preserv)m(e)35 b(the)e(group)f(structure,)i(i.e.,)e(to)h(b)s(e)f(a)h(group)f (homomorphism.)446 3600 y(In)c(this)g(section)g(w)m(e)h(will)d(lo)s(ok) h(at)g(ho)m(w)i(all)d(of)h(this)h(w)m(orks)h(out)f(in)f(the)i(v)m(ery)g (sp)s(ecial)e(case)147 3720 y(of)32 b(the)h(Heisen)m(b)s(erg)h(group.) 43 b(In)33 b(the)g(next)g(section)g(w)m(e)g(will)d(consider)j(the)g (general)f(situation.)147 3986 y FI(Theorem)j(65)49 b FC(Supp)-5 b(ose)45 b FD(X)54 b FC(and)45 b FD(Y)68 b FC(ar)-5 b(e)45 b FD(n)31 b FA(\002)g FD(n)46 b FC(c)-5 b(omplex)45 b(matric)-5 b(es,)48 b(and)d(that)i FD(X)53 b FC(and)46 b FD(Y)147 4107 y FC(c)-5 b(ommute)35 b(with)f(their)h(c)-5 b(ommutator.)44 b(That)35 b(is,)f(supp)-5 b(ose)35 b(that)1344 4341 y FE([)p FD(X)r(;)17 b FE([)p FD(X)r(;)g(Y)22 b FE(]])27 b(=)h([)p FD(Y)5 b(;)17 b FE([)p FD(X)r(;)g(Y)k FE(]])28 b(=)g(0)p FC(.)147 4575 y(Then)1503 4809 y FD(e)1548 4768 y Ft(X)1615 4809 y FD(e)1660 4768 y Ft(Y)1749 4809 y FE(=)g FD(e)1898 4768 y Ft(X)5 b Fz(+)p Ft(Y)15 b Fz(+)2137 4740 y Fr(1)p 2137 4752 31 3 v 2137 4794 a(2)2178 4768 y Fz([)o Ft(X)q(;Y)h Fz(])2357 4809 y FC(.)147 5097 y FI(Remark)147 5292 y FH(This)33 b(is)f(the)h(sp)s(ecial)e(case)j(of)e (\(4.2\))g(in)g(whic)m(h)h(the)g(series)g(terminates)f(after)g(the)h FE([)p FD(X)r(;)17 b(Y)k FE(])33 b FH(term.)p eop %%Page: 73 73 73 72 bop 147 48 a Fx(The)28 b(Bak)n(er-Campb)r(ell-Hausdor\033)c(F)-7 b(orm)n(ula)27 b(for)g(the)h(Heisen)n(b)r(erg)e(Group)1097 b FG(73)147 298 y FI(Pro)s(of)147 482 y FH(Let)33 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FD(e)1305 5250 y Fw(\000)1370 5223 y Fn(t)1394 5202 y Fr(2)p 1370 5235 V 1384 5276 a(2)1438 5250 y Fz([)p Ft(X)q(;Y)15 b Fz(])1645 5292 y FE(=)27 b FD(e)1793 5250 y Ft(tX)1886 5292 y FD(e)1931 5250 y Ft(tY)2018 5292 y FD(e)2063 5250 y Fw(\000)2128 5223 y Fn(t)2152 5202 y Fr(2)p 2128 5235 V 2142 5276 a(2)2196 5250 y Fz([)p Ft(X)q(;Y)15 b Fz(])2392 5292 y FE(\()o FD(X)30 b FE(+)22 b FD(t)17 b FE([)p FD(X)r(;)g(Y)22 b FE(]\))p eop %%Page: 74 74 74 73 bop 147 48 a FK(74)2042 b FG(The)28 b(Bak)n(er-Campb)r (ell-Hausdor\033)c(F)-7 b(orm)n(ula)446 298 y FH(Making)32 b(these)i(substitutions)e(in)m(to)g(\(4.5\))g(giv)m(es)162 512 y FD(dA)p 162 556 124 4 v 181 648 a(dt)324 579 y FE(=)27 b FD(e)472 538 y Ft(tX)565 579 y FD(e)610 538 y Ft(tY)697 579 y FD(e)742 538 y Fw(\000)807 511 y Fn(t)831 490 y Fr(2)p 807 523 59 3 v 821 564 a(2)875 538 y Fz([)p Ft(X)q(;Y)15 b Fz(])1071 579 y FE(\()p FD(X)30 b FE(+)22 b FD(t)17 b FE([)p FD(X)r(;)g(Y)k FE(]\))h(+)g FD(e)1832 538 y Ft(tX)1925 579 y FD(e)1970 538 y Ft(tY)2057 579 y FD(e)2102 538 y Fw(\000)2167 511 y Fn(t)2191 490 y Fr(2)p 2167 523 V 2181 564 a(2)2235 538 y Fz([)p Ft(X)q(;Y)15 b Fz(])2414 579 y FD(Y)43 b FE(+)22 b FD(e)2657 538 y Ft(tX)2750 579 y FD(e)2795 538 y Ft(tY)2882 579 y FD(e)2927 538 y Fw(\000)2992 511 y Fn(t)3016 490 y Fr(2)p 2992 523 V 3006 564 a(2)3060 538 y Fz([)p Ft(X)q(;Y)15 b Fz(])3255 579 y FE(\()p FA(\000)p FD(t)i FE([)q FD(X)r(;)g(Y)k FE(]\))324 802 y(=)27 b FD(e)472 761 y Ft(tX)565 802 y FD(e)610 761 y Ft(tY)697 802 y FD(e)742 761 y Fw(\000)807 734 y Fn(t)831 713 y Fr(2)p 807 746 V 821 787 a(2)875 761 y Fz([)p Ft(X)q(;Y)15 b Fz(])1071 802 y FE(\()p FD(X)30 b FE(+)22 b FD(Y)f FE(\))324 948 y(=)27 b FD(A)17 b FE(\()p FD(t)p FE(\))g(\()p FD(X)30 b FE(+)22 b FD(Y)f FE(\))c FH(.)446 1184 y(Th)m(us)31 b FD(A)17 b FE(\()p FD(t)p FE(\))30 b FH(and)f FD(B)22 b FE(\()p FD(t)p FE(\))30 b FH(satisfy)f(the)h(same)g(di\033eren)m(tial)d(equation.)42 b(Moreo)m(v)m(er,)32 b FD(A)17 b FE(\(0\))27 b(=)147 1304 y FD(B)22 b 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FD(e)2097 2114 y Fl(e)2090 2131 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))147 2405 y FC(for)35 b(al)5 b(l)34 b FD(X)i FA(2)28 b Fk(h)p FC(.)147 2689 y FI(Pro)s(of)147 2882 y FH(Recall)34 b(that)i(the)g (Heisen)m(b)s(erg)h(group)f(has)g(the)g(v)m(ery)h(sp)s(ecial)e(prop)s (ert)m(y)i(that)e(its)h(exp)s(onen)m(tial)147 3003 y(mapping)i(is)g (one-to-one)h(and)g(on)m(to.)63 b(Let)39 b(\020log\021)f(denote)i(the)f (in)m(v)m(erse)i(of)d(this)h(map.)63 b(De\034ne)147 3123 y FD(\036)28 b FE(:)f FD(H)36 b FA(!)27 b FD(G)32 b FH(b)m(y)i(the)f (form)m(ula)1603 3369 y FD(\036)17 b FE(\()o FD(A)p FE(\))28 b(=)f FD(e)2009 3311 y Fl(e)2002 3328 y Ft(\036)q Fz(\(log)13 b Ft(A)p Fz(\))2259 3369 y FH(.)147 3601 y(W)-8 b(e)33 b(will)d(sho)m(w)k(that)e FD(\036)h FH(is)f(a)g(Lie)g(group)g (homomorphism.)446 3726 y(If)45 b FD(X)52 b FH(and)45 b FD(Y)65 b FH(are)45 b(in)f(the)h(Lie)f(algebra)f(of)h(the)h(Heisen)m (b)s(erg)g(group)g(\()p FE(3)30 b FA(\002)h FE(3)44 b FH(strictly)147 3846 y(upp)s(er-triangular)30 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FD(Y)21 b FE(\))1904 4904 y Fv(ii)2026 5014 y FE(=)2139 4988 y Fv(e)2130 5014 y FD(\036)16 b FE(\([)p FD(X)r(;)h FE([)p FD(X)r(;)g(Y)k FE(]])q(\))27 b(=)h(0)994 5116 y Fv(h)1050 5200 y(e)1041 5227 y FD(\036)16 b FE(\()p FD(Y)21 b FE(\))c FD(;)1330 5116 y Fv(h)1386 5200 y(e)1376 5227 y FD(\036)g FE(\()p FD(X)8 b FE(\))16 b FD(;)1685 5200 y Fv(e)1676 5227 y FD(\036)g FE(\()p FD(Y)21 b FE(\))1904 5116 y Fv(ii)2026 5227 y FE(=)2139 5200 y Fv(e)2130 5227 y FD(\036)16 b FE(\([)p FD(Y)5 b(;)17 b FE([)p FD(X)r(;)g(Y)k FE(]]\))28 b(=)f(0)p FH(.)p eop %%Page: 75 75 75 74 bop 147 48 a Fx(The)28 b(General)f(Bak)n(er-Campb)r (ell-Hausdor\033)d(F)-7 b(orm)n(ula)1738 b FG(75)446 298 y FH(W)-8 b(e)33 b(w)m(an)m(t)h(to)f(sho)m(w)h(that)f FD(\036)f FH(is)h(a)g(homomorphism,)d(i.e.,)j(that)f FD(\036)17 b FE(\()p FD(AB)5 b FE(\))28 b(=)h FD(\036)17 b FE(\()o FD(A)p FE(\))g FD(\036)g FE(\()o FD(B)5 b FE(\))p FH(.)147 418 y(W)-8 b(ell,)35 b FD(A)h FH(can)g(b)s(e)g(written)f(as)h FD(e)1339 382 y Ft(X)1442 418 y FH(for)f(a)h(unique)g FD(X)k FA(2)34 b Fk(h)h FH(and)h FD(B)41 b FH(can)36 b(b)s(e)f(written)h(as)g FD(e)3445 382 y Ft(Y)3541 418 y FH(for)f(a)147 539 y(unique)e FD(Y)49 b FA(2)28 b Fk(h)p FH(.)44 b(Th)m(us)34 b(b)m(y)f(Theorem)g(65)1100 762 y FD(\036)17 b FE(\()o FD(AB)5 b FE(\))28 b(=)f FD(\036)1608 681 y Fv(\000)1653 762 y FD(e)1698 721 y Ft(X)1766 762 y FD(e)1811 721 y Ft(Y)1872 681 y Fv(\001)1945 762 y FE(=)h FD(\036)2124 652 y Fv(\020)2182 762 y FD(e)2227 721 y Ft(X)5 b Fz(+)p Ft(Y)16 b Fz(+)2467 694 y Fr(1)p 2467 706 31 3 v 2467 747 a(2)2508 721 y Fz([)o Ft(X)q(;Y)g Fz(])2687 652 y Fv(\021)2763 762 y FH(.)147 1015 y(Using)32 b(the)h(de\034nition)f(of)g FD(\036)g FH(and)h(the)g(fact)f(that)1995 988 y Fv(e)1985 1015 y FD(\036)h FH(is)f(a)g(Lie)g(algebra)f (homomorphism:)842 1268 y FD(\036)17 b FE(\()p FD(AB)5 b FE(\))27 b(=)h(exp)1442 1128 y Fv(\022)1524 1242 y(e)1515 1268 y FD(\036)16 b FE(\()p FD(X)8 b FE(\))22 b(+)1883 1242 y Fv(e)1874 1268 y FD(\036)16 b FE(\()p FD(Y)22 b FE(\))g(+)2233 1201 y(1)p 2233 1245 49 4 v 2233 1337 a(2)2308 1158 y Fv(h)2364 1242 y(e)2355 1268 y FD(\036)16 b FE(\()p FD(X)8 b FE(\))17 b FD(;)2664 1242 y Fv(e)2655 1268 y FD(\036)f FE(\()p FD(Y)21 b FE(\))2883 1158 y Fv(i)2930 1128 y(\023)3020 1268 y FH(.)147 1521 y(Finally)-8 b(,)30 b(using)i(Theorem)h(65)f(again)f(w)m(e)j(ha)m(v)m(e)1200 1737 y FD(\036)17 b FE(\()o FD(AB)5 b FE(\))28 b(=)f FD(e)1685 1678 y Fl(e)1678 1696 y Ft(\036)p Fz(\()q Ft(X)5 b Fz(\))1843 1737 y FD(e)1895 1678 y Fl(e)1888 1696 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))2073 1737 y FE(=)28 b FD(\036)17 b FE(\()o FD(A)p FE(\))g FD(\036)g FE(\()o FD(B)5 b FE(\))17 b FH(.)446 1939 y(Th)m(us)30 b FD(\036)f FH(is)f(a)h(group)g(homomorphism.)39 b(It)29 b(is)f(easy)i(to)f(c)m (hec)m(k)i(that)e FD(\036)f FH(is)h(con)m(tin)m(uous)g(\(b)m(y)147 2070 y(c)m(hec)m(king)35 b(that)e FE(log)p FH(,)g(exp,)i(and)1349 2044 y Fv(e)1339 2070 y FD(\036)e FH(are)h(all)d(con)m(tin)m(uous\),)j (and)f(so)h FD(\036)f FH(is)g(a)g(Lie)g(group)g(homomor-)147 2201 y(phism.)43 b(Moreo)m(v)m(er,)34 b FD(\036)e FH(b)m(y)i (de\034nition)d(has)i(the)g(righ)m(t)f(relationship)e(to)2825 2175 y Fv(e)2815 2201 y FD(\036)p FH(.)44 b(F)-8 b(urthermore,)32 b(since)147 2322 y(the)41 b(exp)s(onen)m(tial)f(mapping)f(is)h (one-to-one)g(and)h(on)m(to,)h(there)f(can)g(b)s(e)g(at)f(most)g(one)h FD(\036)f FH(with)147 2451 y FD(\036)222 2370 y Fv(\000)267 2451 y FD(e)312 2415 y Ft(X)380 2370 y Fv(\001)453 2451 y FE(=)28 b FD(e)609 2397 y Fl(e)602 2415 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))766 2451 y FH(.)44 b(So)32 b(w)m(e)i(ha)m(v)m(e) f(uniqueness.)46 b Fy(\003)147 2711 y FI(4.2)112 b(The)38 b(General)f(Bak)m(er-Campb)s(ell-Hausdor\033)f(F)-9 b(orm)m(ula)147 2878 y FH(The)40 b(imp)s(ortance)d(of)h(the)h(Bak)m(er-Campb)s (ell-Hausdor\033)e(form)m(ula)g(lies)g(not)i(in)f(the)h(details)e(of) 147 2999 y(the)j(form)m(ula,)e(but)i(in)e(the)h(fact)g(that)g(there)h (is)e(one,)j(and)e(the)h(fact)e(that)h(it)f(giv)m(es)i FE(log\()p FD(e)3531 2962 y Ft(X)3598 2999 y FD(e)3643 2962 y Ft(Y)3704 2999 y FE(\))147 3119 y FH(in)c(terms)g(of)g(brac)m(k) m(ets)j(of)d FD(X)44 b FH(and)37 b FD(Y)21 b FH(,)38 b(brac)m(k)m(ets)g(of)e(brac)m(k)m(ets,)k(etc.)56 b(This)37 b(tells)e(us)i(something)147 3239 y(v)m(ery)47 b(imp)s(ortan)m(t,)f (namely)e(that)g(\(at)h(least)f(for)g(elemen)m(ts)i(of)e(the)h(form)f FD(e)3025 3203 y Ft(X)3092 3239 y FH(,)k FD(X)53 b FH(small\))43 b(the)147 3360 y(group)33 b(pro)s(duct)f(for)g(a)h(matrix)e(Lie)h (group)g FD(G)h FH(is)f FC(c)-5 b(ompletely)34 b(expr)-5 b(essible)33 b(in)i(terms)g(of)f(the)h(Lie)147 3480 y(algebr)-5 b(a)p FH(.)42 b(\(This)32 b(is)e(b)s(ecause)j FE(log)1371 3399 y Fv(\000)1416 3480 y FD(e)1461 3444 y Ft(X)1529 3480 y FD(e)1574 3444 y Ft(Y)1635 3399 y Fv(\001)1680 3480 y FH(,)f(and)f(hence)i(also)d FD(e)2436 3444 y Ft(X)2504 3480 y FD(e)2549 3444 y Ft(Y)2641 3480 y FH(itself,)g(can)i(b)s(e)f (computed)h(in)147 3600 y(Lie-algebraic)e(terms)i(b)m(y)i(\(4.2\).\)) 446 3721 y(W)-8 b(e)48 b(will)c(actually)i(state)h(and)h(pro)m(v)m(e)g (an)f(in)m(tegral)e(form)h(of)h(the)g(Bak)m(er-Campb)s(ell-)147 3841 y(Hausdor\033)38 b(form)m(ula,)g(rather)f(than)h(the)g(series)h (form)d(\(4.2\).)59 b(Ho)m(w)m(ev)m(er,)42 b(the)c(in)m(tegral)e(form)g (is)147 3962 y(su\036cien)m(t)k(to)e(obtain)g(the)h(desired)g(result)f (\(4.3\).)61 b(\(See)40 b(Corollary)c(68.\))61 b(The)40 b(series)f(form)e(of)147 4082 y(the)26 b(Bak)m(er-Campb)s (ell-Hausdor\033)e(form)m(ula)f(is)j(stated)g(precisely)f(and)h(pro)m (v)m(ed)h(in)e(V)-8 b(aradara)5 b(jan,)147 4202 y(Sec.)45 b(2.15.)446 4323 y(Consider)33 b(the)g(function)1655 4549 y FD(g)t FE(\()p FD(z)t FE(\))27 b(=)1988 4482 y(log)17 b FD(z)p 1972 4526 227 4 v 1972 4620 a FE(1)22 b FA(\000)2152 4581 y Fz(1)p 2152 4597 36 4 v 2152 4654 a Ft(z)2208 4549 y FH(.)147 4819 y(This)32 b(function)f(is)h(de\034ned)h(and)e (analytic)g(in)g(the)h(disk)g FA(fj)o FD(z)27 b FA(\000)c FE(1)p FA(j)k FD(<)h FE(1)p FA(g)o FH(,)k(and)g(th)m(us)h(for)e FD(z)36 b FH(in)31 b(this)147 4939 y(set,)j FD(g)t FE(\()p FD(z)t FE(\))e FH(can)h(b)s(e)g(expressed)i(as)1453 5217 y FD(g)t FE(\()p FD(z)t FE(\))27 b(=)1801 5092 y Fw(1)1764 5122 y Fv(X)1760 5331 y Ft(m)p Fz(=0)1929 5217 y FD(a)1980 5232 y Ft(m)2047 5217 y FE(\()p FD(z)g FA(\000)22 b FE(1\))2343 5176 y Ft(m)2410 5217 y FH(.)p eop %%Page: 76 76 76 75 bop 147 48 a FK(76)2042 b FG(The)28 b(Bak)n(er-Campb)r (ell-Hausdor\033)c(F)-7 b(orm)n(ula)147 298 y FH(This)33 b(series)g(has)g(radius)f(of)g(con)m(v)m(ergence)j(one.)446 419 y(No)m(w)c(supp)s(ose)i FD(V)52 b FH(is)30 b(a)h (\034nite-dimensional)c(complex)k(v)m(ector)g(space.)44 b(Cho)s(ose)32 b(an)f(arbi-)147 539 y(trary)i(basis)g(for)g FD(V)22 b FH(,)33 b(so)g(that)g FD(V)55 b FH(can)33 b(b)s(e)h(iden)m (ti\034ed)f(with)g Fu(C)2395 503 y Ft(n)2481 539 y FH(and)g(th)m(us)h (the)g(norm)e(of)h(a)g(linear)147 659 y(op)s(erator)g(on)g FD(V)55 b FH(can)34 b(b)s(e)g(de\034ned.)47 b(Then)35 b(for)e(an)m(y)h(op)s(erator)f FD(A)g FH(on)h FD(V)55 b FH(with)33 b FA(k)p FD(A)22 b FA(\000)g FD(I)8 b FA(k)29 b FD(<)g FE(1)p FH(,)34 b(w)m(e)147 780 y(can)f(de\034ne)1429 1052 y FD(g)t FE(\()p FD(A)p FE(\))27 b(=)1801 928 y Fw(1)1764 958 y Fv(X)1760 1167 y Ft(m)p Fz(=0)1929 1052 y FD(a)1980 1067 y Ft(m)2047 1052 y FE(\()p FD(A)22 b FA(\000)h FE(1\))2367 1011 y Ft(m)2433 1052 y FH(.)147 1355 y(W)-8 b(e)23 b(are)g(no)m(w)h(ready)f(to)g(state)g(the)g(in)m (tegral)f(form)f(of)i(the)g(Bak)m(er-Campb)s(ell-Hausdor\033)e(form)m (ula.)147 1588 y FI(Theorem)35 b(67)j(\(Bak)m(er-Campb)s (ell-Hausdor\033)10 b(\))46 b FC(F)-7 b(or)45 b(al)5 b(l)45 b FD(n)31 b FA(\002)g FD(n)46 b FC(c)-5 b(omplex)44 b(matric)-5 b(es)45 b FD(X)147 1708 y FC(and)34 b FD(Y)57 b FC(with)34 b FA(k)p FD(X)8 b FA(k)35 b FC(and)f FA(k)p FD(Y)21 b FA(k)35 b FC(su\036ciently)f(smal)5 b(l,)1068 1995 y FE(log)1210 1914 y Fv(\000)1256 1995 y FD(e)1301 1954 y Ft(X)1368 1995 y FD(e)1413 1954 y Ft(Y)1474 1914 y Fv(\001)1548 1995 y FE(=)27 b FD(X)j FE(+)1860 1859 y Fv(Z)1960 1885 y Fz(1)1916 2085 y(0)2016 1995 y FD(g)t FE(\()p FD(e)2150 1954 y Fz(ad)p Ft(X)2291 1995 y FD(e)2336 1954 y Ft(t)p Fz(ad)q Ft(Y)2497 1995 y FE(\)\()p FD(Y)21 b FE(\))c FD(dt)p FC(.)720 b FH(\(4.6\))147 2273 y FI(Corollary)36 b(68)49 b FC(L)-5 b(et)49 b FD(G)g FC(b)-5 b(e)49 b(a)g(matrix)g(Lie)g (gr)-5 b(oup)49 b(and)f Fk(g)i FC(its)f(Lie)g(algebr)-5 b(a.)86 b(Supp)-5 b(ose)49 b(that)157 2367 y Fv(e)147 2394 y FD(\036)28 b FE(:)f Fk(g)h FA(!)g FF(gl)o FE(\()p FD(n)p FE(;)17 b Fs(C)p FE(\))32 b FC(is)f(a)g(Lie)h(algebr)-5 b(a)30 b(homomorphism.)41 b(Then)31 b(for)g(al)5 b(l)32 b(su\036ciently)f(smal)5 b(l)31 b FD(X)r(;)17 b(Y)147 2514 y FC(in)35 b Fk(g)p FC(,)g FE(log)524 2433 y Fv(\000)570 2514 y FD(e)615 2478 y Ft(X)682 2514 y FD(e)727 2478 y Ft(Y)788 2433 y Fv(\001)869 2514 y FC(is)g(in)f Fk(g)p FC(,)h(and)1223 2747 y Fv(e)1214 2773 y FD(\036)1288 2692 y Fv(\002)1330 2773 y FE(log)1472 2692 y Fv(\000)1518 2773 y FD(e)1563 2732 y Ft(X)1631 2773 y FD(e)1676 2732 y Ft(Y)1737 2692 y Fv(\001)o(\003)1851 2773 y FE(=)28 b(log)2098 2663 y Fv(\020)2157 2773 y FD(e)2209 2714 y Fl(e)2202 2732 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))2367 2773 y FD(e)2419 2714 y Fl(e)2412 2732 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))2569 2663 y Fv(\021)2645 2773 y FC(.)867 b FH(\(4.7\))147 3058 y FI(Remarks)147 3243 y FH(Note)39 b(that)g FD(e)652 3207 y Fz(ad)q Ft(X)794 3243 y FD(e)839 3207 y Ft(t)p Fz(ad)q Ft(Y)1000 3243 y FH(,)h(and)f(hence)i(also)d FD(g)t FE(\()p FD(e)1877 3207 y Fz(ad)p Ft(X)2018 3243 y FD(e)2063 3207 y Ft(t)p Fz(ad)q Ft(Y)2224 3243 y FE(\))p FH(,)i(is)f(a)f(linear)g(op)s(erator)g(on)h(the)g(space)147 3364 y FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))48 b FH(of)41 b(all)f FD(n)29 b FA(\002)h FD(n)42 b FH(complex)f (matrices.)72 b(In)42 b(\(4.6\),)i(this)e(op)s(erator)f(is)h(b)s(eing)f (applied)g(to)147 3484 y(the)35 b(matrix)f FD(Y)21 b FH(.)50 b(The)36 b(fact)f(that)f FD(X)43 b FH(and)35 b FD(Y)56 b FH(are)35 b(assumed)g(small)e(guaran)m(tees)i(that)g FD(e)3394 3448 y Fz(ad)q Ft(X)3536 3484 y FD(e)3581 3448 y Ft(t)p Fz(ad)q Ft(Y)147 3605 y FH(is)43 b(close)g(to)g(the)h(iden)m (tit)m(y)f(op)s(erator)g(on)g FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))49 b FH(for)43 b(all)e FE(0)46 b FA(\024)h FD(t)f FA(\024)h FE(1)p FH(.)75 b(This)44 b(ensures)h(that)147 3725 y FD(g)t FE(\()p FD(e)281 3689 y Fz(ad)p Ft(X)422 3725 y FD(e)467 3689 y Ft(t)p Fz(ad)r Ft(Y)628 3725 y FE(\))33 b FH(is)f(w)m(ell)g(de\034ned.)446 3846 y(If)22 b FD(X)30 b FH(and)22 b FD(Y)44 b FH(comm)m(ute,)24 b(then)e(w)m(e)i(exp)s(ect)f(to)f(ha)m(v)m(e)h FE(log)2472 3765 y Fv(\000)2518 3846 y FD(e)2563 3810 y Ft(X)2630 3846 y FD(e)2675 3810 y Ft(Y)2736 3765 y Fv(\001)2809 3846 y FE(=)28 b(log\()p FD(e)3122 3810 y Ft(X)5 b Fz(+)p Ft(Y)3301 3846 y FE(\))28 b(=)f FD(X)9 b FE(+)q FD(Y)21 b FH(.)147 3966 y(Exercise)27 b(3)f(asks)h(y)m(ou)g(to)e(v)m(erify)h (that)g(the)g(Bak)m(er-Campb)s(ell-Hausdor\033)e(form)m(ula)g(indeed)j (giv)m(es)147 4087 y FD(X)j FE(+)22 b FD(Y)54 b FH(in)32 b(that)g(case.)446 4208 y(F)-8 b(orm)m(ula)33 b(\(4.6\))i(is)f (admittedly)g(horrible-lo)s(oking.)47 b(Ho)m(w)m(ev)m(er,)38 b(w)m(e)e(are)f(in)m(terested)h(not)147 4328 y(in)c(the)i(details)d(of) i(the)g(form)m(ula,)e(but)j(in)e(the)h(fact)g(that)g(it)e(expresses)36 b FE(log)2933 4247 y Fv(\000)2979 4328 y FD(e)3024 4292 y Ft(X)3091 4328 y FD(e)3136 4292 y Ft(Y)3197 4247 y Fv(\001)3276 4328 y FH(\(and)d(hence)147 4448 y FD(e)192 4412 y Ft(X)260 4448 y FD(e)305 4412 y Ft(Y)366 4448 y FH(\))f(in)g(terms)g(of)g(the)h(Lie-algebraic)d(quan)m(tities)i FE(ad)p FD(X)41 b FH(and)32 b FE(ad)p FD(Y)21 b FH(.)446 4569 y(The)30 b(goal)e(of)h(the)h(Bak)m(er-Campb)s(ell-Hausdor\033)e (theorem)h(is)g(to)g(compute)g FE(log)3405 4489 y Fv(\000)3451 4569 y FD(e)3496 4533 y Ft(X)3563 4569 y FD(e)3608 4533 y Ft(Y)3669 4489 y Fv(\001)3715 4569 y FH(.)147 4690 y(Y)-8 b(ou)24 b(ma)m(y)g(w)m(ell)g(ask,)i(\020Wh)m(y)f(don't)f(w)m(e)i (simply)c(expand)k(b)s(oth)e(exp)s(onen)m(tials)f(and)i(the)f (logarithm)147 4810 y(in)30 b(p)s(o)m(w)m(er)h(series)g(and)g(m)m (ultiply)d(ev)m(erything)j(out?\021)43 b(W)-8 b(ell,)29 b(y)m(ou)i(can)g(do)f(this,)h(and)f(if)g(y)m(ou)h(do)f(it)147 4930 y(for)k(the)h(\034rst)g(sev)m(eral)g(terms)f(y)m(ou)h(will)d(get)i (the)h(same)f(answ)m(er)i(as)e(B-C-H.)g(Ho)m(w)m(ev)m(er,)j(there)e(is) 147 5051 y(a)h(serious)g(problem)e(with)i(this)f(approac)m(h,)i (namely:)49 b(Ho)m(w)36 b(do)g(y)m(ou)g(kno)m(w)h(that)f(the)g(terms)g (in)147 5171 y(suc)m(h)27 b(an)e(expansion)h(are)f(expressible)h(in)f (terms)g(of)g(comm)m(utators?)40 b(Consider)25 b(for)g(example)g(the) 147 5292 y(quadratic)30 b(term.)42 b(It)30 b(is)g(clear)g(that)g(this)f (will)f(b)s(e)j(a)f(linear)e(com)m(bination)g(of)i FD(X)3088 5255 y Fz(2)3127 5292 y FH(,)h FD(Y)3263 5255 y Fz(2)3302 5292 y FH(,)g FD(X)8 b(Y)21 b FH(,)31 b(and)p eop %%Page: 77 77 77 76 bop 147 48 a Fx(The)28 b(General)f(Bak)n(er-Campb)r (ell-Hausdor\033)d(F)-7 b(orm)n(ula)1738 b FG(77)147 298 y FD(Y)21 b(X)8 b FH(.)53 b(But)36 b(to)f(b)s(e)h(expressible)h(in) e(terms)g(of)h(comm)m(utators)e(it)h(m)m(ust)h(actually)e(b)s(e)i(a)f (constan)m(t)147 418 y(times)j FE(\()p FD(X)8 b(Y)43 b FA(\000)23 b FD(Y)e(X)8 b FE(\))p FH(.)63 b(Of)38 b(course,)k(for)c (the)i(quadratic)e(term)h(y)m(ou)g(can)g(just)h(m)m(ultiply)c(it)i(out) 147 539 y(and)e(see,)j(and)d(indeed)g(y)m(ou)h(get)1399 499 y Fz(1)p 1399 516 36 4 v 1399 573 a(2)1461 539 y FE(\()p FD(X)8 b(Y)43 b FA(\000)23 b FD(Y)e(X)8 b FE(\))33 b(=)2146 499 y Fz(1)p 2146 516 V 2146 573 a(2)2208 539 y FE([)p FD(X)r(;)17 b(Y)k FE(])p FH(.)54 b(But)37 b(it)e(is)g(far)h (from)f(clear)g(ho)m(w)147 659 y(to)d(pro)m(v)m(e)i(that)e(a)h(similar) c(result)j(o)s(ccurs)i(for)e(all)e(the)j(higher)f(terms.)43 b(See)34 b(Exercise)g(4.)147 932 y FI(Pro)s(of)j(of)h(the)f(Corollary) 147 1121 y FH(W)-8 b(e)35 b(b)s(egin)f(b)m(y)i(pro)m(ving)e(that)h(the) g(corollary)e(follo)m(ws)g(from)h(the)h(in)m(tegral)e(form)g(of)i(the)g (Bak)m(er-)147 1242 y(Campb)s(ell-Hausdor\033)h(form)m(ula.)60 b(The)39 b(pro)s(of)f(is)g(conceptually)g(similar)d(to)j(the)h (reasoning)f(in)147 1362 y(Equation)d(\(4.3\).)51 b(Note)35 b(that)g(if)g FD(X)43 b FH(and)35 b FD(Y)56 b FH(lie)34 b(in)h(some)g(Lie)f(algebra)g Fk(g)i FH(then)g FE(ad)o FD(X)43 b FH(and)36 b FE(ad)p FD(Y)147 1482 y FH(will)28 b(preserv)m(e)k Fk(g)p FH(,)f(and)f(so)h(also)e(will)e FD(g)t FE(\()p FD(e)1627 1446 y Fz(ad)p Ft(X)1769 1482 y FD(e)1814 1446 y Ft(t)p Fz(ad)q Ft(Y)1975 1482 y FE(\)\()p FD(Y)21 b FE(\))p FH(.)42 b(Th)m(us)32 b(whenev)m(er)h(form)m(ula)28 b(\(4.6\))h(holds,)147 1614 y FE(log)290 1533 y Fv(\000)335 1614 y FD(e)380 1578 y Ft(X)448 1614 y FD(e)493 1578 y Ft(Y)554 1533 y Fv(\001)636 1614 y FH(will)35 b(lie)g(in)h Fk(g)p FH(.)56 b(It)37 b(remains)f(only)g(to)g(v)m(erify)h(\(4.7\).)56 b(The)38 b(idea)e(is)g(that)g(if)3423 1588 y Fv(e)3414 1614 y FD(\036)g FH(is)h(Lie)147 1734 y(algebra)c(homomorphism,)f(then) j(it)e(will)f(tak)m(e)k(a)e(big)f(horrible)g(lo)s(oking)f(expression)j (in)m(v)m(olving)147 1855 y(`ad')h(and)f FD(X)44 b FH(and)35 b FD(Y)22 b FH(,)36 b(and)g(turn)f(it)g(in)m(to)g(the)h(same)f (expression)i(with)e FD(X)43 b FH(and)36 b FD(Y)56 b FH(replaced)36 b(b)m(y)157 1949 y Fv(e)147 1975 y FD(\036)17 b FE(\()o FD(X)8 b FE(\))33 b FH(and)618 1949 y Fv(e)608 1975 y FD(\036)17 b FE(\()p FD(Y)k FE(\))p FH(.)446 2109 y(More)33 b(precisely)-8 b(,)33 b(since)1364 2082 y Fv(e)1355 2109 y FD(\036)f FH(is)g(a)g(Lie)g(algebra)f(homomorphism,)1469 2309 y Fv(e)1460 2335 y FD(\036)o FE([)p FD(Y)5 b(;)17 b(X)8 b FE(])28 b(=)f([)1933 2309 y Fv(e)1924 2335 y FD(\036)p FE(\()p FD(Y)21 b FE(\))p FD(;)2189 2309 y Fv(e)2180 2335 y FD(\036)p FE(\()p FD(X)8 b FE(\)])147 2562 y FH(or)1257 2762 y Fv(e)1248 2788 y FD(\036)16 b FE(\(ad)p FD(Y)38 b FE(\()p FD(X)8 b FE(\)\))27 b(=)h(ad)2004 2762 y Fv(e)1995 2788 y FD(\036)16 b FE(\()p FD(Y)21 b FE(\))2240 2678 y Fv(\020)2309 2762 y(e)2299 2788 y FD(\036)c FE(\()p FD(X)8 b FE(\))2538 2678 y Fv(\021)2614 2788 y FH(.)147 3036 y(More)33 b(generally)-8 b(,)1113 3261 y Fv(e)1103 3287 y FD(\036)17 b FE(\(\(ad)p FD(Y)k FE(\))1473 3238 y Ft(n)1536 3287 y FE(\()p FD(X)8 b FE(\)\))28 b(=)1870 3176 y Fv(\020)1930 3287 y FE(ad)2042 3261 y Fv(e)2033 3287 y FD(\036)16 b FE(\()p FD(Y)21 b FE(\))2261 3176 y Fv(\021)2321 3199 y Ft(n)2384 3176 y Fv(\020)2453 3261 y(e)2444 3287 y FD(\036)16 b FE(\()p FD(X)8 b FE(\))2683 3176 y Fv(\021)2759 3287 y FH(.)446 3537 y(This)33 b(b)s(eing)f(the)h (case,)1021 3808 y Fv(e)1012 3834 y FD(\036)1086 3753 y Fv(\000)1132 3834 y FD(e)1177 3793 y Fz(ad)q Ft(Y)1329 3834 y FE(\()p FD(X)8 b FE(\))1494 3753 y Fv(\001)1567 3834 y FE(=)1712 3709 y Fw(1)1675 3739 y Fv(X)1671 3948 y Ft(m)p Fz(=0)1855 3766 y FD(t)1890 3730 y Ft(m)p 1850 3811 113 4 v 1850 3902 a FD(m)p FE(!)1981 3808 y Fv(e)1972 3834 y FD(\036)16 b FE(\(\(ad)p FD(Y)22 b FE(\))2341 3785 y Ft(n)2405 3834 y FE(\()p FD(X)8 b FE(\)\))1567 4155 y(=)1712 4030 y Fw(1)1675 4060 y Fv(X)1671 4269 y Ft(m)p Fz(=0)1855 4087 y FD(t)1890 4051 y Ft(m)p 1850 4132 V 1850 4223 a FD(m)p FE(!)1989 4044 y Fv(\020)2048 4155 y FE(ad)2160 4128 y Fv(e)2151 4155 y FD(\036)16 b FE(\()p FD(Y)22 b FE(\))2380 4044 y Fv(\021)2439 4066 y Ft(n)2503 4044 y Fv(\020)2572 4128 y(e)2562 4155 y FD(\036)17 b FE(\()o FD(X)8 b FE(\))2801 4044 y Fv(\021)1567 4427 y FE(=)28 b FD(e)1716 4386 y Ft(t)p Fz(ad)1823 4368 y Fl(e)1816 4386 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))1990 4316 y Fv(\020)2059 4401 y(e)2049 4427 y FD(\036)p FE(\()p FD(X)8 b FE(\))2272 4316 y Fv(\021)2348 4427 y FH(.)147 4675 y(Similarly)-8 b(,)1032 4897 y Fv(e)1023 4923 y FD(\036)1098 4843 y Fv(\000)1143 4923 y FD(e)1188 4882 y Fz(ad)q Ft(X)1330 4923 y FD(e)1375 4882 y Ft(t)p Fz(ad)q Ft(Y)1536 4923 y FE(\()p FD(X)8 b FE(\))1701 4843 y Fv(\001)1774 4923 y FE(=)28 b FD(e)1923 4882 y Fz(ad)2005 4865 y Fl(e)1997 4882 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))2162 4923 y FD(e)2207 4882 y Ft(t)p Fz(ad)2314 4865 y Fl(e)2307 4882 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))2481 4813 y Fv(\020)2550 4897 y(e)2540 4923 y FD(\036)p FE(\()p FD(X)8 b FE(\))2763 4813 y Fv(\021)2839 4923 y FH(.)147 5171 y(Assume)38 b(no)m(w)g(that)e FD(X)45 b FH(and)37 b FD(Y)58 b FH(are)37 b(small)e(enough)i(that)g(B-C-H)f(applies)g(to)h FD(X)45 b FH(and)37 b FD(Y)21 b FH(,)38 b(and)147 5292 y(to)275 5265 y Fv(e)266 5292 y FD(\036)p FE(\()p FD(X)8 b FE(\))32 b FH(and)719 5265 y Fv(e)710 5292 y FD(\036)o FE(\()p FD(Y)22 b FE(\))p FH(.)43 b(Then,)33 b(using)f(the)h(linearit)m(y)d(of) i(the)g(in)m(tegral)f(and)h(reasoning)g(similar)c(to)p eop %%Page: 78 78 78 77 bop 147 48 a FK(78)2042 b FG(The)28 b(Bak)n(er-Campb)r (ell-Hausdor\033)c(F)-7 b(orm)n(ula)147 298 y FH(the)33 b(ab)s(o)m(v)m(e,)h(w)m(e)f(ha)m(v)m(e:)507 539 y Fv(e)498 565 y FD(\036)572 484 y Fv(\000)618 565 y FE(log)760 484 y Fv(\000)806 565 y FD(e)851 524 y Ft(X)919 565 y FD(e)964 524 y Ft(Y)1025 484 y Fv(\001)o(\001)1144 565 y FE(=)1256 539 y Fv(e)1247 565 y FD(\036)p FE(\()p FD(X)8 b FE(\))22 b(+)1590 429 y Fv(Z)1689 456 y Fz(1)1645 655 y(0)1786 440 y Fw(1)1750 470 y Fv(X)1745 680 y Ft(m)p Fz(=0)1914 565 y FD(a)1965 580 y Ft(m)2042 539 y Fv(e)2032 565 y FD(\036)2107 484 y Fv(\002)o(\000)2194 565 y FD(e)2239 524 y Fz(ad)p Ft(X)2381 565 y FD(e)2426 524 y Ft(t)p Fz(ad)q Ft(Y)2609 565 y FA(\000)h FD(I)2760 484 y Fv(\001)2805 504 y Ft(n)2869 565 y FE(\()p FD(X)8 b FE(\))3034 484 y Fv(\003)3108 565 y FD(dt)1144 886 y FE(=)1256 860 y Fv(e)1247 886 y FD(\036)p FE(\()p FD(X)g FE(\))22 b(+)1590 750 y Fv(Z)1689 777 y Fz(1)1645 976 y(0)1786 761 y Fw(1)1750 791 y Fv(X)1745 1000 y Ft(m)p Fz(=0)1914 886 y FD(a)1965 901 y Ft(m)2049 775 y Fv(\020)2108 886 y FD(e)2153 845 y Fz(ad)2235 827 y Fl(e)2228 845 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))2392 886 y FD(e)2437 845 y Ft(t)p Fz(ad)2545 827 y Fl(e)2537 845 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))2717 886 y FA(\000)23 b FD(I)2868 775 y Fv(\021)2927 798 y Ft(n)2991 886 y FE(\()3038 860 y Fv(e)3029 886 y FD(\036)o FE(\()p FD(X)8 b FE(\)\))17 b FD(dt)1144 1158 y FE(=)27 b(log)1390 1048 y Fv(\020)1449 1158 y FD(e)1501 1099 y Fl(e)1494 1117 y Ft(\036)p Fz(\()p Ft(X)5 b Fz(\))1659 1158 y FD(e)1711 1099 y Fl(e)1704 1117 y Ft(\036)p Fz(\()p Ft(Y)16 b Fz(\))1861 1048 y Fv(\021)1937 1158 y FH(.)147 1376 y(This)33 b(is)f(what)h(w)m(e)g(w)m(an)m(ted)h(to)e(sho)m(w.)45 b Fy(\003)446 1497 y FH(Before)36 b(coming)e(to)i(the)g(pro)s(of)f(Bak) m(er-Campb)s(ell-Hausdor\033)f(form)m(ula)g(itself,)h(w)m(e)i(will)147 1617 y(obtain)44 b(a)h(result)g(concerning)h(deriv)-5 b(ativ)m(es)45 b(of)g(the)g(exp)s(onen)m(tial)g(mapping.)80 b(This)45 b(result)g(is)147 1737 y(v)-5 b(aluable)45 b(in)g(its)g(o)m(wn)i(righ)m(t,)h(and)e(will)e(pla)m(y)h(a)h(cen)m (tral)g(role)f(in)g(our)h(pro)s(of)f(of)g(the)i(Bak)m(er-)147 1858 y(Campb)s(ell-Hausdor\033)31 b(form)m(ula.)446 1978 y(Observ)m(e)j(that)f(if)e FD(X)40 b FH(and)33 b FD(Y)54 b FH(comm)m(ute,)32 b(then)1632 2175 y FD(e)1677 2134 y Ft(X)5 b Fz(+)p Ft(tY)1909 2175 y FE(=)28 b FD(e)2058 2134 y Ft(X)2126 2175 y FD(e)2171 2134 y Ft(tY)147 2372 y FH(and)33 b(so)1221 2530 y Ft(d)p 1208 2546 62 4 v 1208 2603 a(dt)1280 2484 y Fv(\014)1280 2544 y(\014)1313 2608 y Ft(t)p Fz(=0)1449 2569 y FD(e)1494 2528 y Ft(X)5 b Fz(+)p Ft(tY)1727 2569 y FE(=)27 b FD(e)1875 2528 y Ft(X)1992 2530 y(d)p 1979 2546 V 1979 2603 a(dt)2051 2484 y Fv(\014)2051 2544 y(\014)2084 2608 y Ft(t)p Fz(=0)2221 2569 y FD(e)2266 2528 y Ft(tY)2380 2569 y FE(=)g FD(e)2528 2528 y Ft(X)2596 2569 y FD(Y)21 b FH(.)147 2766 y(In)33 b(general,)f FD(X)40 b FH(and)33 b FD(Y)54 b FH(do)32 b(not)h(comm)m(ute,)f(and)1547 2923 y Ft(d)p 1535 2940 V 1535 2997 a(dt)1606 2878 y Fv(\014)1606 2938 y(\014)1640 3002 y Ft(t)p Fz(=0)1776 2963 y FD(e)1821 2921 y Ft(X)5 b Fz(+)p Ft(tY)2053 2963 y FA(6)p FE(=)28 b FD(e)2202 2921 y Ft(X)2269 2963 y FD(Y)22 b FH(.)147 3159 y(This,)33 b(as)g(it)f(turns)h(out,)g(is)f(an)h(imp)s(ortan)m(t)e(p)s(oin)m(t.)43 b(In)33 b(particular,)e(note)i(that)f(in)g(the)h(language)147 3280 y(of)f(m)m(ultiv)-5 b(ariate)30 b(calculus)799 3471 y Ft(d)p 787 3487 V 787 3544 a(dt)858 3425 y Fv(\014)858 3485 y(\014)892 3549 y Ft(t)p Fz(=0)1028 3510 y FD(e)1073 3469 y Ft(X)5 b Fz(+)p Ft(tY)1305 3510 y FE(=)1458 3369 y Fv(\032)1574 3449 y FH(directional)30 b(deriv)-5 b(ativ)m(e)32 b(of)g FC(exp)38 b FH(at)33 b FD(X)8 b FH(,)1574 3569 y(in)32 b(the)h(direction)e(of)h FD(Y)3096 3510 y FH(.)419 b(\(4.8\))147 3765 y(Th)m(us)35 b(computing)d(the)i(left)e(side)h(of)g (\(4.8\))g(is)g(the)g(same)g(as)h(computing)e(all)f(of)i(the)g (directional)147 3885 y(deriv)-5 b(ativ)m(es)32 b(of)g(the)h (\(matrix-v)-5 b(alued\))30 b(function)i FC(exp)p FH(.)42 b(W)-8 b(e)33 b(exp)s(ect)h(the)f(directional)d(deriv)-5 b(ativ)m(e)147 4006 y(to)32 b(b)s(e)h(a)f(linear)f(function)h(of)g FD(Y)22 b FH(,)32 b(for)g(eac)m(h)i(\034xed)f FD(X)8 b FH(.)479 4126 y(No)m(w,)33 b(the)g(function)1248 4331 y FE(1)22 b FA(\000)h FD(e)1464 4295 y Fw(\000)p Ft(z)p 1248 4375 311 4 v 1378 4466 a FD(z)1596 4398 y FE(=)1710 4325 y(1)f FA(\000)g FE(\(1)g FA(\000)h FD(z)k FE(+)2268 4286 y Ft(z)2304 4262 y Fr(2)p 2268 4302 71 4 v 2276 4359 a Fz(2!)2371 4325 y FA(\000)22 b(\001)17 b(\001)g(\001)e FE(\))p 1710 4375 932 4 v 2151 4466 a FD(z)147 4620 y FH(is)32 b(an)h(en)m(tire)f(analytic)f(function)h(of)h FD(z)t FH(,)g(ev)m(en)h(at)e FD(z)h FE(=)27 b(0)p FH(,)33 b(and)f(is)g(giv)m(en)h(b)m(y)g(the)g(p)s(o)m(w)m(er)h(series)925 4820 y FE(1)21 b FA(\000)i FD(e)1140 4784 y Fw(\000)p Ft(z)p 925 4864 311 4 v 1055 4955 a FD(z)1273 4887 y FE(=)1413 4763 y Fw(1)1376 4792 y Fv(X)1382 5002 y Ft(n)p Fz(=1)1520 4887 y FE(\()p FA(\000)p FE(1\))1722 4846 y Ft(n)p Fw(\000)p Fz(1)1869 4820 y FD(z)1918 4784 y Ft(n)p Fw(\000)p Fz(1)p 1869 4864 187 4 v 1920 4955 a FD(n)p FE(!)2094 4887 y(=)k(1)22 b FA(\000)2391 4820 y FD(z)p 2378 4864 76 4 v 2378 4955 a FE(2!)2486 4887 y(+)2594 4820 y FD(z)2643 4784 y Fz(2)p 2594 4864 90 4 v 2601 4955 a FE(3!)2715 4887 y FA(\000)h(\001)17 b(\001)g(\001)e FH(.)147 5171 y(This)34 b(series)g(\(whic)m(h)h(has)f(in\034nite)f (radius)g(of)g(con)m(v)m(ergence\),)k(mak)m(e)d(sense)h(when)g FD(z)k FH(is)33 b(repaced)147 5292 y(b)m(y)h(a)e(linear)f(op)s(erator)h FD(A)g FH(on)h(some)f(\034nite-dimensional)d(v)m(ector)34 b(space.)p eop %%Page: 79 79 79 78 bop 147 48 a Fx(The)28 b(General)f(Bak)n(er-Campb)r (ell-Hausdor\033)d(F)-7 b(orm)n(ula)1738 b FG(79)147 298 y FI(Theorem)35 b(69)j(\(Deriv)-6 b(ativ)m(e)35 b(of)i(Exp)s(onen)m (tial\))47 b FC(L)-5 b(et)32 b FD(X)39 b FC(and)30 b FD(Y)52 b FC(b)-5 b(e)31 b FD(n)14 b FA(\002)g FD(n)32 b FC(c)-5 b(omplex)29 b(matri-)147 418 y(c)-5 b(es.)44 b(Then)829 617 y FD(d)p 811 661 86 4 v 811 752 a(dt)907 540 y Fv(\014)907 599 y(\014)907 659 y(\014)907 719 y(\014)940 783 y Ft(t)p Fz(=0)1077 684 y FD(e)1122 643 y Ft(X)5 b Fz(+)p Ft(tY)1354 684 y FE(=)28 b FD(e)1503 643 y Ft(X)1587 544 y Fv(\032)1672 617 y FD(I)i FA(\000)22 b FD(e)1889 580 y Fw(\000)p Fz(ad)q Ft(X)p 1672 661 415 4 v 1783 752 a FE(ad)p FD(X)2096 684 y FE(\()p FD(Y)f FE(\))2250 544 y Fv(\033)1354 960 y FE(=)28 b FD(e)1503 919 y Ft(X)1587 820 y Fv(\032)1662 960 y FD(Y)43 b FA(\000)1872 893 y FE([)p 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Fv(\001)2793 1400 y(\033)2885 1541 y FC(.)578 b FH(\(4.10\))147 1873 y FI(Remarks)147 2063 y FH(Note)38 b(that)g(the)g(directional)e(deriv)-5 b(ativ)m(e)38 b(in)f(\(4.9\))g (is)h(indeed)g(linear)e(in)i FD(Y)59 b FH(for)37 b(eac)m(h)i(\034xed)g FD(X)8 b FH(.)147 2184 y(Note)38 b(also)e(that)i(\(4.9\))f(is)g(just)h (a)f(sp)s(ecial)g(case)h(of)f(\(4.10\),)h(b)m(y)h(taking)e FD(X)8 b FE(\()p FD(t)p FE(\))36 b(=)g FD(X)d FE(+)26 b FD(tY)21 b FH(,)39 b(and)147 2304 y(ev)-5 b(aluating)31 b(at)h FD(t)c FE(=)g(0)p FH(.)446 2427 y(F)-8 b(urthermore,)38 b(observ)m(e)i(that)d(if)f FD(X)46 b FH(and)37 b FD(Y)59 b FH(comm)m(ute,)38 b(then)h(only)e(the)h(\034rst)g(term)f(in)147 2547 y(the)c(series)g(\(4.9\))f(surviv)m(es.)45 b(In)33 b(that)g(case,)g(w)m(e)h(obtain)2266 2508 y Ft(d)p 2253 2525 62 4 v 2253 2582 a(dt)2325 2463 y Fv(\014)2325 2523 y(\014)2358 2586 y Ft(t)p Fz(=0)2494 2547 y FD(e)2539 2511 y Ft(X)5 b Fz(+)p Ft(tY)2772 2547 y FE(=)27 b FD(e)2920 2511 y Ft(X)2988 2547 y FD(Y)54 b FH(as)32 b(exp)s(ected.)147 2824 y FI(Pro)s(of)37 b(of)h(the)f(Theorem)147 3015 y FH(It)g(is)f(p)s(ossible)g(to)h(pro)m(v)m(e)h(this)e(Theorem)h(b)m(y)h (expanding)f(ev)m(erything)g(in)f(a)h(p)s(o)m(w)m(er)h(series)f(and)147 3135 y(di\033eren)m(tiating)24 b(term-b)m(y-term;)k(w)m(e)f(will)d(not) h(tak)m(e)i(that)f(approac)m(h.)42 b(W)-8 b(e)26 b(will)e(pro)m(v)m(e)k (only)d(form)147 3255 y(\(4.9\))32 b(of)g(the)h(deriv)-5 b(ativ)m(e)32 b(form)m(ula,)f(but)i(the)g(form)e(\(4.10\))h(follo)m(ws) f(b)m(y)i(the)g(c)m(hain)g(rule.)446 3378 y(Let)f(us)g(use)g(the)g(Lie) e(pro)s(duct)i(form)m(ula,)e(and)h(let)g(us)h(assume)g(for)f(the)g (momen)m(t)g(that)g(it)147 3499 y(is)j(legal)e(to)i(in)m(terc)m(hange)g (limit)d(and)j(deriv)-5 b(ativ)m(e.)47 b(\(W)-8 b(e)35 b(will)d(consider)i(this)g(issue)g(at)g(the)g(end.\))147 3619 y(Then)g(w)m(e)f(ha)m(v)m(e)900 3880 y FD(e)945 3839 y Fw(\000)p Ft(X)1122 3813 y FD(d)p 1104 3857 86 4 v 1104 3948 a(dt)1200 3735 y Fv(\014)1200 3795 y(\014)1200 3855 y(\014)1200 3915 y(\014)1233 3979 y Ft(t)p Fz(=0)1370 3880 y FD(e)1415 3839 y Ft(X)5 b Fz(+)p Ft(tY)1647 3880 y FE(=)27 b FD(e)1795 3839 y Fw(\000)p Ft(X)1959 3880 y FE(lim)1935 3940 y Ft(n)p Fw(!1)2173 3813 y FD(d)p 2155 3857 V 2155 3948 a(dt)2251 3735 y Fv(\014)2251 3795 y(\014)2251 3855 y(\014)2251 3915 y(\014)2284 3979 y Ft(t)p Fz(=0)2421 3799 y Fv(\000)2466 3880 y FD(e)2511 3839 y Ft(X)o(=n)2651 3880 y FD(e)2696 3839 y Ft(tY)8 b(=n)2853 3799 y Fv(\001)2898 3815 y Ft(n)2962 3880 y FH(.)147 4170 y(W)-8 b(e)33 b(no)m(w)g(apply)g(the)f(pro)s(duct)h(rule) f(\(generalized)g(to)g FD(n)h FH(factors\))f(to)h(obtain)174 4488 y FD(e)219 4447 y Fw(\000)p Ft(X)396 4421 y FD(d)p 379 4465 V 379 4556 a(dt)474 4344 y Fv(\014)474 4403 y(\014)474 4463 y(\014)474 4523 y(\014)508 4587 y Ft(t)p Fz(=0)644 4488 y FD(e)689 4447 y Ft(X)5 b Fz(+)p Ft(tY)921 4488 y FE(=)28 b FD(e)1070 4447 y Fw(\000)p Ft(X)1233 4488 y FE(lim)1209 4548 y Ft(n)p Fw(!1)1415 4364 y Ft(n)p Fw(\000)p Fz(1)1409 4393 y Fv(X)1417 4606 y Ft(k)r Fz(=0)1570 4377 y Fv(h)1617 4407 y(\000)1663 4488 y FD(e)1708 4447 y Ft(X)o(=n)1847 4488 y FD(e)1892 4447 y Ft(tY)8 b(=n)2049 4407 y Fv(\001)2095 4423 y Ft(n)p Fw(\000)p Ft(k)r Fw(\000)p Fz(1)2342 4407 y Fv(\000)2388 4488 y FD(e)2433 4447 y Ft(X)o(=n)2572 4488 y FD(e)2617 4447 y Ft(tY)g(=n)2774 4488 y FD(Y)j(=n)2949 4407 y Fv(\001)16 b(\000)3056 4488 y FD(e)3101 4447 y Ft(X)o(=n)3241 4488 y FD(e)3286 4447 y Ft(tY)8 b(=n)3443 4407 y Fv(\001)3489 4423 y Ft(k)3531 4377 y Fv(i)3578 4557 y Ft(t)p Fz(=0)921 4826 y FE(=)28 b FD(e)1070 4785 y Fw(\000)p Ft(X)1233 4826 y FE(lim)1209 4886 y Ft(n)p Fw(!1)1415 4702 y Ft(n)p Fw(\000)p Fz(1)1409 4731 y Fv(X)1417 4944 y Ft(k)r Fz(=0)1570 4745 y Fv(\000)1616 4826 y FD(e)1661 4785 y Ft(X)o(=n)1800 4745 y Fv(\001)1846 4761 y Ft(n)p Fw(\000)p Ft(k)r Fw(\000)p Fz(1)2093 4745 y Fv(\000)2139 4826 y FD(e)2184 4785 y Ft(X)o(=n)2324 4826 y FD(Y)11 b(=n)2499 4745 y Fv(\001)k(\000)2606 4826 y FD(e)2651 4785 y Ft(X)o(=n)2791 4745 y Fv(\001)2837 4761 y Ft(k)921 5164 y FE(=)52 b(lim)1025 5224 y Ft(n)p Fw(!1)1240 5097 y FE(1)p 1235 5141 59 4 v 1235 5232 a FD(n)1326 5040 y Ft(n)p Fw(\000)p Fz(1)1320 5069 y Fv(X)1328 5282 y Ft(k)r Fz(=0)1481 5083 y Fv(\000)1526 5164 y FD(e)1571 5123 y Ft(X)o(=n)1711 5083 y Fv(\001)1757 5099 y Fw(\000)p Ft(k)1871 5164 y FD(Y)1966 5083 y Fv(\000)2011 5164 y FD(e)2056 5123 y Ft(X)o(=n)2196 5083 y Fv(\001)2242 5099 y Ft(k)2301 5164 y FH(.)p eop %%Page: 80 80 80 79 bop 147 48 a FK(80)2042 b FG(The)28 b(Bak)n(er-Campb)r (ell-Hausdor\033)c(F)-7 b(orm)n(ula)446 298 y FH(But)1083 423 y Fv(\000)1129 503 y FD(e)1174 462 y Ft(X)o(=n)1314 423 y Fv(\001)1359 439 y Fw(\000)p Ft(k)1473 503 y FD(Y)1568 423 y Fv(\000)1614 503 y FD(e)1659 462 y Ft(X)o(=n)1799 423 y Fv(\001)1844 439 y Ft(k)1915 503 y FE(=)2018 423 y Fv(\002)2060 503 y FE(Ad)2204 423 y Fv(\000)2249 503 y FD(e)2294 462 y Fw(\000)p Ft(X)o(=n)2489 423 y Fv(\001\003)2576 439 y Ft(k)2635 503 y FE(\()p FD(Y)22 b FE(\))1915 683 y(=)2018 603 y Fv(\000)2064 683 y FD(e)2109 642 y Fw(\000)p Fz(ad)p Ft(X)o(=n)2378 603 y Fv(\001)2424 619 y Ft(k)2483 683 y FE(\()p FD(Y)f FE(\))147 889 y FH(\(where)34 b(w)m(e)f(ha)m(v)m (e)h(used)g(the)f(relationship)d(b)s(et)m(w)m(een)35 b FC(A)-5 b(d)43 b FH(and)33 b FC(ad)10 b FH(\).)43 b(So)32 b(w)m(e)i(ha)m(v)m(e)969 1184 y FD(e)1014 1143 y Fw(\000)p Ft(X)1190 1117 y FD(d)p 1173 1161 86 4 v 1173 1253 a(dt)1268 1040 y Fv(\014)1268 1100 y(\014)1268 1159 y(\014)1268 1219 y(\014)1302 1283 y Ft(t)p Fz(=0)1438 1184 y FD(e)1483 1143 y Ft(X)5 b Fz(+)p Ft(tY)1715 1184 y FE(=)52 b(lim)1819 1244 y Ft(n)p Fw(!1)2034 1117 y FE(1)p 2029 1161 59 4 v 2029 1253 a FD(n)2120 1060 y Ft(n)p Fw(\000)p Fz(1)2114 1090 y Fv(X)2122 1302 y Ft(k)r Fz(=0)2275 1104 y Fv(\000)2320 1184 y FD(e)2365 1143 y Fw(\000)p Fz(ad)q Ft(X)o(=n)2635 1104 y Fv(\001)2680 1120 y Ft(k)2739 1184 y FE(\()p FD(Y)22 b FE(\))p FH(.)572 b(\(4.11\))446 1511 y(Observ)m(e)40 b(no)m(w)e(that)1251 1436 y Fv(P)1356 1463 y Ft(n)p Fw(\000)p Fz(1)1356 1540 y Ft(k)r Fz(=0)1510 1430 y Fv(\000)1556 1511 y FD(e)1601 1475 y Fw(\000)p Fz(ad)p Ft(X)o(=n)1870 1430 y Fv(\001)1916 1451 y Ft(k)1996 1511 y FH(is)g(a)f(geometric)g (series.)60 b(Let)38 b(us)h(no)m(w)g(reason)147 1631 y(for)27 b(a)g(momen)m(t)f(at)h(the)h(purely)f(formal)e(lev)m(el.)42 b(Using)27 b(the)g(usual)g(form)m(ula)f(for)g(geometric)g(series,)147 1752 y(w)m(e)34 b(get)626 2027 y FD(e)671 1985 y Fw(\000)p Ft(X)847 1959 y FD(d)p 830 2004 86 4 v 830 2095 a(dt)926 1882 y Fv(\014)926 1942 y(\014)926 2002 y(\014)926 2061 y(\014)959 2125 y Ft(t)p Fz(=0)1095 2027 y FD(e)1140 1985 y Ft(X)5 b Fz(+)p Ft(tY)1373 2027 y FE(=)51 b(lim)1476 2086 y Ft(n)p Fw(!1)1691 1959 y FE(1)p 1687 2004 59 4 v 1687 2095 a FD(n)1765 1953 y(I)30 b FA(\000)1937 1872 y Fv(\000)1983 1953 y FD(e)2028 1917 y Fw(\000)p Fz(ad)q Ft(X)o(=n)2297 1872 y Fv(\001)2343 1893 y Ft(n)p 1765 2004 626 4 v 1834 2095 a FD(I)g FA(\000)23 b FD(e)2052 2066 y Fw(\000)p Fz(ad)p Ft(X)o(=n)2400 2027 y FE(\()p FD(Y)e FE(\))1373 2308 y(=)51 b(lim)1476 2368 y Ft(n)p Fw(!1)2186 2241 y FD(I)30 b FA(\000)22 b FD(e)2403 2205 y Fw(\000)p Fz(ad)q Ft(X)p 1687 2285 1413 4 v 1687 2412 a FD(n)1762 2301 y Fv(h)1809 2412 y FD(I)29 b FA(\000)1981 2301 y Fv(\020)2041 2412 y FD(I)h FA(\000)2223 2373 y Fz(ad)q Ft(X)p 2223 2389 138 4 v 2271 2446 a(n)2393 2412 y FE(+)2501 2364 y Fz(\(ad)q Ft(X)5 b Fz(\))2693 2345 y Fr(2)p 2501 2389 228 4 v 2548 2446 a Ft(n)2591 2427 y Fr(2)2626 2446 y Fz(2!)2760 2412 y FA(\000)23 b(\001)17 b(\001)g(\001)2993 2301 y Fv(\021i)3109 2308 y FE(\()p FD(Y)k FE(\))1373 2659 y(=)51 b(lim)1476 2719 y Ft(n)p Fw(!1)1878 2592 y FD(I)30 b FA(\000)22 b FD(e)2095 2556 y Fw(\000)p Fz(ad)q Ft(X)p 1687 2637 798 4 v 1687 2747 a FE(ad)p FD(X)30 b FA(\000)2010 2700 y Fz(\(ad)q Ft(X)5 b Fz(\))2202 2681 y Fr(2)p 2010 2724 228 4 v 2075 2781 a Ft(n)p Fz(2!)2269 2747 y FE(+)22 b FA(\001)17 b(\001)g(\001)2494 2659 y FE(\()p FD(Y)k FE(\))1373 2964 y(=)1486 2897 y FD(I)30 b FA(\000)23 b FD(e)1704 2861 y Fw(\000)p Fz(ad)p Ft(X)p 1486 2941 415 4 v 1597 3033 a FE(ad)p FD(X)1911 2964 y FE(\()p FD(Y)e FE(\))p FH(.)147 3195 y(This)33 b(is)f(what)h(w)m(e)g(w)m(an)m(ted)h(to)e(sho)m(w!)446 3315 y(Do)s(es)44 b(this)f(argumen)m(t)h(mak)m(e)f(sense)i(at)f(an)m(y) g(rigorous)e(lev)m(el?)76 b(In)44 b(fact)f(it)g(do)s(es.)76 b(As)147 3435 y(usual,)50 b(let)45 b(us)i(consider)g(\034rst)f(the)h (diagonalizable)c(case.)85 b(That)47 b(is,)i(assume)e(that)f FE(ad)o FD(X)54 b FH(is)147 3556 y(diagonalizable)36 b(as)k(an)f(op)s(erator)g(on)g FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)i FE(\))p FH(,)48 b(and)39 b(assume)h(that)f FD(Y)61 b FH(is)39 b(an)g(eigen)m(v)m(ector)i(for)147 3676 y FE(ad)p FD(X)8 b FH(.)45 b(This)33 b(means)g(that)f FE(ad)p FD(X)8 b FE(\()p FD(Y)21 b FE(\))29 b(=)f([)p FD(X)r(;)17 b(Y)k FE(])29 b(=)f FD(\025Y)21 b FH(,)33 b(for)g(some)f FD(\025)d FA(2)f Fu(C)20 b FH(.)51 b(No)m(w,)34 b(there)f(are)g(t)m(w)m(o)147 3797 y(cases,)43 b FD(\025)c FE(=)g(0)g FH(and)h FD(\025)f FA(6)p FE(=)g(0)p FH(.)64 b(The)40 b FD(\025)f FE(=)g(0)g FH(case)i(corresp)s(onds)f(to)f(the)h (case)g(in)f(whic)m(h)h FD(X)47 b FH(and)147 3917 y FD(Y)56 b FH(comm)m(ute,)34 b(and)h(w)m(e)h(ha)m(v)m(e)f(already)f(observ)m(ed) j(that)d(the)h(Theorem)g(holds)f(trivially)d(in)j(that)147 4037 y(case.)446 4158 y(The)k(in)m(teresting)e(case,)j(then,)f(is)e (the)h(case)h FD(\025)d FA(6)p FE(=)f(0)p FH(.)56 b(Note)37 b(that)f FE(\(ad)p FD(X)8 b FE(\))3170 4109 y Ft(n)3233 4158 y FE(\()p FD(Y)22 b FE(\))34 b(=)h FD(\025)3590 4121 y Ft(n)3637 4158 y FD(Y)21 b FH(,)147 4278 y(and)33 b(so)1292 4403 y Fv(\000)1338 4483 y FD(e)1383 4442 y Fw(\000)p Fz(ad)p Ft(X)o(=n)1652 4403 y Fv(\001)1697 4419 y Ft(k)1757 4483 y FE(\()p FD(Y)21 b FE(\))27 b(=)2042 4403 y Fv(\000)2088 4483 y FD(e)2133 4442 y Fw(\000)p Ft(\025=n)2311 4403 y Fv(\001)2357 4419 y Ft(k)2416 4483 y FE(\()p FD(Y)21 b FE(\))p FH(.)147 4689 y(Th)m(us)34 b(the)f(geometric)e(series)h(in)g(\(4.11\))f(b)s(ecomes)i(an)f (ordinary)g(complex-v)-5 b(alued)30 b(series,)j(with)147 4809 y(ratio)e FD(e)425 4773 y Fw(\000)p Ft(\025=n)604 4809 y FH(.)44 b(Since)33 b FD(\025)28 b FA(6)p FE(=)g(0)p FH(,)k(this)h(ratio)e(will)f(b)s(e)j(di\033eren)m(t)g(from)f(one)h(for) f(all)f(su\036cien)m(tly)i(large)147 4930 y FD(n)p FH(.)44 b(Th)m(us)34 b(w)m(e)g(get)963 5217 y FD(e)1008 5176 y Fw(\000)p Ft(X)1185 5149 y FD(d)p 1167 5194 86 4 v 1167 5285 a(dt)1263 5072 y Fv(\014)1263 5132 y(\014)1263 5192 y(\014)1263 5252 y(\014)1296 5316 y Ft(t)p Fz(=0)1433 5217 y FD(e)1478 5176 y Ft(X)5 b Fz(+)p Ft(tY)1710 5217 y FE(=)1813 5046 y Fv( )1916 5217 y FE(lim)1892 5277 y Ft(n)p Fw(!1)2107 5149 y FE(1)p 2103 5194 59 4 v 2103 5285 a FD(n)2181 5143 y(I)30 b FA(\000)2354 5062 y Fv(\000)2399 5143 y FD(e)2444 5107 y Fw(\000)p Ft(\025=n)2623 5062 y Fv(\001)2668 5083 y Ft(n)p 2181 5194 535 4 v 2250 5285 a FD(I)g FA(\000)23 b FD(e)2468 5256 y Fw(\000)p Ft(\025=n)2725 5046 y Fv(!)2821 5217 y FD(Y)e FH(.)p eop %%Page: 81 81 81 80 bop 147 48 a Fx(The)28 b(General)f(Bak)n(er-Campb)r (ell-Hausdor\033)d(F)-7 b(orm)n(ula)1738 b FG(81)147 298 y FH(There)34 b(is)e(no)m(w)h(no)g(trouble)f(in)f(taking)h(the)h (limit)c(as)k(w)m(e)g(did)f(formally)e(ab)s(o)m(v)m(e)j(to)g(get)1212 575 y FD(e)1257 534 y Fw(\000)p Ft(X)1433 507 y FD(d)p 1416 552 86 4 v 1416 643 a(dt)1511 430 y Fv(\014)1511 490 y(\014)1511 550 y(\014)1511 610 y(\014)1545 674 y Ft(t)p Fz(=0)1681 575 y FD(e)1726 534 y Ft(X)5 b Fz(+)p Ft(tY)1958 575 y FE(=)2072 507 y(1)22 b FA(\000)g FD(e)2287 471 y Fw(\000)p Ft(\025)p 2072 552 316 4 v 2201 643 a FD(\025)2398 575 y(Y)1958 856 y FE(=)2072 789 y FD(I)30 b FA(\000)22 b FD(e)2289 753 y Fw(\000)p Fz(ad)q Ft(X)p 2072 834 415 4 v 2183 925 a FE(ad)p FD(X)2496 856 y FE(\()p FD(Y)g FE(\))p FH(.)446 1102 y(W)-8 b(e)39 b(see)h(then)f(that)f(the)h (Theorem)g(holds)f(in)g(the)h(case)h(that)e FE(ad)p FD(X)47 b FH(is)38 b(diagonalizable)147 1222 y(and)28 b FD(Y)49 b FH(is)28 b(an)g(eigen)m(v)m(ector)h(of)e FE(ad)p FD(X)8 b FH(.)42 b(If)28 b FE(ad)p FD(X)36 b FH(is)27 b(diagonalizable)d(but) 29 b FD(Y)49 b FH(is)27 b(not)h(an)g(eigen)m(v)m(ector,)147 1343 y(then)36 b FD(Y)56 b FH(is)35 b(a)g(linear)f(com)m(bination)f(of) h(eigen)m(v)m(ectors)j(and)e(applying)f(the)i(ab)s(o)m(v)m(e)g (computation)147 1463 y(to)c(eac)m(h)i(of)e(those)h(eigen)m(v)m(ectors) h(giv)m(es)f(the)g(desired)g(result.)446 1584 y(W)-8 b(e)43 b(need,)i(then,)h(to)c(consider)g(the)h(case)g(where)g FE(ad)p FD(X)50 b FH(is)42 b(not)g(diagonalizable.)69 b(But)147 1704 y(\(Exercise)43 b(20\),)h(if)d FD(X)49 b FH(is)42 b(a)f(diagonalizable)d(matrix,)43 b(then)f FE(ad)p FD(X)50 b FH(will)40 b(b)s(e)i(diagonalizable)c(as)147 1824 y(an)45 b(op)s(erator)f(on)g FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)86 b(Since,)48 b(as)d(w)m(e)h(ha)m(v)m(e) g(already)e(observ)m(ed,)50 b(ev)m(ery)c(matrix)d(is)i(the)147 1945 y(limit)c(of)j(diagonalizable)c(matrices,)47 b(w)m(e)e(are)f (essen)m(tially)g(done.)79 b(F)-8 b(or)44 b(it)f(is)h(easy)h(to)f(see)h (b)m(y)147 2065 y(di\033eren)m(tiating)32 b(the)i(p)s(o)m(w)m(er)h 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5292 y FD(e)2210 5250 y Ft(tY)2296 5211 y Fv(\001)p eop %%Page: 82 82 82 81 bop 147 48 a FK(82)2042 b FG(The)28 b(Bak)n(er-Campb)r (ell-Hausdor\033)c(F)-7 b(orm)n(ula)147 298 y FH(If)34 b FD(X)42 b FH(and)34 b FD(Y)55 b FH(are)34 b(su\036cien)m(tly)h (small,)d(then)i FD(Z)24 b FE(\()p FD(t)p FE(\))34 b FH(is)f(de\034ned)j(for)d FE(0)d FA(\024)h FD(t)f FA(\024)h FE(1)p FH(.)47 b(It)34 b(is)g(left)f(as)h(an)147 418 y(exercise)g(to)e(v)m(erify)h(that)f FD(Z)7 b FE(\()p FD(t)p FE(\))33 b FH(is)f(smo)s(oth.)43 b(Our)32 b(goal)f(is)h(to)g (compute)h FD(Z)7 b FE(\(1\))p FD(:)446 539 y FH(By)33 b(de\034nition)1666 746 y FD(e)1711 704 y Ft(Z)5 b Fz(\()p Ft(t)p Fz(\))1876 746 y FE(=)27 b FD(e)2024 704 y Ft(X)2092 746 y FD(e)2137 704 y Ft(tY)147 953 y FH(so)33 b(that)1114 1184 y FD(e)1159 1143 y Fw(\000)p Ft(Z)5 b Fz(\()p Ft(t)p Fz(\))1379 1117 y FD(d)p 1361 1161 86 4 v 1361 1253 a(dt)1457 1184 y(e)1502 1143 y Ft(Z)g Fz(\()p Ft(t)p Fz(\))1667 1184 y FE(=)1770 1104 y Fv(\000)1816 1184 y FD(e)1861 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2746 3655 a(12)2870 3587 y([)p FD(Y)5 b(;)17 b FE([)p FD(X)r(;)g(Y)k FE(]])1058 3773 y(+)54 b FH(higher)32 b(order)h(terms.)147 4002 y(This)g(is)f(the)h(expression)h(in)d(\(4.2\).)147 4275 y FI(4.4)112 b(Subgroups)39 b(and)f(Subalgebras)147 4446 y FH(Supp)s(ose)29 b(that)e FD(G)h FH(is)f(a)h(matrix)e(Lie)h (group,)i FD(H)35 b FH(another)28 b(matrix)e(Lie)h(group,)h(and)g(supp) s(ose)h(that)147 4566 y FD(H)35 b FA(\032)29 b FD(G)p FH(.)42 b(Then)29 b(certainly)f(the)h(Lie)f(algebra)g Fk(h)g FH(of)g FD(H)36 b FH(will)27 b(b)s(e)h(a)h(subalgebra)f(of)g (the)h(Lie)f(algebra)147 4687 y Fk(g)38 b FH(of)e FD(G)p FH(.)58 b(Do)s(es)38 b(this)f(go)g(the)h(other)f(w)m(a)m(y)h(around?)58 b(That)38 b(is)e(giv)m(en)i(a)f(Lie)g(group)g FD(G)g FH(with)g(Lie)147 4807 y(algebra)32 b Fk(g)p FH(,)i(and)f(a)g (subalgebra)g Fk(h)g FH(of)g Fk(g)p FH(,)h(is)e(there)i(a)f(matrix)f (Lie)g(group)i FD(H)40 b FH(whose)35 b(Lie)d(algebra)147 4927 y(is)g Fk(h)p FH(?)446 5051 y(In)43 b(the)g(case)g(of)f(the)h 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b(there)h(is)f(going)f(to)h(b)s(e)h(a)f(matrix)f(Lie)g(group)i FD(H)42 b FH(with)35 b(Lie)g(algebra)147 1113 y Fk(h)p FH(,)e(then)g FD(H)40 b FH(w)m(ould)32 b(con)m(tain)g(the)h(set)1304 1410 y FD(H)1385 1425 y Fz(0)1452 1410 y FE(=)1555 1270 y Fv(\032)10 b(\022)1755 1349 y FD(e)1800 1313 y Ft(it)1981 1349 y FE(0)1780 1469 y(0)108 b FD(e)1982 1433 y Ft(ita)2115 1270 y Fv(\023)2188 1266 y(\014)2188 1325 y(\014)2188 1385 y(\014)2188 1445 y(\014)2238 1410 y FD(t)28 b FA(2)g Fu(R)2461 1270 y Fv(\033)2558 1410 y FH(.)147 1713 y(T)-8 b(o)37 b(b)s(e)g(a)g(matrix)e(Lie)h(group,)i FD(H)45 b FH(w)m(ould)36 b(ha)m(v)m(e)j(to)d(b)s(e)h(closed)g(in)f FF(GL)17 b FE(\(2;)g Fu(C)i FE(\))6 b FH(,)38 b(and)f(so)g(it)f(w)m (ould)147 1833 y(con)m(tain)c(the)h(closure)g(of)f FD(H)1177 1848 y Fz(0)1216 1833 y FH(,)h(whic)m(h)g(\(see)g(\))g(is)f(the)h(set) 1274 2136 y FD(H)1355 2151 y Fz(1)1422 2136 y FE(=)1526 1995 y Fv(\032)9 b(\022)1725 2075 y FD(e)1770 2039 y Ft(it)1936 2075 y FE(0)1750 2195 y(0)108 b 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b(closed)h(subset)h(of)e Fk(g)p FH(.)44 b(Th)m(us)879 1570 y FE([)p FD(X)r(;)17 b(Y)22 b FE(])27 b(=)1308 1503 y FD(d)p 1290 1547 86 4 v 1290 1638 a(dt)1386 1426 y Fv(\014)1386 1485 y(\014)1386 1545 y(\014)1386 1605 y(\014)1419 1669 y Ft(t)p Fz(=0)1555 1570 y FD(e)1600 1529 y Ft(tX)1693 1570 y FD(Y)22 b(e)1817 1529 y Fw(\000)p Ft(tX)1992 1570 y FE(=)33 b(lim)2096 1633 y Ft(h)p Fw(!)p Fz(0)2269 1416 y Fv(\000)2315 1496 y FD(e)2360 1460 y Ft(hX)2468 1496 y FD(Y)21 b(e)2591 1460 y Fw(\000)p Ft(hX)2776 1496 y FA(\000)i FD(Y)2954 1416 y Fv(\001)p 2269 1547 732 4 v 2606 1638 a FD(h)147 1852 y FH(is)34 b(in)f Fk(h)p FH(.)48 b(\(This)35 b(argumen)m(t)e(is)h(precisely)g(the)h(one)f(w)m(e) h(used)h(to)d(sho)m(w)j(that)e(the)g(Lie)g(algebra)e(of)147 1972 y(a)38 b(matrix)f(Lie)g(group)h(is)g(a)g(closed)h(under)g(brac)m (k)m(ets,)i(once)e(w)m(e)h(had)e(established)g(that)g(it)g(is)f(a)147 2093 y(subspace.\))45 b Fy(\003)446 2213 y FH(W)-8 b(e)45 b(are)g(no)m(w)g(ready)g(to)g(state)g(the)g(main)e(theorem)h(of)h(this) f(section,)k(whic)m(h)d(is)f(our)147 2334 y(second)34 b(ma)5 b(jor)31 b(application)f(of)i(the)h(Bak)m(er-Campb)s (ell-Hausdor\033)e(form)m(ula.)147 2563 y FI(Theorem)k(73)49 b FC(L)-5 b(et)36 b FD(G)g FC(b)-5 b(e)35 b(a)h(matrix)f(Lie)h(gr)-5 b(oup)35 b(with)h(Lie)f(algebr)-5 b(a)35 b Fk(g)p FC(.)47 b(L)-5 b(et)36 b Fk(h)g FC(b)-5 b(e)35 b(a)h(Lie)g(sub)-5 b(al-)147 2683 y(gebr)g(a)36 b(of)g Fk(g)p FC(.)50 b(Then)36 b(ther)-5 b(e)36 b(exists)g(a)h(unique)f(c)-5 b(onne)g(cte)g(d)36 b(Lie)g(sub)-5 b(gr)g(oup)36 b FD(H)44 b FC(of)36 b FD(G)h FC(such)f(that)h(the)147 2804 y(Lie)e(algebr)-5 b(a)34 b(of)g FD(H)43 b FC(is)34 b Fk(h)p FC(.)147 3064 y FI(Remark)147 3249 y FH(Giv)m(en)k(a)g(matrix)e(Lie)h(group)h FD(G)g FH(and)f(a)h(subalgebra)g Fk(h)f FH(of)h Fk(g)p FH(,)h(the)g(asso)s (ciated)e(connected)j(Lie)147 3369 y(subgroup)31 b FD(H)38 b FC(might)i FH(b)s(e)30 b(a)h(matrix)e(Lie)g(group.)43 b(This)31 b(will)d(happ)s(en)j(precisely)f(if)g FD(H)38 b FH(is)30 b(a)g(closed)147 3490 y(subset)d(of)e FD(G)p FH(.)41 b(There)27 b(are)e(v)-5 b(arious)25 b(conditions)g(under)h (whic)m(h)g(y)m(ou)g(can)g(pro)m(v)m(e)h(that)e FD(H)33 b FH(is)25 b(closed.)147 3610 y(F)-8 b(or)34 b(example,)h(if)f FD(G)e FE(=)f FF(GL)17 b FE(\()o FD(n)p FE(;)g Fu(C)j FE(\))6 b FH(,)36 b(and)f Fk(h)g FH(is)f(semisimple,)f(then)j FD(H)42 b FH(is)34 b(automatically)e(closed,)147 3731 y(and)44 b(hence)h(a)f(matrix)e(Lie)h(group.)77 b(\(See)44 b(Helgason,)i(Chapter)f(I)s(I,)f(Exercises)h(and)f(F)-8 b(urther)147 3851 y(Results,)33 b(D.\))147 4111 y FI(Pro)s(of)147 4296 y FH(Not)g(written)f(at)g(this)g(time.)42 b Fy(\003)147 4559 y FI(4.5)112 b(Exercises)266 4727 y FH(1.)49 b(The)38 b FI(cen)m(ter)f FH(of)f(a)h(Lie)f(algebra)g Fk(g)i FH(is)e(de\034ned)j (to)d(b)s(e)i(the)f(set)h(of)f(all)e FD(X)43 b FA(2)36 b Fk(g)h FH(suc)m(h)h(that)391 4847 y FE([)p FD(X)r(;)17 b(Y)22 b FE(])27 b(=)h(0)k FH(for)g(all)f FD(Y)49 b FA(2)28 b Fk(g)p FH(.)43 b(No)m(w)33 b(consider)g(the)g(Heisen)m(b)s(erg)g (group)1347 5186 y FD(H)i FE(=)1567 4981 y Fv(8)1567 5071 y(<)1567 5250 y(:)1655 4985 y(0)1655 5165 y(@)1784 5064 y FE(1)83 b FD(a)k(b)1784 5185 y FE(0)d(1)j FD(c)1784 5305 y FE(0)d(0)g(1)2140 4985 y Fv(1)2140 5165 y(A)2244 5186 y FA(j)p FD(a;)17 b(b;)g(c)28 b FA(2)g Fu(R)2698 4981 y Fv(9)2698 5071 y(=)2698 5250 y(;)p eop %%Page: 87 87 87 86 bop 147 48 a Fx(Exercises)3179 b FG(87)391 298 y FH(with)32 b(Lie)g(algebra)1315 609 y Fk(h)c FE(=)1487 405 y Fv(8)1487 495 y(<)1487 674 y(:)1576 409 y(0)1576 588 y(@)1704 488 y FE(0)83 b FD(\013)h(\014)1704 608 y FE(0)90 b(0)i FD(\015)1704 729 y FE(0)e(0)96 b(0)2084 409 y Fv(1)2084 588 y(A)2188 609 y FA(j)o FD(\013)q(;)17 b(\014)6 b(;)17 b(\015)32 b FA(2)c Fu(R)2686 405 y Fv(9)2686 495 y(=)2686 674 y(;)2791 609 y FH(.)391 926 y(Determine)37 b(the)g(cen)m(ter)i FD(Z)7 b FE(\()p Fk(h)p FE(\))36 b FH(of)h Fk(h)p FH(.)57 b(F)-8 b(or)36 b(an)m(y)h FD(X)r(;)17 b(Y)57 b FA(2)36 b 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y(Bak)m(er-Campb)s(ell-Hausdor\033)d(form)m(ula.)266 4186 y(5.)49 b(Using)e(the)g(tec)m(hniques)i(in)d(Section)h(4.3,)k (compute)c(the)g(series)h(form)e(of)g(the)i(Bak)m(er-)391 4307 y(Campb)s(ell-Hausdor\033)39 b(form)m(ula)h(up)h(through)h (fourth-order)e(brac)m(k)m(ets.)72 b(\(W)-8 b(e)42 b(ha)m(v)m(e)h(al-) 391 4427 y(ready)33 b(computed)g(up)g(through)f(third-order)g(brac)m(k) m(ets.\))266 4625 y(6.)49 b(Let)38 b Fk(a)f FH(b)s(e)h(a)g(subalgebra)f (of)g(the)h(Lie)f(algebra)g(of)g(the)h(Heisen)m(b)s(erg)g(group.)59 b(Sho)m(w)38 b(that)391 4745 y FE(exp)18 b(\()p Fk(a)p FE(\))30 b FH(is)g(a)g(connected)i(Lie)e(subgroup)h(of)f(the)g(Heisen)m (b)s(erg)i(group.)42 b(Sho)m(w)31 b(that)g(in)e(fact)391 4865 y FE(exp)18 b(\()p Fk(a)p FE(\))32 b FH(is)g(a)g(matrix)f(Lie)h (group.)266 5063 y(7.)49 b(Sho)m(w)35 b(that)g(ev)m(ery)h(connected)g (Lie)e(subgroup)h(of)f FF(SU)17 b FE(\(2\))34 b FH(is)g(closed.)50 b(Sho)m(w)35 b(that)f(this)g(is)391 5183 y(not)e(the)h(case)h(for)e FF(SU)17 b FE(\(3\))p FH(.)p eop %%Page: 88 88 88 87 bop 147 48 a FK(88)2042 b FG(The)28 b(Bak)n(er-Campb)r (ell-Hausdor\033)c(F)-7 b(orm)n(ula)p eop %%Page: 89 89 89 88 bop 1688 1048 a FL(Chapter)37 b(5)920 1223 y(BASIC)h(REPRESENT) -10 b(A)g(TION)37 b(THEOR)-10 b(Y)147 1493 y FI(5.1)112 b(Represen)m(tations)147 1662 y(De\034nition)36 b(74)49 b FC(L)-5 b(et)34 b FD(G)g FC(b)-5 b(e)34 b(a)f(matrix)h(Lie)f(gr)-5 b(oup.)45 b(Then)33 b(a)g FB(\034nite-dimensional)40 b(c)-6 b(omplex)147 1782 y(r)g(epr)g(esentation)37 b FC(of)e FD(G)g FC(is)f(a)h(Lie)g(gr)-5 b(oup)34 b(homomorphism)1567 2005 y FE(\005)28 b(:)g FD(G)f FA(!)h FF(GL)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))147 2228 y FC(\()p FD(n)28 b FA(\025)g FE(1)p FC(\))35 b(or)f(mor)-5 b(e)35 b(gener)-5 b(al)5 b(ly)34 b(a)g(Lie)h(gr)-5 b(oup)35 b(homomorphism)1615 2451 y FE(\005)28 b(:)f FD(G)h FA(!)f FF(GL)p FE(\()p FD(V)22 b FE(\))147 2674 y FC(wher)-5 b(e)37 b FD(V)60 b FC(is)38 b(a)f(\034nite-dimensional)f(c)-5 b(omplex)36 b(ve)-5 b(ctor)38 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b(a)h(\034nite-dimensional)d(c)-5 b(omplex)33 b(ve)-5 b(ctor)35 b(sp)-5 b(ac)g(e.)43 b(If)35 b Fk(g)g FC(is)f(a)h(r)-5 b(e)g(al)34 b(Lie)h(algebr)-5 b(a,)33 b(then)147 3398 y(a)j FB(\034nite-dimensional)42 b(r)-6 b(e)g(al)43 b(r)-6 b(epr)g(esentation)38 b FC(of)e Fk(g)g FC(is)g(a)f(Lie)h(algebr)-5 b(a)35 b(homomorphism)f FD(\031)147 3518 y FC(of)h Fk(g)g FC(into)f FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))41 b FC(or)35 b(into)f FF(gl)p FE(\()p FD(V)21 b FE(\))p FC(.)446 3639 y(If)45 b FE(\005)g FC(or)g FD(\031)k FC(is)c(a)g(one-to-one)f(homomorphism,)h (then)g(the)g(r)-5 b(epr)g(esentation)44 b(is)h(c)-5 b(al)5 b(le)-5 b(d)147 3760 y FB(faithful)p FC(.)147 4026 y FI(Remarks)147 4213 y FH(Y)d(ou)28 b(should)g(think)g(of)g(a)g (repren)m(tation)g(as)g(a)g(\(linear\))e FI(action)i FH(of)f(a)h(group)g(or)g(Lie)f(algebra)g(on)h(a)147 4333 y(v)m(ector)33 b(space.)45 b(\(Since,)32 b(sa)m(y)-8 b(,)33 b(to)e(ev)m(ery)j FD(g)d FA(2)d FD(G)k FH(there)h(is)e(asso)s 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FD(V)21 b FC(.)65 b(A)42 b(subsp)-5 b(ac)g(e)40 b FD(W)55 b FC(of)41 b FD(V)63 b FC(is)41 b(c)-5 b(al)5 b(le)-5 b(d)41 b FB(invariant)h FC(if)147 5292 y FE(\005\()p FD(A)p FE(\))p FD(w)51 b FA(2)e FD(W)60 b FC(for)46 b(al)5 b(l)45 b FD(w)51 b FA(2)e FD(W)60 b FC(and)45 b(al)5 b(l)46 b FD(A)i FA(2)h FD(G)p FC(.)78 b(A)n(n)46 b(invariant)f(subsp)-5 b(ac)g(e)45 b FD(W)60 b FC(is)46 b(c)-5 b(al)5 b(le)-5 b(d)p eop %%Page: 90 90 90 89 bop 147 48 a FK(90)2452 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FB(non-trivial)35 b FC(if)e FD(W)41 b FA(6)p FE(=)28 b FA(f)p FE(0)p FA(g)33 b FC(and)f FD(W)42 b FA(6)p FE(=)27 b FD(V)22 b FC(.)44 b(A)34 b(r)-5 b(epr)g(esentation) 32 b(with)h(no)g(non-trivial)g(invariant)147 418 y(subsp)-5 b(ac)g(es)34 b(is)h(c)-5 b(al)5 b(le)-5 b(d)33 b FB(irr)-6 b(e)g(ducible)p FC(.)446 544 y(The)37 b(terms)g FB(invariant)p FC(,)h FB(non-trivial)p FC(,)h(and)e FB(irr)-6 b(e)g(ducible)37 b FC(ar)-5 b(e)37 b(de\034ne)-5 b(d)36 b(analo)-5 b(gously)147 665 y(for)35 b(r)-5 b(epr)g(esentations)34 b(of)g(Lie)h(algebr)-5 b(as.)147 933 y FI(De\034nition)36 b(76)49 b FC(L)-5 b(et)31 b FD(G)g FC(b)-5 b(e)30 b(a)h(matrix)f(Lie)h(gr)-5 b(oup,)31 b(let)g FE(\005)g FC(b)-5 b(e)30 b(a)h(r)-5 b(epr)g(esentation)30 b(of)g FD(G)h FC(acting)f(on)147 1053 y(the)i(sp)-5 b(ac)g(e)31 b FD(V)22 b FC(,)32 b(and)f(let)i FE(\006)f FC(b)-5 b(e)32 b(a)f(r)-5 b(epr)g(esentation)31 b(of)h FD(G)g FC(acting)f(on)h(the)g(sp)-5 b(ac)g(e)31 b FD(W)14 b FC(.)43 b(A)33 b(line)-5 b(ar)31 b(map)147 1174 y FD(\036)d FE(:)f FD(V)50 b FA(!)27 b FD(W)48 b FC(is)35 b(c)-5 b(al)5 b(le)-5 b(d)34 b(a)h FB(morphism)g FC(\(or)g FB(intertwining)k(map)p FC(\))d(of)e(r)-5 b(epr)g (esentations)34 b(if)1474 1411 y FD(\036)p FE(\(\005\()p FD(A)p FE(\))p FD(v)t FE(\))27 b(=)h(\006\()p FD(A)p FE(\))p FD(\036)p FE(\()p FD(v)t FE(\))147 1648 y FC(for)34 b(al)5 b(l)34 b FD(A)28 b FA(2)g FD(G)34 b FC(and)g(al)5 b(l)34 b FD(v)e FA(2)c FD(V)21 b FC(.)45 b(The)33 b(analo)-5 b(gous)34 b(pr)-5 b(op)g(erty)34 b(de\034nes)f(morphisms)g(of)h(r)-5 b(epr)g(esen-)147 1768 y(tations)35 b(of)f(a)h(Lie)g(algebr)-5 b(a.)446 1894 y(If)43 b FD(\036)g FC(is)g(a)g(morphism)e(of)i(r)-5 b(epr)g(esentations,)44 b(and)f(in)g(addition)f FD(\036)h FC(is)g(invertible,)h(then)147 2015 y FD(\036)37 b FC(is)h(said)e(to)i (b)-5 b(e)37 b(an)g FB(isomorphism)h FC(of)f(r)-5 b(epr)g(esentations.) 52 b(If)37 b(ther)-5 b(e)37 b(exists)g(an)g(isomorphism)147 2135 y(b)-5 b(etwe)g(en)36 b FD(V)59 b FC(and)36 b FD(W)14 b FC(,)37 b(then)f(the)h(r)-5 b(epr)g(esentations)36 b(ar)-5 b(e)36 b(said)g(to)h(b)-5 b(e)37 b FB(isomorphic)g FC(\(or)g FB(e)-6 b(quiva-)147 2255 y(lent)p FC(\).)147 2550 y FI(Remarks)147 2746 y FH(Tw)m(o)34 b(isomorphic)d(represen)m (tations)i(should)g(b)s(e)g(regarded)g(as)g(b)s(eing)f(\020the)h (same\021)f(represen)m(ta-)147 2867 y(tion.)42 b(A)30 b(t)m(ypical)f(problem)f(in)i(represen)m(tation)g(theory)h(is)e(to)h (determine,)g(up)g(to)f(isomorphism,)147 2987 y(all)39 b(the)j(irreducible)e(represen)m(tations)i(of)f(a)g(particular)f(group) h(or)g(Lie)f(algebra.)69 b(In)41 b(Section)147 3108 y(5.4)e(w)m(e)h (will)c(determine)j(all)e(the)i(\034nite-dimensional)d(complex)i (irreducible)g(represen)m(tations)147 3228 y(of)32 b(the)h(Lie)f (algebra)f FF(su)q FE(\(2\))p FH(.)147 3502 y FI(Prop)s(osition)36 b(77)48 b FC(L)-5 b(et)46 b FD(G)g FC(b)-5 b(e)45 b(a)g(matrix)h(Lie)f (gr)-5 b(oup)45 b(with)h(Lie)f(algebr)-5 b(a)45 b Fk(g)p FC(,)j(and)d(let)g FE(\005)h FC(b)-5 b(e)45 b(a)147 3622 y(\(\034nite-dimensional)26 b(r)-5 b(e)g(al)28 b(or)h(c)-5 b(omplex\))27 b(r)-5 b(epr)g(esentation)28 b(of)g FD(G)p FC(,)i(acting)e(on)g(the)h(sp)-5 b(ac)g(e)28 b FD(V)21 b FC(.)43 b(Then)147 3743 y(ther)-5 b(e)35 b(is)g(a)f(unique)h(r)-5 b(epr)g(esentation)34 b FD(\031)39 b FC(of)34 b Fk(g)h FC(acting)g(on)f(the)h(same)f(sp)-5 b(ac)g(e)34 b(such)h(that)1643 3980 y FE(\005\()p FD(e)1799 3939 y Ft(X)1867 3980 y FE(\))27 b(=)h FD(e)2081 3939 y Ft(\031)r Fz(\()p Ft(X)5 b Fz(\))147 4217 y FC(for)35 b(al)5 b(l)34 b FD(X)i FA(2)28 b Fk(g)p FC(.)45 b(The)34 b(r)-5 b(epr)g(esentation)34 b FD(\031)39 b FC(c)-5 b(an)34 b(b)-5 b(e)34 b(c)-5 b(ompute)g(d)35 b(as)1465 4506 y FD(\031)t FE(\()p FD(X)8 b FE(\))27 b(=)1857 4438 y FD(d)p 1840 4483 86 4 v 1840 4574 a(dt)1936 4361 y Fv(\014)1936 4421 y(\014)1936 4481 y(\014)1936 4540 y(\014)1969 4604 y Ft(t)p Fz(=0)2105 4506 y FE(\005)2195 4425 y Fv(\000)2241 4506 y FD(e)2286 4464 y Ft(tX)2379 4425 y Fv(\001)147 4798 y FC(and)34 b(satis\034es)1250 5035 y FD(\031)1325 4954 y Fv(\000)1371 5035 y FD(AX)8 b(A)1606 4994 y Fw(\000)p Fz(1)1700 4954 y Fv(\001)1774 5035 y FE(=)27 b(\005\()p FD(A)p FE(\))p FD(\031)t FE(\()p FD(X)8 b FE(\)\005\()p FD(A)p FE(\))2545 4994 y Fw(\000)p Fz(1)147 5272 y FC(for)35 b(al)5 b(l)34 b FD(X)i FA(2)28 b Fk(g)35 b FC(and)f(al)5 b(l)35 b FD(A)27 b FA(2)h FD(G)p FC(.)p eop %%Page: 91 91 91 90 bop 147 48 a Fx(Represen)n(tations)2934 b FG(91)147 298 y FI(Pro)s(of)147 505 y FH(Theorem)35 b(46)e(in)h(Chapter)h(3)f (states)h(that)f(for)g(eac)m(h)h(Lie)e(group)h(homomorphism)e FD(\036)e FE(:)h FD(G)f FA(!)h FD(H)147 625 y FH(there)37 b(is)e(an)g(asso)s(ciated)h(Lie)f(algebra)f(homomorphism)2319 599 y Fv(e)2310 625 y FD(\036)f FE(:)g Fk(g)g FA(!)f Fk(h)p FH(.)53 b(T)-8 b(ak)m(e)37 b FD(H)j FE(=)33 b FF(GL)p FE(\()p FD(V)21 b FE(\))36 b FH(and)147 745 y FD(\036)28 b FE(=)f(\005)p FH(.)42 b(Since)28 b(the)h(Lie)e(algebra)g (of)g FF(GL)p FE(\()p FD(V)21 b FE(\))28 b FH(is)g FF(gl)o FE(\()p FD(V)22 b FE(\))28 b FH(\(since)g(the)g(exp)s(onen)m(tial)g(of) f(an)m(y)i(op)s(erator)147 877 y(is)f(in)m(v)m(ertible\),)h(the)g(asso) s(ciated)f(Lie)g(algebra)g(homomorphism)2550 850 y Fv(e)2541 877 y FD(\036)f FE(=)h FD(\031)k FH(maps)c(from)g Fk(g)h FH(to)f FF(gl)p FE(\()p FD(V)21 b FE(\))p FH(,)147 997 y(and)33 b(so)g(constitutes)g(a)f(represen)m(tation)h(of)f Fk(g)p FH(.)446 1128 y(The)i(prop)s(erties)e(of)g FD(\031)k FH(follo)m(w)31 b(from)g(the)i(prop)s(erties)g(of)2566 1102 y Fv(e)2557 1128 y FD(\036)f FH(giv)m(en)h(in)f(Theorem)g(6.)44 b Fy(\003)147 1444 y FI(Prop)s(osition)36 b(78)48 b FC(L)-5 b(et)32 b Fk(g)g FC(b)-5 b(e)32 b(a)f(r)-5 b(e)g(al)32 b(Lie)f(algebr)-5 b(a,)32 b(and)f Fk(g)2302 1459 y Fi(C)2386 1444 y FC(its)h(c)-5 b(omplexi\034c)g(ation.)41 b(Then)31 b(every)147 1564 y(\034nite-dimensional)23 b(c)-5 b(omplex)25 b(r)-5 b(epr)g(esentation)25 b FD(\031)k FC(of)d Fk(g)f FC(has)h(a)f(unique)h(extension)e(to)i(a)g(\(c)-5 b(omplex-)147 1684 y(line)g(ar\))34 b(r)-5 b(epr)g(esentation)34 b(of)h Fk(g)1262 1699 y Fi(C)1314 1684 y FC(,)g(also)f(denote)-5 b(d)34 b FD(\031)t FC(.)45 b(The)34 b(r)-5 b(epr)g(esentation)34 b(of)g Fk(g)3066 1699 y Fi(C)3154 1684 y FC(satis\034es)1356 1937 y FD(\031)t FE(\()p FD(X)c FE(+)22 b FD(iY)g FE(\))27 b(=)h FD(\031)t FE(\()p FD(X)8 b FE(\))22 b(+)g FD(i\031)t FE(\()p FD(Y)f FE(\))147 2190 y FC(for)35 b(al)5 b(l)34 b FD(X)i FA(2)28 b Fk(g)p FC(.)147 2516 y FI(Pro)s(of)147 2723 y FH(This)33 b(follo)m(ws)e(from)g(Exercise)j(14)e(of)g(Chapter)i (3.)147 3038 y FI(De\034nition)i(79)49 b FC(L)-5 b(et)36 b FD(G)g FC(b)-5 b(e)36 b(a)g(matrix)f(Lie)h(gr)-5 b(oup,)36 b(let)g FA(H)h FC(b)-5 b(e)36 b(a)f(Hilb)-5 b(ert)37 b(sp)-5 b(ac)g(e,)35 b(and)g(let)h FD(U)10 b FE(\()p FA(H)q FE(\))147 3159 y FC(denote)36 b(the)g(gr)-5 b(oup)36 b(of)g(unitary)h(op)-5 b(er)g(ators)35 b(on)h FA(H)q FC(.)49 b(Then)35 b(a)h(homomorphism)e FE(\005)c(:)h FD(G)f FA(!)f FD(U)10 b FE(\()p FA(H)q FE(\))147 3279 y FC(is)35 b(c)-5 b(al)5 b(le)-5 b(d)34 b(a)h FB(unitary)41 b(r)-6 b(epr)g(esentation)38 b FC(of)d FD(G)g FC(if)f FE(\005)i FC(satis\034es)e(the)h(fol)5 b(lowing)33 b(c)-5 b(ontinuity)36 b(c)-5 b(on-)147 3399 y(dition:)44 b(If)35 b FD(A)644 3414 y Ft(n)691 3399 y FD(;)17 b(A)27 b FA(2)h FD(G)35 b FC(and)f FD(A)1303 3414 y Ft(n)1378 3399 y FA(!)28 b FD(A)p FC(,)34 b(then)1571 3652 y FE(\005\()p FD(A)1755 3667 y Ft(n)1802 3652 y FE(\))p FD(v)d FA(!)d FE(\005\()p FD(A)p FE(\))p FD(v)147 3905 y FC(for)i(al)5 b(l)29 b FD(v)j FA(2)c(H)q FC(.)43 b(A)30 b(unitary)g(r)-5 b(epr)g(esentation)29 b(with)h(no)f(non-trivial)g(close)-5 b(d)29 b(invariant)g(subsp)-5 b(ac)g(es)147 4025 y(is)35 b(c)-5 b(al)5 b(le)-5 b(d)34 b FB(irr)-6 b(e)g(ducible)p FC(.)147 4352 y FI(Remarks)147 4558 y FH(This)37 b(con)m(tin)m(uit)m(y) f(condition)f(is)h(called)f FI(strong)42 b(con)m(tin)m(uit)m(y)p FH(.)52 b(One)37 b(could)f(require)h(the)f(ev)m(en)147 4679 y(stronger)d(condition)e(that)i FA(k)p FE(\005\()p FD(A)1401 4694 y Ft(n)1448 4679 y FE(\))22 b FA(\000)h FE(\005\()p FD(A)p FE(\))p FA(k)k(!)h FE(0)p FH(,)33 b(but)g(this)f(turns)h(out)g(to)f(b)s(e)h(to)s(o)f(stringen)m(t)147 4799 y(a)i(requiremen)m(t.)47 b(In)34 b(practice,)g(an)m(y)g (homomorphism)d(of)i FD(G)h FH(in)m(to)f FD(U)10 b FE(\()p FA(H)q FE(\))34 b FH(y)m(ou)h(can)f(write)f(do)m(wn)147 4920 y(explicitly)e(will)f(b)s(e)j(strongly)f(con)m(tin)m(uous.)446 5051 y(One)39 b(could)f(try)h(to)f(de\034ne)i(some)e(analog)f(of)h (unitary)g(represen)m(tations)i(for)e(Lie)g(alge-)147 5171 y(bras,)k(but)f(there)f(are)g(serious)g(tec)m(hnical)g (di\036culties)f(asso)s(ciated)h(with)f(getting)g(the)h(\020righ)m (t\021)147 5292 y(de\034nition.)p eop %%Page: 92 92 92 91 bop 147 48 a FK(92)2452 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FI(5.2)112 b(Wh)m(y)38 b(Study)g(Represen)m(tations?) 147 501 y FH(If)50 b(a)f(represen)m(tation)h FE(\005)f FH(is)g(a)g(faithful)e(represen)m(tation)j(of)f(a)g(matrix)f(Lie)h (group)g FD(G)p FH(,)k(then)147 622 y FA(f)p FE(\005\()p FD(A)p FE(\))17 b FA(j)o FD(A)28 b FA(2)g FD(G)10 b FA(g)38 b FH(is)f(a)g(group)h(of)f(matrices)g(whic)m(h)h(is)f(isomorphic)f(to)h (the)h(original)d(group)i FD(G)p FH(.)147 742 y(Th)m(us)h FE(\005)e FH(allo)m(ws)e(us)j(to)e FC(r)-5 b(epr)g(esent)45 b FD(G)36 b FH(as)g(a)g(group)f(of)h(matrices.)52 b(This)36 b(is)g(the)g(motiv)-5 b(ation)33 b(for)147 862 y(the)i(term)f(represen) m(tation.)50 b(\(Of)34 b(course,)i(w)m(e)f(still)d(call)h FE(\005)h FH(a)g(represen)m(tation)i(ev)m(en)g(if)d(it)g(is)h(not)147 983 y(faithful.\))446 1139 y(Despite)40 b(the)f(origin)d(of)j(the)g (term,)h(the)f(p)s(oin)m(t)f(of)g(represen)m(tation)i(theory)f(is)f FC(not)48 b FH(\(at)147 1260 y(least)42 b(in)g(this)g(course\))h(to)f (represen)m(t)j(a)d(group)g(as)h(a)f(group)g(of)g(matrices.)73 b(After)42 b(all,)h(all)d(of)147 1380 y(our)47 b(groups)h(are)f (already)g(matrix)f(groups!)88 b(While)46 b(it)g(migh)m(t)g(seem)h (redundan)m(t)i(to)e(study)147 1500 y(represen)m(tations)37 b(of)f(a)g(group)g(whic)m(h)h(is)e(already)h(represen)m(ted)i(as)f(a)f (group)f(of)h(matrices,)g(this)147 1621 y(is)c(precisely)h(what)g(w)m (e)g(are)g(going)e(to)h(do.)446 1777 y(The)44 b(reason)f(for)f(this)g (is)h(that)f(a)h(represen)m(tation)g(can)g(b)s(e)g(though)m(t)g(of)f (\(as)h(w)m(e)h(ha)m(v)m(e)147 1897 y(already)33 b(noted\))g(as)g(an)g (action)g(of)f(our)h(group)g(on)g(some)g(v)m(ector)h(space.)46 b(Suc)m(h)34 b(actions)f(\(repre-)147 2018 y(sen)m(tations\))k(arise)e (naturally)g(in)g(man)m(y)h(branc)m(hes)i(of)d(b)s(oth)h(mathematics)e (and)i(ph)m(ysics,)j(and)147 2138 y(it)32 b(is)g(imp)s(ortan)m(t)e(to)j (understand)g(them.)446 2295 y(A)d(t)m(ypical)g(example)f(w)m(ould)h(b) s(e)h(a)f(di\033eren)m(tial)e(equation)i(in)f(three-dimensional)f (space)147 2415 y(whic)m(h)40 b(has)f(rotational)e(symmetry)-8 b(.)63 b(If)39 b(the)g(equation)g(has)h(rotational)c(symmetry)-8 b(,)41 b(then)f(the)147 2535 y(space)35 b(of)e(solutions)g(will)e(b)s (e)j(in)m(v)-5 b(arian)m(t)32 b(under)j(rotations.)45 b(Th)m(us)36 b(the)e(space)h(of)e(solutions)f(will)147 2656 y(constitute)h(a)g(represen)m(tation)h(of)e(the)i(rotation)d (group)i FF(SO)p FE(\(3\))p FH(.)45 b(If)33 b(y)m(ou)h(kno)m(w)g(what)f (all)e(of)i(the)147 2776 y(represen)m(tations)39 b(of)f FF(SO)p FE(\(3\))g FH(are,)h(this)f(can)g(help)g(immensely)e(in)h (narro)m(wing)h(do)m(wn)g(what)h(the)147 2896 y(space)i(of)f(solutions) f(can)h(b)s(e.)66 b(\(As)40 b(w)m(e)h(will)d(see,)43 b FF(SO)p FE(\(3\))d FH(has)g(lots)f(of)h(other)g(represen)m(tations) 147 3017 y(b)s(esides)33 b(the)g(ob)m(vious)g(one)g(in)f(whic)m(h)h FF(SO)p FE(\(3\))f FH(acts)h(on)g Fu(R)2260 2981 y Fz(3)2305 3017 y FH(.\))446 3173 y(In)46 b(fact,)k(one)c(of)g(the)g(c)m(hief)g (applications)e(of)i(represen)m(tation)g(theory)h(is)e(to)h(exploit)147 3294 y(symmetry)-8 b(.)48 b(If)34 b(a)g(system)h(has)g(symmetry)-8 b(,)34 b(then)h(the)g(set)f(of)g(symmetries)g(will)d(form)i(a)h(group,) 147 3414 y(and)c(understanding)h(the)f(represen)m(tations)h(of)f(the)g (symmetry)g(group)g(allo)m(ws)f(y)m(ou)i(to)e(use)i(that)147 3534 y(symmetry)i(to)f(simplify)e(the)j(problem.)446 3691 y(In)g(addition,)e(studying)h(the)h(represen)m(tations)h(of)e(a)g (group)g FD(G)h FH(\(or)f(of)g(a)g(Lie)g(algebra)f Fk(g)p FH(\))147 3811 y(can)42 b(giv)m(e)f(information)d(ab)s(out)j(the)g (group)g(\(or)g(Lie)g(algebra\))f(itself.)68 b(F)-8 b(or)40 b(example,)j(if)d FD(G)h FH(is)147 3931 y(a)f FC(\034nite)47 b FH(group,)42 b(then)f(asso)s(ciated)f(to)f FD(G)h FH(is)g(something)f (called)g(the)h FI(group)47 b(algebra)p FH(.)66 b(The)147 4052 y(structure)27 b(of)f(this)g(group)f(algebra)g(can)h(b)s(e)h (describ)s(ed)f(v)m(ery)i(nicely)d(in)h(terms)f(of)h(the)g(irreducible) 147 4172 y(represen)m(tations)34 b(of)e FD(G)p FH(.)446 4328 y(In)22 b(this)f(course,)k(w)m(e)e(will)d(b)s(e)h(in)m(terested)i (primarily)c(in)i(computing)f(the)i(\034nite-dimensional)147 4449 y(irreducible)50 b(complex)h(represen)m(tations)i(of)e(matrix)f (Lie)h(groups.)101 b(As)52 b(w)m(e)h(shall)d(see,)57 b(this)147 4569 y(problem)36 b(can)i(b)s(e)g(reduced)h(almost)c (completely)i(to)g(the)h(problem)e(of)h(computing)f(the)i(\034nite-)147 4690 y(dimensional)25 b(irreducible)h(complex)h(represen)m(tations)h (of)f(the)h(asso)s(ciated)f(Lie)g(algebra.)41 b(In)27 b(this)147 4810 y(c)m(hapter,)33 b(w)m(e)f(will)d(discuss)j(the)g (theory)f(at)g(an)g(elemen)m(tary)h(lev)m(el,)f(and)g(will)e(consider)i (in)g(detail)147 4930 y(the)j(example)f(of)g FF(SO)p FE(\(3\))h FH(and)f FF(SU)q FE(\(2\))p FH(.)46 b(In)34 b(Chapter)g(6,)g(w)m(e)h(will)c(study)k(the)f(represen)m(tations)g(of) 147 5051 y FF(SU)q FE(\(3\))p FH(,)c(whic)m(h)g(is)f(substan)m(tially)f (more)h(in)m(v)m(olv)m(ed)h(than)g(that)f(of)g FF(SU)q FE(\(2\))p FH(,)h(and)g(giv)m(e)f(an)h(o)m(v)m(erview)147 5171 y(of)45 b(the)g(represen)m(tation)h(theory)f(of)f(a)h(v)m(ery)i (imp)s(ortan)m(t)42 b(class)j(of)g(Lie)f(groups,)49 b(namely)-8 b(,)47 b(the)147 5292 y(semisimple)30 b(ones.)p eop %%Page: 93 93 93 92 bop 147 48 a Fx(Examples)28 b(of)f(Represen)n(tations)2462 b FG(93)147 298 y FI(5.3)112 b(Examples)35 b(of)j(Represen)m(tations) 147 467 y(The)g(Standard)g(Represen)m(tation)147 656 y FH(A)g(matrix)f(Lie)g(group)h FD(G)g FH(is)g(b)m(y)h(de\034nition)e (a)h(subset)i(of)d(some)h FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))44 b FH(or)37 b FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)66 b(The)147 776 y(inclusion)35 b(map)h(of)g FD(G)h FH(in)m(to)f FF(GL)o FE(\()p FD(n)p FE(\))h FH(\(i.e.,)g FE(\005\()p FD(A)p FE(\))e(=)g FD(A)p FH(\))h(is)h(a)f(represen)m(tation)h(of)f FD(G)p FH(,)i(called)e(the) 147 896 y FI(standard)45 b(represen)m(tation)38 b FH(of)f FD(G)p FH(.)60 b(Th)m(us)40 b(for)d(example)h(the)g(standard)h (represen)m(tation)f(of)147 1017 y FF(SO)p FE(\(3\))j FH(is)f(the)h(one)f(in)g(whic)m(h)h FF(SO)p FE(\(3\))g FH(acts)g(in)f(the)h(usual)f(w)m(a)m(y)i(on)e Fu(R)2793 981 y Fz(3)2838 1017 y FH(.)68 b(If)41 b FD(G)f FH(is)g(a)g(subgroup) 147 1137 y(of)i FF(GL)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))48 b FH(or)43 b FF(GL)o FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(,)51 b(then)43 b(its)g(Lie)e(algebra)h Fk(g)h FH(will)d(b)s(e)j(a)f(subalgebra)g(of)g FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))48 b FH(or)147 1258 y FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))p FH(.)67 b(The)39 b(inclusion)e(of)h Fk(g)g FH(in)m(to)g FF(gl)p FE(\()p FD(n)p FE(;)17 b Fu(R)5 b FE(\))44 b FH(or)38 b FF(gl)p FE(\()p FD(n)p 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FI(trivial)i(represen)m(tation)d FH(of)g Fk(g)p FH(,)h FD(\031)e FE(:)d Fk(g)g FA(!)f FF(gl)p FE(\(1;)17 b Fu(C)i FE(\))p FH(,)39 b(b)m(y)1743 2875 y FD(\031)t FE(\()p FD(X)8 b FE(\))27 b(=)h(0)147 3101 y FH(for)k(all)f FD(X)k FA(2)28 b Fk(g)p FH(.)44 b(This)33 b(is)f(an)g(irreducible)f(represen)m(tation.)147 3372 y FI(The)38 b(Adjoin)m(t)e(Represen)m(tation)147 3560 y FH(Let)30 b FD(G)f FH(b)s(e)h(a)f(matrix)f(Lie)h(group)g(with)g (Lie)g(algebra)f Fk(g)p FH(.)43 b(W)-8 b(e)30 b(ha)m(v)m(e)g(already)f (de\034ned)i(the)f(adjoin)m(t)147 3681 y(mapping)1602 3906 y FE(Ad)e(:)g FD(G)f FA(!)h FF(GL)o FE(\()p Fk(g)p FE(\))147 4132 y FH(b)m(y)34 b(the)f(form)m(ula)1525 4357 y FF(Ad)p FD(A)p FE(\()p FD(X)8 b FE(\))28 b(=)f FD(AX)8 b(A)2243 4316 y Fw(\000)p Fz(1)2338 4357 y FH(.)147 4583 y(Recall)23 b(that)h FC(A)-5 b(d)35 b FH(is)24 b(a)h(Lie)f(group)g (homomorphism.)38 b(Since)24 b FC(A)-5 b(d)35 b FH(is)24 b(a)h(Lie)f(group)g(homomorphism)147 4703 y(in)m(to)34 b(a)g(group)g(of)g(in)m(v)m(ertible)f(op)s(erators,)i(w)m(e)g(see)h (that)e(in)f(fact)h FC(A)-5 b(d)45 b FH(is)34 b(a)g(represen)m(tation)h (of)f FD(G)p FH(,)147 4823 y(acting)46 b(on)g(the)h(space)h Fk(g)p FH(.)85 b(Th)m(us)48 b(w)m(e)g(can)f(no)m(w)g(giv)m(e)f FC(A)-5 b(d)57 b FH(its)46 b(prop)s(er)h(name,)i(the)e FI(adjoin)m(t)147 4944 y(represen)m(tation)32 b FH(of)g FD(G)p FH(.)44 b(The)33 b(adjoin)m(t)f(represen)m(tation)h(is)f(a)g (real)g(represen)m(tation)h(of)f FD(G)p FH(.)446 5066 y(Similarly)-8 b(,)29 b(if)i Fk(g)i FH(is)f(a)g(Lie)g(algebra,)f(w)m(e) j(ha)m(v)m(e)1650 5292 y FE(ad)28 b(:)g Fk(g)g FA(!)f FF(gl)p FE(\()p Fk(g)p FE(\))p eop %%Page: 94 94 94 93 bop 147 48 a FK(94)2452 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FH(de\034ned)34 b(b)m(y)g(the)f(form)m(ula)1566 505 y FF(ad)p FD(X)8 b FE(\()p FD(Y)21 b FE(\))27 b(=)h([)p FD(X)r(;)17 b(Y)k FE(])p FH(.)147 712 y(W)-8 b(e)29 b(kno)m(w)h(that)e FC(ad)38 b FH(is)28 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FH(Consider)i(the)h(space)f FD(V)1061 1892 y Ft(m)1168 1877 y FH(of)f(homogeneous)h(p)s(olynomials)c(in)j(t)m(w)m(o)i(complex) e(v)-5 b(ariables)38 b(with)147 1998 y(total)31 b(degree)j FD(m)e FH(\()p FD(m)c FA(\025)h FE(0)p FH(\).)43 b(That)32 b(is,)h FD(V)1641 2013 y Ft(m)1740 1998 y FH(is)f(the)h(space)g(of)f (functions)h(of)f(the)h(form)854 2205 y FD(f)11 b FE(\()p FD(z)996 2220 y Fz(1)1035 2205 y FD(;)17 b(z)1124 2220 y Fz(2)1164 2205 y FE(\))27 b(=)h FD(a)1384 2220 y Fz(0)1423 2205 y FD(z)1472 2164 y Ft(m)1468 2230 y Fz(1)1562 2205 y FE(+)22 b FD(a)1711 2220 y Fz(1)1751 2205 y FD(z)1800 2164 y Ft(m)p Fw(\000)p Fz(1)1796 2230 y(1)1957 2205 y FD(z)2002 2220 y Fz(2)2064 2205 y FE(+)g FD(a)2213 2220 y Fz(2)2253 2205 y FD(z)2302 2164 y Ft(m)p Fw(\000)p Fz(2)2298 2230 y(1)2459 2205 y FD(z)2508 2164 y Fz(2)2504 2230 y(2)2565 2205 y FA(\001)17 b(\001)g(\001)j FE(+)i FD(a)2852 2220 y Ft(m)2919 2205 y FD(z)2968 2164 y Ft(m)2964 2230 y Fz(2)3542 2205 y FH(\(5.1\))147 2412 y(with)27 b 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(and)g(that)g(ev)m(ery)h(\034nite-dimensional)c(irreducible)h(represen) m(tation)147 5171 y(of)e FF(SU)p FE(\(2\))g FH(is)f(equiv)-5 b(alen)m(t)36 b(to)g(one)g(\(and)g(only)f(one\))h(of)g(the)g FE(\005)2443 5186 y Ft(m)2510 5171 y FH('s.)54 b(\(Of)36 b(course,)i(no)d(t)m(w)m(o)i(of)f(the)147 5292 y FE(\005)220 5307 y Ft(m)287 5292 y FH('s)d(are)g(equiv)-5 b(alen)m(t,)32 b(since)h(they)g(don't)g(ev)m(en)h(ha)m(v)m(e)g(the)f(same)f (dimension.\))p eop %%Page: 95 95 95 94 bop 147 48 a Fx(Examples)28 b(of)f(Represen)n(tations)2462 b FG(95)446 298 y FH(Let)43 b(us)g(no)m(w)g(compute)f(the)h(corresp)s (onding)g(Lie)e(algebra)h(represen)m(tation)h FD(\031)3426 313 y Ft(m)3493 298 y FH(.)73 b(Ac-)147 418 y(cording)32 b(to)g(Prop)s(osition)f(77,)h FD(\031)1350 433 y Ft(m)1449 418 y FH(can)h(b)s(e)g(computed)g(as)1378 669 y FD(\031)1433 684 y Ft(m)1500 669 y FE(\()p FD(X)8 b FE(\))28 b(=)1834 602 y FD(d)p 1816 646 86 4 v 1816 738 a(dt)1912 525 y Fv(\014)1912 584 y(\014)1912 644 y(\014)1912 704 y(\014)1945 768 y Ft(t)p Fz(=0)2082 669 y FE(\005)2155 684 y Ft(m)2238 588 y Fv(\000)2284 669 y FD(e)2329 628 y Ft(tX)2422 588 y Fv(\001)2484 669 y FH(.)147 924 y(So)1228 1155 y FE(\()p FD(\031)1321 1170 y Ft(m)1388 1155 y FE(\()p FD(X)8 b FE(\))p FD(f)j FE(\))16 b(\()p FD(z)t FE(\))28 b(=)1960 1088 y FD(d)p 1943 1133 V 1943 1224 a(dt)2038 1011 y Fv(\014)2038 1071 y(\014)2038 1131 y(\014)2038 1190 y(\014)2072 1254 y Ft(t)p Fz(=0)2208 1155 y FD(f)2283 1075 y Fv(\000)2329 1155 y FD(e)2374 1114 y Fw(\000)p Ft(tX)2522 1155 y FD(z)2571 1075 y Fv(\001)2634 1155 y FH(.)147 1423 y(No)m(w)k(let)f FD(z)t FE(\()p FD(t)p FE(\))h FH(b)s(e)f(the)h(curv)m(e)h(in)d Fu(C)1435 1387 y Fz(2)1512 1423 y FH(de\034ned)i(as)g FD(z)t FE(\()p FD(t)p FE(\))c(=)g FD(e)2302 1387 y Fw(\000)p Ft(tX)2450 1423 y FD(z)t FH(,)k(so)g(that)f FD(z)t FE(\(0\))d(=)f FD(z)t FH(.)44 b(Of)31 b(course,)147 1543 y FD(z)t FE(\()p FD(t)p FE(\))j FH(can)e(b)s(e)h(written)f(as)h FD(z)t FE(\()p FD(t)p FE(\))c(=)e(\()p FD(z)1488 1558 y Fz(1)1528 1543 y FE(\()p FD(t)p FE(\))p FD(;)17 b(z)1728 1558 y Fz(2)1768 1543 y FE(\()p FD(t)p FE(\)\))p FH(,)32 b(with)g FD(z)2243 1558 y Ft(i)2272 1543 y FE(\()p FD(t)p FE(\))c FA(2)g Fu(C)20 b FH(.)49 b(By)33 b(the)g(c)m(hain)f(rule,)1128 1800 y FD(\031)1183 1815 y Ft(m)1250 1800 y FE(\()p FD(X)8 b FE(\))p FD(f)38 b FE(=)1627 1733 y FD(@)5 b(f)p 1614 1777 142 4 v 1614 1868 a(@)g(z)1715 1883 y Fz(1)1803 1733 y FD(dz)1899 1748 y Fz(1)p 1803 1777 136 4 v 1827 1868 a FD(dt)1948 1655 y Fv(\014)1948 1715 y(\014)1948 1775 y(\014)1948 1835 y(\014)1981 1899 y Ft(t)p Fz(=0)2123 1800 y FE(+)2244 1733 y FD(@)g(f)p 2231 1777 142 4 v 2231 1868 a(@)g(z)2332 1883 y Fz(2)2419 1733 y FD(dz)2515 1748 y Fz(2)p 2419 1777 136 4 v 2444 1868 a FD(dt)2565 1655 y Fv(\014)2565 1715 y(\014)2565 1775 y(\014)2565 1835 y(\014)2598 1899 y Ft(t)p Fz(=0)2734 1800 y FH(.)147 2061 y(But)43 b FD(dz)t(=dt)p FA(j)614 2090 y Ft(t)p Fz(=0)761 2061 y FE(=)28 b FA(\000)p FD(X)8 b(z)t FH(,)33 b(so)g(w)m(e)h(obtain)d(the)i(follo)m(wing)d(form)m(ula)g(for)i FD(\031)2848 2076 y Ft(m)2915 2061 y FE(\()p FD(X)8 b FE(\))730 2314 y FD(\031)785 2329 y Ft(m)852 2314 y FE(\()p FD(X)g FE(\))p FD(f)38 b FE(=)27 b FA(\000)1307 2246 y FD(@)5 b(f)p 1293 2291 142 4 v 1293 2382 a(@)g(z)1394 2397 y Fz(1)1462 2314 y FE(\()p FD(X)1581 2329 y Fz(11)1656 2314 y FD(z)1701 2329 y Fz(1)1763 2314 y FE(+)22 b FD(X)1942 2329 y Fz(12)2016 2314 y FD(z)2061 2329 y Fz(2)2101 2314 y FE(\))g FA(\000)2284 2246 y FD(@)5 b(f)p 2271 2291 V 2271 2382 a(@)g(z)2372 2397 y Fz(2)2439 2314 y FE(\()p FD(X)2558 2329 y Fz(21)2632 2314 y FD(z)2677 2329 y Fz(1)2739 2314 y FE(+)22 b FD(X)2918 2329 y Fz(22)2993 2314 y FD(z)3038 2329 y Fz(2)3078 2314 y FE(\))16 b FH(.)383 b(\(5.3\))446 2553 y(No)m(w,)33 b(according)e(to)h(Prop)s(osition)e(78,)i(ev)m(ery)i (\034nite-dimensional)28 b(complex)k(represen-)147 2673 y(tation)38 b(of)g(the)h(Lie)g(algebra)e FF(su)q FE(\(2\))h FH(extends)j(uniquely)e(to)g(a)f(complex-linear)e(represen)m(tation)147 2793 y(of)d(the)g(complexi\034cation)d(of)j FF(su)p FE(\(2\))p FH(.)44 b(But)33 b(the)g(complexi\034cation)e(of)h FF(su)q FE(\(2\))g FH(is)h(\(isomorphic)d(to\))147 2914 y FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))42 b FH(\(Chapter)37 b(3,)g(Prop)s(osition)d (64\).)54 b(T)-8 b(o)36 b(see)h(that)f(this)g(is)g(so,)h(note)f(that)g FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))42 b FH(is)36 b(the)147 3034 y(space)e(of)e(all)e FE(2)22 b FA(\002)h FE(2)32 b FH(complex)g(matrices)g(with)g(trace)h(zero.)44 b(But)32 b(if)g FD(X)40 b FH(is)32 b(in)g FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(,)38 b(then)916 3282 y FD(X)e FE(=)1146 3214 y FD(X)30 b FA(\000)23 b FD(X)1446 3178 y Fw(\003)p 1146 3259 339 4 v 1291 3350 a FE(2)1517 3282 y(+)1625 3214 y FD(X)30 b FE(+)22 b FD(X)1923 3178 y Fw(\003)p 1625 3259 338 4 v 1769 3350 a FE(2)2000 3282 y(=)2114 3214 y FD(X)29 b FA(\000)23 b FD(X)2413 3178 y Fw(\003)p 2114 3259 339 4 v 2259 3350 a FE(2)2484 3282 y(+)f FD(i)2625 3214 y(X)31 b FE(+)22 b FD(X)2924 3178 y Fw(\003)p 2625 3259 338 4 v 2753 3350 a FE(2)p FD(i)147 3512 y FH(where)31 b(b)s(oth)e FE(\()p FD(X)23 b FA(\000)16 b FD(X)977 3476 y Fw(\003)1016 3512 y FE(\))p FD(=)p FE(2)29 b FH(and)h FE(\()p FD(X)23 b FE(+)16 b FD(X)1691 3476 y Fw(\003)1730 3512 y FE(\))p FD(=)p FE(2)p FD(i)29 b FH(are)g(in)f FF(su)q FE(\(2\))p FH(.)42 b(\(Chec)m(k!\))i(It)30 b(is)e(easy)j(to)e (see)h(that)147 3632 y(this)36 b(decomp)s(osition)e(is)h(unique,)i(so)f (that)g(ev)m(ery)i FD(X)j FA(2)34 b FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))42 b FH(can)36 b(b)s(e)g(written)g(uniquely)g(as)147 3753 y FD(X)41 b FE(=)33 b FD(X)459 3768 y Fz(1)522 3753 y FE(+)24 b FD(iY)712 3768 y Fz(1)787 3753 y FH(with)36 b FD(X)1094 3768 y Fz(1)1133 3753 y FD(;)17 b(Y)1234 3768 y Fz(1)1306 3753 y FA(2)33 b FF(su)p FE(\(2\))p FH(.)53 b(Th)m(us)37 b FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))41 b FH(is)35 b(isomomorphic)e(as)j(a)f(v)m(ector)i(space)147 3873 y(to)32 b FF(su)q FE(\(2\))479 3888 y Fg(C)541 3873 y FH(.)44 b(But)33 b(this)f(is)g(in)g(fact)g(an)g(isomorphism)e(of)i (Lie)g(algebras,)g(since)h(in)f(b)s(oth)g(cases)566 4072 y FE([)p FD(X)674 4087 y Fz(1)735 4072 y FE(+)22 b FD(iY)923 4087 y Fz(1)963 4072 y FD(;)17 b(X)1088 4087 y Fz(2)1149 4072 y FE(+)22 b FD(iY)1337 4087 y Fz(2)1376 4072 y FE(])28 b(=)g([)p FD(X)1643 4087 y Fz(1)1682 4072 y FD(;)17 b(X)1807 4087 y Fz(2)1846 4072 y FE(])22 b FA(\000)h FE([)p FD(Y)2079 4087 y Fz(1)2118 4072 y FD(;)17 b(Y)2219 4087 y Fz(2)2258 4072 y FE(])22 b(+)g FD(i)17 b FE(\([)p FD(X)2601 4087 y Fz(1)2641 4072 y FD(;)g(Y)2742 4087 y Fz(2)2780 4072 y FE(])22 b(+)g([)q FD(X)3036 4087 y Fz(2)3075 4072 y FD(;)17 b(Y)3176 4087 y Fz(1)3215 4072 y FE(]\))f FH(.)147 4272 y(\(See)34 b(Exercise)f(5.\))446 4392 y(So,)j(the)g(represen)m (tation)g FD(\031)1478 4407 y Ft(m)1580 4392 y FH(of)f FF(su)p FE(\(2\))g FH(giv)m(en)h(b)m(y)g(\(5.3\))f(extends)i(to)e(a)g (represen)m(tation)147 4512 y(of)26 b FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(,)34 b(whic)m(h)27 b(w)m(e)h(will)d(also)g(call)g FD(\031)1615 4527 y Ft(m)1682 4512 y FH(.)42 b(I)27 b(assert)g(that)g (in)f(fact)g(form)m(ula)f(\(5.3\),)j(still)c(holds)j(for)147 4633 y FD(X)36 b FA(2)28 b FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))p FH(.)49 b(Wh)m(y)32 b(is)f(this?)42 b(W)-8 b(ell,)30 b(\(5.3\))g(is)g(undoubtedly)i(\(complex\))e(linear,)g(and)h(it)e (agrees)147 4753 y(with)36 b(the)h(original)c FD(\031)956 4768 y Ft(m)1059 4753 y FH(for)j FD(X)42 b FA(2)35 b FF(su)p FE(\(2\))p FH(.)55 b(But)36 b(there)h(is)f(only)g(one)h (complex)f(linear)e(extension)147 4874 y(of)e FD(\031)313 4889 y Ft(m)413 4874 y FH(from)f FF(su)p FE(\(2\))i FH(to)f FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))p FH(,)39 b(so)33 b(this)f(m)m(ust)h (b)s(e)f(it!)446 4994 y(So,)h(for)f(example,)g(consider)h(the)f(elemen) m(t)1591 5246 y FD(H)j FE(=)1811 5105 y Fv(\022)1925 5185 y FE(1)122 b(0)1925 5305 y(0)83 b FA(\000)p FE(1)2225 5105 y Fv(\023)p eop %%Page: 96 96 96 95 bop 147 48 a FK(96)2452 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FH(in)32 b(the)h(Lie)f(algebra)f FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(.)49 b(Applying)32 b(form)m(ula)e (\(5.3\))i(giv)m(es)1239 578 y FE(\()p FD(\031)1332 593 y Ft(m)1399 578 y FE(\()p FD(H)8 b FE(\))p FD(f)j FE(\))k(\()p FD(z)t FE(\))29 b(=)e FA(\000)2034 511 y FD(@)5 b(f)p 2020 555 142 4 v 2020 647 a(@)g(z)2121 662 y Fz(1)2172 578 y FD(z)2217 593 y Fz(1)2279 578 y FE(+)2400 511 y FD(@)g(f)p 2387 555 V 2387 647 a(@)g(z)2488 662 y Fz(2)2539 578 y FD(z)2584 593 y Fz(2)2623 578 y FH(.)147 849 y(Th)m(us)34 b(w)m(e)g(see)g(that)1377 1105 y FD(\031)1432 1120 y Ft(m)1499 1105 y FE(\()p FD(H)8 b FE(\))27 b(=)h FA(\000)p FD(z)1917 1120 y Fz(1)2009 1037 y FD(@)p 1967 1082 V 1967 1173 a(@)5 b(z)2068 1188 y Fz(1)2141 1105 y FE(+)22 b FD(z)2284 1120 y Fz(2)2376 1037 y FD(@)p 2333 1082 V 2333 1173 a(@)5 b(z)2434 1188 y Fz(2)2485 1105 y FH(.)1030 b(\(5.4\))147 1394 y(Applying)32 b FD(\031)625 1409 y Ft(m)692 1394 y FE(\()p FD(H)8 b FE(\))32 b FH(to)g(a)g(basis)h(elemen) m(t)f FD(z)1738 1358 y Ft(k)1734 1419 y Fz(1)1781 1394 y FD(z)1830 1353 y Ft(m)p Fw(\000)p Ft(k)1826 1419 y Fz(2)2024 1394 y FH(w)m(e)h(get)565 1626 y FD(\031)620 1641 y Ft(m)687 1626 y FE(\()p FD(H)8 b FE(\))p FD(z)901 1584 y Ft(k)897 1650 y Fz(1)944 1626 y FD(z)993 1584 y Ft(m)p Fw(\000)p Ft(k)989 1650 y Fz(2)1181 1626 y FE(=)28 b FA(\000)p FD(k)s(z)1465 1584 y Ft(k)1461 1650 y Fz(1)1509 1626 y FD(z)1558 1584 y Ft(m)p Fw(\000)p Ft(k)1554 1650 y Fz(2)1741 1626 y FE(+)22 b(\()p FD(m)g FA(\000)h FD(k)s FE(\))p FD(z)2225 1584 y Ft(k)2221 1650 y Fz(1)2268 1626 y FD(z)2317 1584 y Ft(m)p Fw(\000)p Ft(k)2313 1650 y Fz(2)2506 1626 y FE(=)k(\()p FD(m)22 b FA(\000)h FE(2)p FD(k)s FE(\))p FD(z)3044 1584 y Ft(k)3040 1650 y Fz(1)3087 1626 y FD(z)3136 1584 y Ft(m)p Fw(\000)p Ft(k)3132 1650 y Fz(2)3297 1626 y FH(.)147 1857 y(Th)m(us)41 b FD(z)450 1820 y Ft(k)446 1881 y Fz(1)493 1857 y FD(z)542 1815 y Ft(m)p Fw(\000)p Ft(k)538 1881 y Fz(2)742 1857 y FH(is)e(an)g(eigen)m (v)m(ector)h(for)f FD(\031)1713 1872 y Ft(m)1780 1857 y FE(\()p FD(H)8 b FE(\))38 b FH(with)h(eigen)m(v)-5 b(alue)38 b FE(\()p FD(m)27 b FA(\000)g FE(2)p FD(k)s FE(\))p FH(.)63 b(In)40 b(particular,)147 1977 y FD(\031)202 1992 y Ft(m)269 1977 y FE(\()p FD(H)8 b FE(\))32 b FH(is)g (diagonalizable.)446 2101 y(Let)h FD(X)40 b FH(and)33 b FD(Y)54 b FH(b)s(e)32 b(the)h(elemen)m(ts)1256 2365 y FD(X)j FE(=)1476 2225 y Fv(\022)1591 2304 y FE(0)83 b(1)1591 2425 y(0)g(0)1813 2225 y Fv(\023)1903 2365 y FE(;)g FD(Y)49 b FE(=)2223 2225 y Fv(\022)2338 2304 y FE(0)82 b(0)2338 2425 y(1)g(0)2560 2225 y Fv(\023)147 2655 y FH(in)32 b FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(.)49 b(Then)34 b(\(5.3\))e(tells)f(us)i(that)1186 2882 y FD(\031)1241 2897 y Ft(m)1308 2882 y FE(\()p FD(X)8 b FE(\))28 b(=)f FA(\000)p FD(z)1726 2897 y Fz(2)1810 2842 y Ft(@)p 1776 2859 109 4 v 1776 2916 a(@)t(z)1850 2925 y Fr(1)1895 2882 y FE(;)83 b FD(\031)2060 2897 y Ft(m)2127 2882 y FE(\()p FD(Y)21 b FE(\))28 b(=)f FA(\000)p 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y(1)2321 4709 y FD(z)2370 4668 y Fz(2)2366 4734 y(2)2427 4709 y FA(\001)17 b(\001)g(\001)k FE(+)h FD(a)2715 4724 y Ft(m)2781 4709 y FD(z)2830 4668 y Ft(m)2826 4734 y Fz(2)147 4940 y FH(with)42 b(at)g(least)g(one)g(of)g(the)h FD(a)1284 4955 y Ft(k)1327 4940 y FH('s)f(non-zero.)73 b(Let)42 b FD(k)2132 4955 y Fz(0)2214 4940 y FH(b)s(e)g(the)h(largest)e(v)-5 b(alue)42 b(of)g FD(k)j FH(for)d(whic)m(h)147 5060 y FD(a)198 5075 y Ft(k)269 5060 y FA(6)p FE(=)27 b(0)p FH(,)33 b(and)f(consider)1714 5292 y FD(\031)1769 5307 y Ft(m)1836 5292 y FE(\()p FD(X)8 b FE(\))2001 5250 y Ft(k)2038 5259 y Fr(0)2076 5292 y FD(w)s FH(.)p eop %%Page: 97 97 97 96 bop 147 48 a Fx(Examples)28 b(of)f(Represen)n(tations)2462 b FG(97)147 298 y FH(Since)44 b(\(b)m(y)h(\(5.5\)\))e(eac)m(h)i (application)d(of)h FD(\031)1804 313 y Ft(m)1871 298 y FE(\()p FD(X)8 b FE(\))44 b FH(lo)m(w)m(ers)g(the)g(p)s(o)m(w)m(er)h (of)f FD(z)3027 313 y Fz(1)3110 298 y FH(b)m(y)h(1,)i FD(\031)3435 313 y Ft(m)3502 298 y FE(\()p FD(X)8 b FE(\))3667 262 y Ft(k)3704 271 y Fr(0)147 418 y FH(will)35 b(kill)f(all)h(the)i (terms)g(in)f FD(w)j FH(whose)f(p)s(o)m(w)m(er)f(of)g FD(z)2061 433 y Fz(1)2137 418 y FH(is)g(less)g(than)f FD(k)2706 433 y Fz(0)2746 418 y FH(,)i(that)e(is,)i(all)c(except)39 b(the)147 539 y FD(a)198 554 y Ft(k)235 563 y Fr(0)274 539 y FD(z)323 495 y Ft(k)360 504 y Fr(0)319 563 y Fz(1)399 539 y FD(z)448 495 y Ft(m)p Fw(\000)p Ft(k)602 504 y Fr(0)444 563 y Fz(2)674 539 y FH(term.)k(On)33 b(the)g(other)f(hand,)h (w)m(e)h(compute)e(easily)g(that)1064 736 y FD(\031)1119 751 y Ft(m)1186 736 y FE(\()p FD(X)8 b FE(\))1351 695 y Ft(k)1388 704 y Fr(0)1443 656 y Fv(\000)1489 736 y FD(a)1540 751 y Ft(k)1577 760 y Fr(0)1615 736 y FD(z)1664 693 y Ft(k)1701 702 y Fr(0)1660 761 y Fz(1)1741 736 y FD(z)1790 693 y Ft(m)p Fw(\000)p Ft(k)1944 702 y Fr(0)1786 761 y Fz(2)1983 656 y Fv(\001)2056 736 y FE(=)28 b FD(k)2211 751 y Fz(0)2250 736 y FE(!\()p FA(\000)p FE(1\))2479 695 y Ft(k)2516 704 y Fr(0)2555 736 y FD(a)2606 751 y Ft(k)2643 760 y Fr(0)2682 736 y FD(z)2731 695 y Ft(m)2727 761 y Fz(2)2798 736 y FH(.)446 934 y(W)-8 b(e)26 b(see,)h(then,)h(that) c FD(\031)1287 949 y Ft(m)1354 934 y FE(\()p FD(X)8 b FE(\))1519 898 y Ft(k)1556 907 y Fr(0)1595 934 y FD(w)27 b FH(is)e(a)f FC(non-zer)-5 b(o)30 b FH(m)m(ultiple)23 b(of)i FD(z)2780 898 y Ft(m)2776 959 y Fz(2)2847 934 y FH(.)41 b(Since)25 b FD(W)39 b FH(is)24 b(assumed)147 1055 y(in)m(v)-5 b(arian)m(t,)31 b FD(W)47 b FH(m)m(ust)32 b(con)m(tain)g(this)g(m)m(ultiple)e(of)i FD(z)2039 1019 y Ft(m)2035 1079 y Fz(2)2107 1055 y FH(,)g(and)h(so)g(also)e FD(z)2720 1019 y Ft(m)2716 1079 y Fz(2)2820 1055 y FH(itself.)446 1175 y(But)d(no)m(w)g(it)e(follo)m(ws)g(from)h(\(5.5\))g(that)g FD(\031)1955 1190 y Ft(m)2022 1175 y FE(\()p FD(Y)21 b FE(\))2176 1139 y Ft(k)2219 1175 y FD(z)2268 1139 y Ft(m)2264 1200 y Fz(2)2362 1175 y FH(is)27 b(a)h FC(non-zer)-5 b(o)32 b FH(m)m(ultiple)25 b(of)i FD(z)3462 1139 y Ft(k)3458 1200 y Fz(1)3505 1175 y FD(z)3554 1134 y Ft(m)p Fw(\000)p 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y FE(\()p FD(R)q FE(\))p FD(f)11 b FE(])16 b(\()p FD(x)p FE(\))28 b(=)g FD(f)2130 2095 y Fv(\000)2176 2175 y FD(R)2251 2134 y Fw(\000)p Fz(1)2345 2175 y FD(x)2400 2095 y Fv(\001)2463 2175 y FH(.)147 2373 y(Since)47 b(Leb)s(esgue)h (measure)f FD(dx)f FH(is)g(rotationally)e(in)m(v)-5 b(arian)m(t,)48 b FE(\005)2579 2388 y Fz(1)2619 2373 y FE(\()p FD(R)q FE(\))e FH(is)g(a)h(unitary)f(op)s(erator)147 2494 y(for)34 b(eac)m(h)h FD(R)d FA(2)f FF(SO)p FE(\(3\))p FH(.)49 b(The)35 b(calculation)d(of)i(the)h(previous)g(subsection)g(sho)m(ws)h (that)e(the)h(map)147 2614 y FD(R)29 b FA(!)e FE(\005)450 2629 y Fz(1)490 2614 y FE(\()p FD(R)q FE(\))d FH(is)f(a)h(homomorphism) d(of)j FF(SO)p FE(\(3\))f FH(in)m(to)h FD(U)10 b FE(\()p FA(H)q FE(\))p FH(.)41 b(This)24 b(map)f(is)h(strongly)f(con)m(tin)m (uous,)147 2734 y(and)33 b(hence)h(constitutes)f(a)f(unitary)g (represen)m(tation)h(of)f FF(SO)q FE(\(3\))p FH(.)446 2855 y(Similarly)-8 b(,)37 b(w)m(e)j(ma)m(y)f(consider)h(the)f(unit)g (sphere)i FD(S)2407 2819 y Fz(2)2485 2855 y FA(\032)e Fu(R)2667 2819 y Fz(3)2713 2855 y FH(,)i(with)e(the)g(usual)g(surface) 147 2975 y(measure)f FE(\012)p FH(.)58 b(Of)37 b(course,)i(an)m(y)f FD(R)f FA(2)f FF(SO)p FE(\(3\))h FH(maps)g FD(S)2177 2939 y Fz(2)2253 2975 y FH(in)m(to)g FD(S)2522 2939 y Fz(2)2561 2975 y FH(.)57 b(F)-8 b(or)37 b(eac)m(h)h FD(R)g FH(w)m(e)g(can)g(de\034ne)147 3096 y FE(\005)220 3111 y Fz(2)260 3096 y FE(\()p FD(R)q FE(\))32 b FH(acting)g(on)g FD(L)937 3059 y Fz(2)977 3096 y FE(\()p FD(S)1081 3059 y Fz(2)1120 3096 y FD(;)17 b(d)p FE(\012\))33 b FH(b)m(y)1399 3294 y FE([\005)1499 3309 y Fz(2)1539 3294 y FE(\()p FD(R)q FE(\))p FD(f)11 b FE(])16 b(\()p FD(x)p FE(\))28 b(=)g FD(f)2130 3213 y Fv(\000)2176 3294 y FD(R)2251 3252 y Fw(\000)p Fz(1)2345 3294 y FD(x)2400 3213 y Fv(\001)2463 3294 y FH(.)147 3491 y(Then)34 b FE(\005)475 3506 y Fz(2)547 3491 y FH(is)e(a)g(unitary)g(represen)m(tation)h(of)g FF(SO)p FE(\(3\))p FH(.)446 3612 y(Neither)39 b(of)f(the)h(unitary)f (represen)m(tations)i FE(\005)2198 3627 y Fz(1)2276 3612 y FH(and)f FE(\005)2545 3627 y Fz(2)2623 3612 y FH(is)f(irreducible.)61 b(In)39 b(the)g(case)147 3732 y(of)46 b FE(\005)345 3747 y Fz(2)384 3732 y FH(,)k FD(L)527 3696 y Fz(2)567 3732 y FE(\()p FD(S)671 3696 y Fz(2)710 3732 y FD(;)17 b(d)p FE(\012\))45 b FH(has)i(a)f(v)m(ery)h(nice)f(decomp)s(osition)e(as)i (the)h(orthogonal)d(direct)i(sum)f(of)147 3853 y(\034nite-dimensional) 23 b(in)m(v)-5 b(arian)m(t)24 b(subspaces.)44 b(This)26 b(decomp)s(osition)d(is)j(the)g(theory)h(of)e(\020spherical)147 3973 y(harmonics,\021)32 b(whic)m(h)h(are)f(w)m(ell)g(kno)m(wn)i(in)e (the)g(ph)m(ysics)i(\(and)f(mathematics\))e(literature.)147 4229 y FI(A)37 b(Unitary)g(Represen)m(tation)g(of)g(the)h(Reals)147 4414 y FH(Let)33 b FA(H)c FE(=)e FD(L)604 4378 y Fz(2)644 4414 y FE(\()p Fu(R)5 b FD(;)17 b(dx)p FE(\))p FH(.)49 b(F)-8 b(or)32 b(eac)m(h)h FD(a)28 b FA(2)g Fu(R)5 b FH(,)39 b(de\034ne)34 b FD(T)2050 4429 y Ft(a)2119 4414 y FE(:)28 b 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y(A)n(nalysis)p FH(,)32 b(Theorem)h(9.17\).)p eop %%Page: 98 98 98 97 bop 147 48 a FK(98)2452 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FI(The)38 b(Unitary)f(Represen)m(tations)g(of)g(the)g (Real)g(Heisen)m(b)s(erg)g(Group)147 495 y FH(Consider)c(the)g(Heisen)m (b)s(erg)g(group)1203 846 y FD(H)i FE(=)1423 642 y Fv(8)1423 732 y(<)1423 911 y(:)1512 646 y(0)1512 825 y(@)1640 725 y FE(1)83 b FD(a)k(b)1640 845 y FE(0)d(1)k FD(c)1640 966 y FE(0)c(0)g(1)1997 646 y Fv(1)1997 825 y(A)2100 846 y FA(j)p FD(a;)17 b(b;)g(c)28 b FA(2)g Fu(R)2554 642 y Fv(9)2554 732 y(=)2554 911 y(;)2659 846 y FH(.)147 1197 y(No)m(w)j(consider)g(a)f(real,)g(non-zero)h(constan)m(t,)g(whic)m (h)g(for)f(reasons)h(of)f(historical)e(con)m(v)m(en)m(tion)k(w)m(e)147 1317 y(will)k(call)h Fu(~)p FH(\(\020aitc)m(h-bar\021\).)60 b(No)m(w)39 b(for)f(eac)m(h)h Fu(~)f FA(2)g Fu(R)5 b FA(nf)p FE(0)p FA(g)p FH(,)45 b(de\034ne)40 b(a)e(unitary)g(op)s (erator)g FE(\005)3558 1332 y Fi(~)3639 1317 y FH(on)147 1438 y FD(L)213 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FC(describ)-5 b(e)g(d)35 b(in)i(Se)-5 b(ction)147 4251 y(5.3.)147 4505 y FI(Pro)s(of)147 4690 y FH(Let)29 b FD(\031)k FH(b)s(e)c(an)g(irreducible)f(represen)m(tation)i(of)e FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))35 b FH(acting)28 b(on)h(a)g (\(\034nite-dimensional)c(com-)147 4810 y(plex\))30 b(space)h FD(V)22 b FH(.)42 b(Our)30 b(strategy)h(is)e(to)h(diagonalize)d(the)k (op)s(erator)e FD(\031)t FE(\()p FD(H)8 b FE(\))p FH(.)42 b(Of)30 b(course,)h FC(a)h(priori)p FH(,)147 4930 y(w)m(e)39 b(don't)f(kno)m(w)h(that)e FD(\031)t FE(\()p FD(H)8 b FE(\))37 b FH(is)g(diagonalizable.)56 b(Ho)m(w)m(ev)m(er,)41 b(b)s(ecause)e(w)m(e)g(are)e(w)m(orking)h(o)m(v)m(er)147 5051 y(the)46 b(\(algebraically)c(closed\))j(\034eld)g(of)g(complex)g (n)m(um)m(b)s(ers,)k FD(\031)t FE(\()p FD(H)8 b FE(\))45 b FH(m)m(ust)g(ha)m(v)m(e)i(at)e(least)g(one)147 5171 y(eigen)m(v)m(ector.)446 5292 y(The)34 b(follo)m(wing)29 b(lemma)i(is)h(the)h(k)m(ey)h(to)e(the)h(en)m(tire)f(pro)s(of.)p eop %%Page: 101 101 101 100 bop 147 48 a Fx(The)28 b(Irreducible)f(Represen)n(tations)f(of) h Ff(su)p Fe(\(2\))2003 b FG(101)147 298 y FI(Lemma)33 b(83)49 b FC(L)-5 b(et)35 b FD(u)g FC(b)-5 b(e)34 b(an)h(eigenve)-5 b(ctor)33 b(of)i FD(\031)t FE(\()p FD(H)8 b FE(\))34 b FC(with)h(eigenvalue)e FD(\013)c FA(2)f Fu(C)20 b FC(.)50 b(Then)1320 522 y FD(\031)t FE(\()p FD(H)8 b FE(\))p FD(\031)t FE(\()p FD(X)g FE(\))p FD(u)26 b FE(=)h(\()p FD(\013)c FE(+)f(2\))p FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)p FC(.)147 746 y(Thus)41 b(either)h FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)39 b FE(=)h(0)p FC(,)i(or)g(else)f FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)40 b FC(is)h(an)h(eigenve)-5 b(ctor)40 b(for)h FD(\031)t FE(\()p FD(H)8 b FE(\))41 b FC(with)g(eigenvalue)147 866 y FD(\013)23 b FE(+)f(2)p FC(.)45 b(Similarly,)1344 1091 y FD(\031)t FE(\()p FD(H)8 b FE(\))p FD(\031)t FE(\()p FD(Y)21 b FE(\))p FD(u)27 b FE(=)g(\()p FD(\013)c FA(\000)g FE(2\))p FD(\031)t FE(\()p FD(Y)e FE(\))p FD(u)147 1315 y FC(so)38 b(that)g(either)f FD(\031)t FE(\()p 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FH(As)j(w)m(e)h(ha)m(v)m(e)g(observ)m(ed,)h FD(\031)t FE(\()p FD(H)8 b FE(\))28 b FH(m)m(ust)h(ha)m(v)m(e)h(at)e (least)h(one)g(eigen)m(v)m(ector)g FD(u)g FH(\()p FD(u)e FA(6)p FE(=)g(0)p FH(\),)i(with)147 418 y(some)k(eigen)m(v)-5 b(alue)32 b FD(\013)c FA(2)g Fu(C)20 b FH(.)49 b(By)33 b(the)g(lemma,)1335 661 y FD(\031)t FE(\()p FD(H)8 b FE(\))p FD(\031)t FE(\()p FD(X)g FE(\))p FD(u)26 b FE(=)h(\()p FD(\013)c FE(+)f(2\))p FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)147 904 y FH(and)33 b(more)f(generally)1245 1146 y FD(\031)t FE(\()p FD(H)8 b FE(\))p FD(\031)t FE(\()p FD(X)g FE(\))1693 1105 y Ft(n)1739 1146 y FD(u)27 b FE(=)g(\()p FD(\013)c FE(+)f(2)p FD(n)p FE(\))p FD(\031)t FE(\()p FD(X)8 b FE(\))2515 1105 y Ft(n)2562 1146 y FD(u)p FH(.)147 1389 y(This)34 b(means)g(that)g(either)g FD(\031)t FE(\()p FD(X)8 b FE(\))1386 1353 y Ft(n)1432 1389 y FD(u)30 b FE(=)g(0)p FH(,)k(or)f(else)i FD(\031)t FE(\()p FD(X)8 b FE(\))2265 1353 y Ft(n)2311 1389 y FD(u)34 b FH(is)f(an)h(eigen)m(v)m 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y(non-zero.)147 4867 y FI(Lemma)33 b(84)49 b FC(With)35 b(the)g(ab)-5 b(ove)34 b(notation,)1105 5110 y FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)1385 5125 y Ft(k)1455 5110 y FE(=)27 b([)p FD(k)s(\025)22 b FA(\000)h FD(k)s FE(\()p FD(k)i FA(\000)e FE(1\)])16 b FD(u)2272 5125 y Ft(k)r Fw(\000)p Fz(1)2504 5110 y FE(\()p FD(k)31 b(>)c FE(0\))1075 5255 y FD(\031)21 b FE(\()p FD(X)8 b FE(\))16 b FD(u)1388 5270 y Fz(0)1455 5255 y FE(=)27 b(0)p FD(:)p eop %%Page: 103 103 103 102 bop 147 48 a Fx(The)28 b(Irreducible)f(Represen)n(tations)f(of) h Ff(su)p Fe(\(2\))2003 b FG(103)147 298 y FI(Pro)s(of)37 b(of)h(the)f(lemma)147 482 y FH(W)-8 b(e)35 b(pro)s(ceed)h(b)m(y)g (induction)e(on)g FD(k)s FH(.)51 b(In)35 b(the)g(case)h FD(k)f FE(=)c(1)k FH(w)m(e)g(note)g(that)g FD(u)2937 497 y Fz(1)3008 482 y FE(=)c FD(\031)t FE(\()p FD(Y)21 b FE(\))p FD(u)3384 497 y Fz(0)3423 482 y FH(.)50 b(Using)147 603 y(the)33 b(comm)m(utation)d(relation)h FE([)p FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(;)17 b(\031)t FE(\()p FD(Y)k FE(\)])27 b(=)h FD(\031)t FE(\()p FD(H)8 b FE(\))32 b FH(w)m(e)h(ha)m(v)m(e)891 818 y FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)1171 833 y Fz(1)1237 818 y FE(=)28 b FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(\031)t FE(\()p FD(Y)20 b FE(\))p FD(u)1833 833 y Fz(0)1900 818 y FE(=)28 b(\()p FD(\031)t FE(\()p FD(Y)21 b FE(\))p FD(\031)t FE(\()p FD(X)8 b FE(\))21 b(+)h FD(\031)t FE(\()p FD(H)8 b FE(\)\))16 b FD(u)2932 833 y Fz(0)2971 818 y FH(.)147 1033 y(But)33 b FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)621 1048 y Fz(0)687 1033 y FE(=)28 b(0)p FH(,)k(so)h(w)m(e)g(get)1644 1247 y FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)1924 1262 y Fz(1)1990 1247 y FE(=)28 b FD(\025u)2207 1262 y Fz(0)147 1462 y FH(whic)m(h)33 b(is)f(the)h(lemma)e(in)g(the)i(case)h FD(k)c FE(=)e(1)p FH(.)446 1583 y(No)m(w,)33 b(b)m(y)h(de\034nition)d FD(u)1320 1598 y Ft(k)r Fz(+1)1480 1583 y FE(=)d FD(\031)t FE(\()p FD(Y)21 b FE(\))p FD(u)1853 1598 y Ft(k)1895 1583 y FH(.)43 b(Using)33 b(\(5.9\))f(and)g(induction)g(w)m(e)h(ha)m(v) m(e)847 1797 y FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)1127 1812 y Ft(k)r Fz(+1)1287 1797 y FE(=)27 b FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(\031)t FE(\()p FD(Y)21 b FE(\))p FD(u)1883 1812 y Ft(k)1287 1943 y FE(=)27 b(\()p FD(\031)t FE(\()p FD(Y)21 b FE(\))p FD(\031)t FE(\()p FD(X)8 b FE(\))22 b(+)g FD(\031)t FE(\()p FD(H)8 b FE(\)\))16 b FD(u)2319 1958 y Ft(k)1287 2088 y FE(=)27 b FD(\031)t FE(\()p FD(Y)21 b FE(\))c([)p FD(k)s(\025)22 b FA(\000)h FD(k)s FE(\()p FD(k)i FA(\000)e FE(1\)])16 b FD(u)2334 2103 y Ft(k)r Fw(\000)p Fz(1)2489 2088 y FE(+)22 b(\()p FD(\025)g FA(\000)g FE(2)p FD(k)s FE(\))p FD(u)3000 2103 y Ft(k)1287 2233 y FE(=)27 b([)p FD(k)s(\025)22 b FA(\000)h FD(k)s FE(\()p FD(k)i FA(\000)e FE(1\))f(+)g(\()p FD(\025)g FA(\000)g FE(2)p FD(k)s FE(\)])17 b FD(u)2582 2248 y Ft(k)2624 2233 y FH(.)147 2448 y(Simplifying)29 b(the)k(last)f (expression)h(giv)m(e)g(the)g(Lemma.)42 b Fy(\003)446 2568 y FH(Since)h FD(\031)t FE(\()p FD(H)8 b FE(\))42 b FH(can)h(ha)m(v)m(e)h(only)e(\034nitely)f(man)m(y)i(eigen)m(v)-5 b(alues,)45 b(the)e FD(u)3016 2583 y Ft(k)3058 2568 y FH('s)h(cannot)e(all)f(b)s(e)147 2689 y(non-zero.)j(There)33 b(m)m(ust)g(therefore)g(b)s(e)g(an)f(in)m(teger)h FD(m)28 b FA(\025)g FE(0)k FH(suc)m(h)i(that)1565 2904 y FD(u)1621 2919 y Ft(k)1690 2904 y FE(=)28 b FD(\031)t FE(\()p FD(Y)21 b FE(\))2007 2863 y Ft(k)2050 2904 y FD(u)2106 2919 y Fz(0)2172 2904 y FA(6)p FE(=)28 b(0)147 3119 y FH(for)k(all)f FD(k)f FA(\024)f FD(m)p FH(,)j(but)1437 3333 y FD(u)1493 3348 y Ft(m)p Fz(+1)1677 3333 y FE(=)27 b FD(\031)t FE(\()p FD(Y)22 b FE(\))1994 3292 y Ft(m)p Fz(+1)2150 3333 y FD(u)2206 3348 y Fz(0)2273 3333 y FE(=)27 b(0)p FH(.)446 3548 y(No)m(w)33 b(if)f FD(u)814 3563 y Ft(m)p Fz(+1)998 3548 y FE(=)27 b(0)p FH(,)32 b(then)i(certainly)d FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)2115 3563 y Ft(m)p Fz(+1)2299 3548 y FE(=)27 b(0)p FH(.)44 b(Then)33 b(b)m(y)h(Lemma)d(84,)537 3763 y FE(0)d(=)f FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)997 3778 y Ft(m)p Fz(+1)1181 3763 y FE(=)27 b([\()p FD(m)c FE(+)f(1\))p FD(\025)g FA(\000)g FD(m)p FE(\()p FD(m)h FE(+)f(1\)])16 b FD(u)2335 3778 y Ft(m)2429 3763 y FE(=)28 b(\()p FD(m)22 b FE(+)g(1\)\()p FD(\025)g FA(\000)g FD(m)p FE(\))p FD(u)3258 3778 y Ft(m)3325 3763 y FH(.)147 3978 y(But)32 b FD(u)396 3993 y Ft(m)490 3978 y FA(6)p FE(=)27 b(0)p FH(,)32 b(and)g FD(m)20 b FE(+)g(1)28 b FA(6)p FE(=)f(0)32 b FH(\(since)g FD(m)c FA(\025)g FE(0)p FH(\).)43 b(Th)m(us)33 b(in)e(order)g(to)h(ha)m(v)m(e)h FE(\()p FD(m)20 b FE(+)g(1\)\()p FD(\025)g FA(\000)h FD(m)p FE(\))p FD(u)3676 3993 y Ft(m)147 4098 y FH(equal)33 b(to)f(zero,)h(w)m(e)g(m)m(ust)g(ha)m(v)m(e)h FD(\025)27 b FE(=)h FD(m)p FH(.)446 4219 y(W)-8 b(e)40 b(ha)m(v)m(e)h(made)f (considerable)f(progress.)66 b(Giv)m(en)39 b(a)h(\034nite-dimensional)c (irreducible)147 4339 y(represen)m(tation)g FD(\031)k FH(of)35 b FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(,)42 b(acting)35 b(on)g(a)g(space)i FD(V)21 b FH(,)37 b(there)f(exists)g(an) g(in)m(teger)f FD(m)e FA(\025)h FE(0)h FH(and)147 4459 y(non-zero)e(v)m(ectors)h FD(u)930 4474 y Fz(0)969 4459 y FD(;)17 b FA(\001)g(\001)g(\001)d FD(u)1201 4474 y Ft(m)1300 4459 y FH(suc)m(h)34 b(that)e(\(putting)g FD(\025)g FH(equal)h(to)f FD(m)p FH(\))1084 4674 y FD(\031)t FE(\()p FD(H)8 b FE(\))p FD(u)1364 4689 y Ft(k)1433 4674 y FE(=)28 b(\()p FD(m)22 b FA(\000)h FE(2)p FD(k)s FE(\))p FD(u)1979 4689 y Ft(k)1094 4820 y FD(\031)t FE(\()p FD(Y)e FE(\))p FD(u)1363 4835 y Ft(k)1433 4820 y FE(=)28 b FD(u)1593 4835 y Ft(k)r Fz(+1)1823 4820 y FE(\()p FD(k)i(<)e(m)p FE(\))1070 4965 y FD(\031)t FE(\()p FD(Y)21 b FE(\))p FD(u)1339 4980 y Ft(m)1433 4965 y FE(=)28 b(0)1084 5110 y FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)1364 5125 y Ft(k)1433 5110 y FE(=)28 b([)p FD(k)s(m)22 b FA(\000)h FD(k)s FE(\()p FD(k)i FA(\000)e FE(1\)])16 b FD(u)2279 5125 y Ft(k)r Fw(\000)p Fz(1)2509 5110 y FE(\()p FD(k)31 b(>)c FE(0\))1087 5255 y FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(u)1367 5270 y Fz(0)1433 5255 y FE(=)28 b(0)1907 b FH(\(5.10\))p eop %%Page: 104 104 104 103 bop 147 48 a FK(104)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)446 298 y FH(The)37 b(v)m(ectors)g FD(u)1040 313 y Fz(0)1079 298 y FD(;)17 b FA(\001)g(\001)g(\001)d FD(u)1311 313 y Ft(m)1413 298 y FH(m)m(ust)36 b(b)s(e)g(linearly)d (indep)s(enden)m(t,)38 b(since)e(they)g(are)g(eigen)m(v)m(ec-)147 418 y(tors)j(of)f FD(\031)t FE(\()p FD(H)8 b FE(\))39 b FH(with)f(distinct)g(eigen)m(v)-5 b(alues.)62 b(Moreo)m(v)m(er,)42 b(the)e FE(\()p FD(m)26 b FE(+)h(1\))p FH(-dimensional)35 b(span)k(of)147 539 y FD(u)203 554 y Fz(0)242 539 y FD(;)17 b FA(\001)g(\001)g(\001)e FD(u)475 554 y Ft(m)572 539 y FH(is)31 b(explicitly)f(in)m(v)-5 b(arian)m(t)30 b(under)i FD(\031)t FE(\()p FD(H)8 b FE(\))p FH(,)32 b FD(\031)t FE(\()p FD(X)8 b FE(\))p FH(,)31 b(and)h FD(\031)t FE(\()p FD(Y)21 b FE(\))p FH(,)31 b(and)h(hence)h(under)f FD(\031)t FE(\()p FD(Z)7 b FE(\))147 659 y FH(for)32 b(all)f FD(Z)j FA(2)28 b FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(.)49 b(Since)33 b FD(\031)j FH(is)d(irreducible,)e(this)h(space)i(m)m(ust)e (b)s(e)h(all)e(of)h FD(V)21 b FH(.)446 784 y(W)-8 b(e)35 b(ha)m(v)m(e)h(no)m(w)g(sho)m(wn)g(that)f(ev)m(ery)h(irreducible)e (represen)m(tation)h(of)f FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))41 b FH(is)34 b(of)h(the)147 904 y(form)26 b(\(5.10\).)40 b(It)27 b(remains)f(to)g(sho)m(w)i(that)e(ev)m(erything)i(of)e(the)h (form)f(\(5.10\))g(is)g(a)g(represen)m(tation,)147 1024 y(and)f(that)f(it)g(is)g(irreducible.)39 b(That)25 b(is,)h(if)d(w)m(e)i FC(de\034ne)31 b FD(\031)t FE(\()p FD(H)8 b FE(\))p FH(,)26 b FD(\031)t FE(\()p FD(X)8 b FE(\))p FH(,)26 b(and)e FD(\031)t FE(\()p FD(Y)d FE(\))k FH(b)m(y)g(\(5.10\))f(\(where)147 1145 y(the)30 b FD(u)368 1160 y Ft(k)410 1145 y FH('s)g(are)f(basis)g (elemen)m(ts)h(for)e(some)h FE(\()p FD(m)15 b FE(+)g(1\))p FH(-dimensional)27 b(v)m(ector)j(space\),)h(then)f(w)m(e)g(w)m(an)m(t) 147 1265 y(to)35 b(sho)m(w)i(that)e(they)i(ha)m(v)m(e)g(the)f(righ)m(t) e(comm)m(utation)g(relations)g(to)h(form)f(a)h(represen)m(tation)h(of) 147 1386 y FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(,)38 b(and)33 b(that)f(this)h(represen)m(tation)g(is)f(irreducible.)446 1510 y(The)41 b(computation)c(of)i(the)h(comm)m(utation)e(relations)g (of)h FD(\031)t FE(\()p FD(H)8 b FE(\))p FH(,)40 b FD(\031)t FE(\()p FD(X)8 b FE(\))p FH(,)41 b(and)f FD(\031)t FE(\()p FD(Y)21 b FE(\))39 b FH(is)147 1631 y(straigh)m(tforw)m(ard,)g(and)f (is)g(left)f(as)h(an)g(exercise.)62 b(Note)38 b(that)g(when)h(dealing)d (with)i FD(\031)t FE(\()p FD(Y)21 b FE(\))p FH(,)39 b(y)m(ou)147 1751 y(should)34 b(treat)g(separately)h(the)f(v)m(ectors)i FD(u)1722 1766 y Ft(k)1764 1751 y FH(,)f FD(k)f(<)c(m)p FH(,)35 b(and)f FD(u)2411 1766 y Ft(m)2477 1751 y FH(.)49 b(Irreducibilit)m(y)32 b(is)i(also)f(easy)i(to)147 1871 y(c)m(hec)m(k,)g(b)m(y)f(imitating)28 b(the)33 b(pro)s(of)f(of)g(Prop)s (osition)f(80.)43 b(\(See)33 b(Exercise)h(6.\))446 1996 y(W)-8 b(e)38 b(ha)m(v)m(e)i(no)m(w)e(sho)m(wn)i(that)d(there)i(is)f (an)f(irreducible)g(represen)m(tation)i(of)e FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))44 b FH(in)147 2116 y(eac)m(h)k(dimension)c FD(m)32 b FE(+)g(1)p FH(,)50 b(b)m(y)d(explicitly)d(writing)h(do)m(wn)i (ho)m(w)g FD(H)8 b FH(,)50 b FD(X)8 b FH(,)50 b(and)c FD(Y)68 b FH(should)46 b(act)147 2237 y(\(Equation)33 b(5.10\))f(in)h(a)f(basis.)45 b(But)34 b(w)m(e)g(ha)m(v)m(e)g(sho)m(wn) h(more)d(than)h(this.)45 b(W)-8 b(e)33 b(also)f(ha)m(v)m(e)j(sho)m(wn) 147 2357 y(that)25 b(an)m(y)g FE(\()p FD(m)7 b FE(+)g(1\))p FH(-dimensional)21 b(irreducible)j(represen)m(tation)h(of)g FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))30 b FH(m)m(ust)25 b(b)s(e)h(of)e(the)h(form)147 2478 y(\(5.10\).)72 b(It)43 b(follo)m(ws)e(that)h(an)m(y)h(t)m(w)m(o)g(irreducible)e(represen)m (tations)j(of)e FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))49 b FH(of)41 b(dimension)147 2598 y FE(\()p FD(m)25 b FE(+)g(1\))36 b FH(m)m(ust)h(b)s(e)g(equiv)-5 b(alen)m(t.)55 b(F)-8 b(or)35 b(if)h FD(\031)1742 2613 y Fz(1)1818 2598 y FH(and)h FD(\031)2067 2613 y Fz(2)2143 2598 y FH(are)f(t)m(w)m(o)i(irreducible)d (represen)m(tations)i(of)147 2718 y(dimension)e FE(\()p FD(m)26 b FE(+)e(1\))p FH(,)38 b(acting)e(on)g(spaces)j FD(V)1810 2733 y Fz(1)1885 2718 y FH(and)e FD(V)2136 2733 y Fz(2)2176 2718 y FH(,)g(then)h FD(V)2524 2733 y Fz(1)2600 2718 y FH(has)f(a)f(basis)h FD(u)3163 2733 y Fz(0)3202 2718 y FD(;)17 b FA(\001)g(\001)g(\001)d FD(u)3434 2733 y Ft(m)3537 2718 y FH(as)37 b(in)147 2839 y(\(5.10\))i(and)g FD(V)689 2854 y Fz(2)768 2839 y FH(has)h(a)f (similar)d(basis)42 b Fv(e)-58 b FD(u)1666 2854 y Fz(0)1705 2839 y FD(;)17 b FA(\001)g(\001)g(\001)g Fv(e)-58 b FD(u)1937 2854 y Ft(m)2004 2839 y FH(.)63 b(But)40 b(then)g(the)g(map)e FD(\036)h FE(:)g FD(V)3142 2854 y Fz(1)3221 2839 y FA(!)g FD(V)3417 2854 y Fz(2)3496 2839 y FH(whic)m(h)147 2959 y(sends)34 b FD(u)464 2974 y Ft(k)539 2959 y FH(to)h Fv(e)-58 b FD(u)714 2974 y Ft(k)789 2959 y FH(will)30 b(b)s(e)j(an)f(isomorphism)e(of)i(represen)m(tations.)45 b(\(Think)33 b(ab)s(out)f(it.\))446 3084 y(In)45 b(particular,)i(the)e FE(\()p FD(m)31 b FE(+)g(1\))p FH(-dimensional)41 b(represen)m(tation) 46 b FD(\031)2887 3099 y Ft(m)2999 3084 y FH(describ)s(ed)f(in)g(Sec-) 147 3204 y(tion)35 b(5.3)h(m)m(ust)g(b)s(e)g(equiv)-5 b(alen)m(t)35 b(to)h(\(5.10\).This)g(can)g(b)s(e)g(seen)h(explicitly)d (b)m(y)j(in)m(tro)s(ducing)e(the)147 3325 y(follo)m(wing)30 b(basis)i(for)g FD(V)1004 3340 y Ft(m)1071 3325 y FH(:)723 3601 y FD(u)779 3616 y Ft(k)849 3601 y FE(=)27 b([)p FD(\031)1034 3616 y Ft(m)1101 3601 y FE(\()p FD(Y)21 b FE(\)])1282 3553 y Ft(k)1342 3601 y FE(\()p FD(z)1429 3560 y Ft(m)1425 3626 y Fz(2)1496 3601 y FE(\))28 b(=)f(\()p FA(\000)p FE(1\))1867 3560 y Ft(k)2046 3534 y FD(m)p FE(!)p 1920 3578 365 4 v 1920 3670 a(\()p FD(m)22 b FA(\000)h FD(k)s FE(\)!)2294 3601 y FD(z)2343 3560 y Ft(k)2339 3626 y Fz(1)2386 3601 y FD(z)2435 3560 y Ft(m)p Fw(\000)p Ft(k)2431 3626 y Fz(2)2791 3601 y 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FI(Prop)s(osition)k(87)48 b FC(A)33 b(\034nite-dimensional)e(c) -5 b(ompletely)32 b(r)-5 b(e)g(ducible)32 b(r)-5 b(epr)g(esentation)31 b(of)i(a)f(gr)-5 b(oup)147 4121 y(or)38 b(Lie)g(algebr)-5 b(a)37 b(is)h(e)-5 b(quivalent)37 b(to)h(a)g(dir)-5 b(e)g(ct)38 b(sum)g(of)f(\(one)h(or)f(mor)-5 b(e\))38 b(irr)-5 b(e)g(ducible)37 b(r)-5 b(epr)g(esenta-)147 4242 y(tions.)147 4504 y FI(Pro)s(of)147 4689 y FH(The)46 b(pro)s(of)d(is)h(b)m(y)i(induction)d(on)i(the)f (dimension)g(of)g(the)g(space)i FD(V)22 b FH(.)79 b(If)44 b FE(dim)16 b FD(V)69 b FE(=)48 b(1)p FH(,)f(then)147 4810 y(automatically)42 b(the)j(represen)m(tation)h(is)e(irreducible,)j (since)f(then)f FD(V)67 b FH(is)45 b(has)g(no)g(non-trivial)147 4930 y(subspaces,)39 b(let)c(alone)f(non-trivial)e(in)m(v)-5 b(arian)m(t)34 b(subspaces.)55 b(Th)m(us)37 b FD(V)57 b FH(is)35 b(a)g(direct)g(sum)g(of)g(irre-)147 5050 y(ducible)d (represen)m(tations,)i(with)e(just)h(one)g(summand,)f(namely)f FD(V)54 b FH(itself.)446 5171 y(Supp)s(ose,)34 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4832 y(\014)1509 4891 y(\014)1509 4951 y(\014)1542 5015 y Ft(t)p Fz(=0)1679 4916 y FE(\005)1752 4931 y Fz(1)1791 4916 y FE(\()p FD(e)1874 4875 y Ft(tX)1967 4916 y FE(\))p FD(u)f FA(\012)g FE(\005)2255 4931 y Fz(2)2295 4916 y FE(\()p FD(e)2378 4875 y Ft(tY)2464 4916 y FE(\))p FD(v)1290 5193 y FE(=)1393 5052 y Fv(\022)1504 5125 y FD(d)p 1486 5170 V 1486 5261 a(dt)1582 5048 y Fv(\014)1582 5108 y(\014)1582 5168 y(\014)1582 5228 y(\014)1616 5292 y Ft(t)p Fz(=0)1752 5193 y FE(\005)1825 5208 y Fz(1)1865 5193 y FE(\()p FD(e)1948 5152 y Ft(tX)2041 5193 y FE(\))p FD(u)2135 5052 y Fv(\023)2229 5193 y FA(\012)h FD(v)j FE(+)c FD(u)g FA(\012)2677 5052 y Fv(\022)2788 5125 y FD(d)p 2771 5170 V 2771 5261 a(dt)2866 5048 y Fv(\014)2866 5108 y(\014)2866 5168 y(\014)2866 5228 y(\014)2900 5292 y Ft(t)p Fz(=0)3036 5193 y FE(\005)3109 5208 y Fz(2)3149 5193 y FE(\()p FD(e)3232 5152 y Ft(tY)3318 5193 y FE(\))p FD(v)3407 5052 y Fv(\023)3497 5193 y FH(.)p eop %%Page: 113 113 113 112 bop 147 48 a Fx(T)-7 b(ensor)27 b(Pro)r(ducts)g(of)h(Represen)n (tations)2173 b FG(113)147 298 y FH(This)28 b(sho)m(ws)h(that)e FD(\031)901 313 y Fz(1)953 298 y FA(\012)12 b FD(\031)1097 313 y Fz(2)1137 298 y FE(\()p FD(X)r(;)17 b(Y)k FE(\))28 b(=)f FD(\031)1604 313 y Fz(1)1644 298 y FE(\()p FD(X)8 b FE(\))k FA(\012)g FD(I)20 b FE(+)12 b FD(I)19 b FA(\012)12 b FD(\031)2267 313 y Fz(2)2307 298 y FE(\()p FD(Y)22 b FE(\))27 b FH(on)g(elemen)m(ts)h(of)f(the)h(form)e FD(u)12 b FA(\012)g FD(v)t FH(,)147 418 y(and)33 b(therefore)g(on)f (the)h(whole)g(space)g FD(U)g FA(\012)23 b FD(V)e FH(.)44 b Fy(\003)147 590 y FI(De\034nition)36 b(96)49 b FC(L)-5 b(et)32 b Fk(g)h FC(and)e Fk(h)h FC(b)-5 b(e)32 b(Lie)f(algebr)-5 b(as,)32 b(and)f(let)h FD(\031)2406 605 y Fz(1)2478 590 y FC(and)g FD(\031)2720 605 y Fz(2)2791 590 y FC(b)-5 b(e)32 b(r)-5 b(epr)g(esentations)31 b(of)h Fk(g)147 710 y FC(and)37 b Fk(h)p FC(,)g(acting)g(on)f(sp)-5 b(ac)g(es)36 b FD(U)48 b FC(and)37 b FD(V)21 b FC(.)52 b(Then)36 b(the)h FB(tensor)44 b(pr)-6 b(o)g(duct)39 b FC(of)e FD(\031)2990 725 y Fz(1)3067 710 y FC(and)g FD(\031)3314 725 y Fz(2)3353 710 y FC(,)h(denote)-5 b(d)147 830 y FD(\031)202 845 y Fz(1)264 830 y FA(\012)23 b FD(\031)419 845 y Fz(2)459 830 y FC(,)34 b(is)h(a)f(r)-5 b(epr)g(esentation)34 b(of)h Fk(g)22 b FA(\010)h Fk(h)35 b FC(acting)f(on)h FD(U)d FA(\012)23 b FD(V)e FC(,)35 b(given)f(by)1096 1000 y FD(\031)1151 1015 y Fz(1)1213 1000 y FA(\012)22 b FD(\031)1367 1015 y Fz(2)1407 1000 y FE(\()p FD(X)r(;)17 b(Y)k FE(\))28 b(=)g FD(\031)1875 1015 y Fz(1)1914 1000 y FE(\()p FD(X)8 b FE(\))22 b FA(\012)h FD(I)30 b FE(+)22 b FD(I)30 b FA(\012)22 b FD(\031)2599 1015 y Fz(2)2639 1000 y FE(\()p FD(Y)f FE(\))147 1171 y FC(for)35 b(al)5 b(l)34 b FD(X)i FA(2)28 b Fk(g)35 b FC(and)f FD(Y)49 b FA(2)28 b Fk(h)p FD(:)147 1423 y FI(Remarks)147 1608 y FH(It)40 b(is)g(easy)h(to)e(c)m (hec)m(k)k(that)c(this)h(indeed)g(de\034nes)i(a)e(represen)m(tation)g (of)g Fk(g)27 b FA(\010)h Fk(h)p FH(.)65 b(Note)40 b(that)g(if)147 1728 y(w)m(e)k(de\034ned)h FD(\031)703 1743 y Fz(1)772 1728 y FA(\012)30 b FD(\031)934 1743 y Fz(2)973 1728 y FE(\()p FD(X)r(;)17 b(Y)22 b FE(\))45 b(=)h FD(\031)1477 1743 y Fz(1)1516 1728 y FE(\()p FD(X)8 b FE(\))29 b FA(\012)h FD(\031)1872 1743 y Fz(2)1912 1728 y FE(\()p FD(Y)21 b FE(\))p FH(,)46 b(this)d(w)m(ould)g FC(not)52 b FH(b)s(e)43 b(a)g(represn)m(tation)g(of)147 1849 y Fk(g)16 b FA(\010)g Fk(h)p FH(,)31 b(for)e(this)g(is)g(not)g(ev)m(en)j(a)d(linear)f(map.)41 b(\(E.g.,)31 b(w)m(e)f(w)m(ould)g(then)g(ha)m(v)m(e)h FD(\031)3016 1864 y Fz(1)3072 1849 y FA(\012)16 b FD(\031)3220 1864 y Fz(2)3260 1849 y FE(\(2)p FD(X)r(;)h FE(2)p FD(Y)k FE(\))27 b(=)147 1969 y(4)p FD(\031)251 1984 y Fz(1)309 1969 y FA(\012)19 b FD(\031)460 1984 y Fz(2)500 1969 y FE(\()p FD(X)r(;)e(Y)k FE(\))p FH(!\))43 b(Note)30 b(also)g(that)g(the)i(ab)s(o)m(v)m(e)f(de\034nition)f(applies)f(ev)m (en)k(if)c FD(\031)3084 1984 y Fz(1)3154 1969 y FH(and)i FD(\031)3397 1984 y Fz(2)3468 1969 y FH(do)f(not)147 2089 y(come)i(from)g(a)g(represen)m(tation)h(of)f(an)m(y)h(matrix)e (Lie)h(group.)147 2261 y FI(De\034nition)k(97)49 b FC(L)-5 b(et)41 b FD(G)g FC(b)-5 b(e)40 b(a)g(matrix)g(Lie)h(gr)-5 b(oup,)42 b(and)e(let)g FE(\005)2548 2276 y Fz(1)2629 2261 y FC(and)g FE(\005)2897 2276 y Fz(2)2977 2261 y FC(b)-5 b(e)40 b(r)-5 b(epr)g(esentations)147 2381 y(of)46 b FD(G)p FC(,)i(acting)e(on)f(sp)-5 b(ac)g(es)45 b FD(V)1243 2396 y Fz(1)1328 2381 y FC(and)h FD(V)1586 2396 y Fz(2)1625 2381 y FC(.)78 b(Then)45 b(the)h FB(tensor)54 b(pr)-6 b(o)g(duct)49 b FC(of)c FE(\005)3132 2396 y Fz(1)3218 2381 y FC(and)g FE(\005)3491 2396 y Fz(2)3577 2381 y FC(is)g(a)147 2501 y(r)-5 b(epr)g(esentation)34 b(of)h FD(G)g FC(acting)f(on)g FD(V)1496 2516 y Fz(1)1558 2501 y FA(\012)22 b FD(V)1714 2516 y Fz(2)1789 2501 y FC(de\034ne)-5 b(d)33 b(by)1309 2672 y FE(\005)1382 2687 y Fz(1)1443 2672 y FA(\012)23 b FE(\005)1616 2687 y Fz(2)1656 2672 y FE(\()p FD(A)p FE(\))k(=)h(\005)2009 2687 y Fz(1)2048 2672 y FE(\()p FD(A)p FE(\))22 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a(dt)1892 3989 y Fv(\014)1892 4048 y(\014)1892 4108 y(\014)1892 4168 y(\014)1926 4232 y Ft(t)p Fz(=0)2062 4133 y FE(\005)2135 4148 y Fz(1)2191 4052 y Fv(\000)2237 4133 y FD(e)2282 4092 y Ft(tX)2375 4052 y Fv(\001)2437 4133 y FD(u)22 b FA(\012)g FE(\005)2687 4148 y Fz(2)2744 4052 y Fv(\000)2789 4133 y FD(e)2834 4092 y Ft(tX)2927 4052 y Fv(\001)2990 4133 y FD(v)1673 4349 y FE(=)28 b FD(\031)1832 4364 y Fz(1)1888 4349 y FE(\()p FD(X)8 b FE(\))16 b FD(u)22 b FA(\012)h FD(v)i FE(+)d FD(v)k FA(\012)d FD(\031)2645 4364 y Fz(2)2701 4349 y FE(\()p FD(X)8 b FE(\))17 b FD(u)p FH(.)147 4520 y(This)33 b(is)f(what)h(w)m(e)g(w)m(an)m(ted)h(to)e(sho)m (w.)45 b Fy(\003)147 4691 y FI(De\034nition)36 b(99)49 b FC(If)39 b Fk(g)h FC(is)g(a)f(Lie)h(algebr)-5 b(a,)40 b(and)f FD(\031)2018 4706 y Fz(1)2097 4691 y FC(and)g FD(\031)2346 4706 y Fz(2)2426 4691 y FC(ar)-5 b(e)39 b(r)-5 b(epr)g(esentations)39 b(of)h Fk(g)g FC(acting)147 4811 y(on)35 b(sp)-5 b(ac)g(es)35 b FD(V)639 4826 y Fz(1)714 4811 y FC(and)f FD(V)960 4826 y Fz(2)1000 4811 y FC(,)h(then)g(the)h FB(tensor)42 b(pr)-6 b(o)g(duct)38 b FC(of)d FD(\031)2355 4826 y Fz(1)2431 4811 y FC(and)f FD(\031)2675 4826 y Fz(2)2751 4811 y FC(is)h(a)g(r)-5 b(epr)g(esentation)35 b(of)g Fk(g)147 4932 y FC(acting)g(on)f(the)h(sp)-5 b(ac)g(e)34 b FD(V)1051 4947 y Fz(1)1112 4932 y FA(\012)23 b FD(V)1269 4947 y Fz(2)1343 4932 y FC(de\034ne)-5 b(d)34 b(by)1149 5102 y FD(\031)1204 5117 y Fz(1)1266 5102 y FA(\012)22 b FD(\031)1420 5117 y Fz(2)1460 5102 y FE(\()p FD(X)8 b FE(\))28 b(=)f FD(\031)1811 5117 y Fz(1)1851 5102 y FE(\()p FD(X)8 b FE(\))22 b FA(\012)g FD(I)30 b FE(+)22 b FD(I)30 b FA(\012)23 b FD(\031)2536 5117 y Fz(2)2576 5102 y FE(\()p FD(X)8 b FE(\))147 5272 y FC(for)35 b(al)5 b(l)34 b FD(X)i FA(2)28 b Fk(g)p FD(:)p eop %%Page: 114 114 114 113 bop 147 48 a FK(114)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FI(Remarks)147 493 y FH(It)32 b(is)f(easy)h(to)f(c)m (hec)m(k)j(that)d FE(\005)1225 508 y Fz(1)1284 493 y FA(\012)21 b FE(\005)1455 508 y Fz(2)1526 493 y FH(and)31 b FD(\031)1769 508 y Fz(1)1829 493 y FA(\012)20 b FD(\031)1981 508 y Fz(2)2052 493 y FH(are)32 b(actually)e(represen)m(tations)i(of)f FD(G)g FH(and)h Fk(g)p FH(,)147 613 y(resp)s(ectiv)m(ely)-8 b(.)52 b(There)37 b(is)d(some)h(am)m(biguit)m(y)f(in)g(the)i(notation,) e(sa)m(y)-8 b(,)37 b FE(\005)2780 628 y Fz(1)2843 613 y FA(\012)25 b FE(\005)3018 628 y Fz(2)3057 613 y FH(.)51 b(F)-8 b(or)35 b(ev)m(en)h(if)e FE(\005)3702 628 y Fz(1)147 734 y FH(and)27 b FE(\005)404 749 y Fz(2)470 734 y FH(are)g(b)s(oth)g (represen)m(tations)h(of)e(the)h(same)g(group)f FD(G)p FH(,)i(w)m(e)g(could)f(still)d(regard)j FE(\005)3378 749 y Fz(1)3428 734 y FA(\012)10 b FE(\005)3588 749 y Fz(2)3655 734 y FH(as)147 854 y(a)30 b(represen)m(tation)g(of)g FD(G)17 b FA(\002)g FD(G)p FH(,)31 b(b)m(y)f(taking)f FD(H)36 b FE(=)27 b FD(G)j FH(in)f(de\034nition)g(94.)42 b(W)-8 b(e)31 b(will)c(rely)j(on)g(con)m(text)147 975 y(to)h(mak)m(e)g(clear)g(whether)h(w)m(e)g(are)f(thinking)f(of)h FE(\005)1994 990 y Fz(1)2053 975 y FA(\012)19 b FE(\005)2222 990 y Fz(2)2293 975 y FH(as)31 b(a)g(represen)m(tation)h(of)e FD(G)19 b FA(\002)h FD(G)31 b FH(or)g(as)147 1095 y(represen)m(tation)i (of)f FD(G)p FH(.)446 1221 y(Supp)s(ose)43 b FE(\005)911 1236 y Fz(1)993 1221 y FH(and)g FE(\005)1266 1236 y Fz(2)1347 1221 y FH(are)g FC(irr)-5 b(e)g(ducible)49 b FH(represen)m(tations)43 b(of)f(a)g(group)g FD(G)p FH(.)73 b(If)42 b(w)m(e)h(re-)147 1341 y(gard)37 b FE(\005)447 1356 y Fz(1)511 1341 y FA(\012)26 b FE(\005)687 1356 y Fz(2)763 1341 y FH(as)37 b(a)g(represen)m(tation)g (of)f FD(G)p FH(,)i(it)e(ma)m(y)h(no)f(longer)g(b)s(e)h(irreducible.)55 b(If)37 b(it)f(is)g(not)147 1461 y(irreducible,)31 b(one)g(can)h (attempt)e(to)h(decomp)s(ose)h(it)e(as)i(a)f(direct)g(sum)g(of)g (irreducible)f(represen-)147 1582 y(tations.)56 b(This)37 b(pro)s(cess)h(is)f(called)e FI(Clebsc)m(h-Gordan)j FH(theory)-8 b(.)57 b(In)37 b(the)h(case)g(of)e FF(SU)q FE(\(2\))p FH(,)h(this)147 1702 y(theory)29 b(is)e(relativ)m(ely)g(simple.)40 b(\(In)28 b(the)h(ph)m(ysics)g(literature,)e(the)i(problem)d(of)i (analyzing)e(tensor)147 1823 y(pro)s(ducts)k(of)f(represen)m(tations)h (of)f FF(SU)q FE(\(2\))g FH(is)g(called)f(\020addition)f(of)i(angular)f (momen)m(tum.\021\))40 b(See)147 1943 y(Exercise)34 b(15.)147 2224 y FI(5.7)112 b(Sc)m(h)m(ur's)38 b(Lemma)147 2397 y FH(Let)j FE(\005)g FH(and)g FE(\006)g FH(b)s(e)g(represen)m(tations)g (of)g(a)f(matrix)f(Lie)h(group)h FD(G)p FH(,)h(acting)e(on)h(spaces)h FD(V)62 b FH(and)147 2517 y FD(W)14 b FH(.)43 b(Recall)30 b(that)h(a)h FI(morphism)26 b FH(of)31 b(represen)m(tations)h(is)f(a)h (linear)e(map)h FD(\036)c FE(:)h FD(V)49 b FA(!)27 b FD(W)45 b FH(with)32 b(the)147 2637 y(prop)s(ert)m(y)h(that)1419 2873 y FD(\036)17 b FE(\(\005\()p FD(A)p FE(\))p FD(v)t FE(\))27 b(=)h(\006\()p FD(A)p FE(\))17 b(\()p FD(\036)p FE(\()p FD(v)t FE(\)\))147 3109 y FH(for)33 b(all)e FD(v)h FA(2)d FD(V)55 b FH(and)33 b(all)e FD(A)e FA(2)g FD(G)p FH(.)46 b(Sc)m(h)m(ur's)35 b(Lemma)c(is)i(an)g(extremely)g(imp)s(ortan) m(t)e(result)j(whic)m(h)147 3229 y(tells)g(us)i(ab)s(out)e(morphisms)g (of)g(irreducible)g(represen)m(tations.)52 b(P)m(art)35 b(of)f(Sc)m(h)m(ur's)j(Lemma)d(ap-)147 3349 y(plies)28 b(to)h(b)s(oth)g(real)f(and)h(complex)g(represen)m(tations,)i(but)e (part)g(of)f(it)g(applies)g(only)h(to)g(complex)147 3470 y(represen)m(tations.)446 3595 y(It)f(is)f(desirable)g(to)g(b)s(e)h (able)f(to)g(state)h(Sc)m(h)m(ur's)i(lemma)25 b(sim)m(ultaneously)h (for)h(groups)h(and)147 3716 y(Lie)35 b(algebras.)51 b(In)36 b(order)f(to)g(do)g(so,)i(w)m(e)f(need)h(to)e(indulge)f(in)g(a) h(common)f(abuse)i(of)f(notation.)147 3836 y(If,)45 b(sa)m(y)-8 b(,)46 b FE(\005)c FH(is)g(a)g(represen)m(tation)h(of)f FD(G)g FH(acting)g(on)g(a)g(space)i FD(V)21 b FH(,)45 b(w)m(e)e(will)d(refer)j(to)f FD(V)64 b FH(as)43 b(the)147 3957 y(represen)m(tation,)33 b(without)f(explicit)g(reference)i(to)e FE(\005)p FH(.)147 4227 y FI(Theorem)j(100)j(\(Sc)m(h)m(ur's)f(Lemma\)) 159 b FC(1.)49 b(L)-5 b(et)27 b FD(V)49 b FC(and)26 b FD(W)41 b FC(b)-5 b(e)27 b(irr)-5 b(e)g(ducible)26 b(r)-5 b(e)g(al)26 b(or)h(c)-5 b(omplex)391 4347 y(r)g(epr)g(esentations)37 b(of)g(a)h(gr)-5 b(oup)38 b(or)g(Lie)f(algebr)-5 b(a,)38 b(and)f(let)h FD(\036)33 b FE(:)g FD(V)55 b FA(!)33 b FD(W)51 b FC(b)-5 b(e)38 b(a)g(morphism.)391 4467 y(Then)c(either)h FD(\036)27 b FE(=)h(0)34 b FC(or)h FD(\036)g FC(is)f(an)h(isomorphism.) 263 4697 y(2.)48 b(L)-5 b(et)39 b FD(V)61 b FC(b)-5 b(e)39 b(an)g(irr)-5 b(e)g(ducible)38 b(c)-5 b(omplex)38 b(r)-5 b(epr)g(esentation)38 b(of)h(a)g(gr)-5 b(oup)39 b(or)g(Lie)g(algebr)-5 b(a,)39 b(and)391 4817 y(let)c FD(\036)27 b FE(:)h FD(V)49 b FA(!)28 b FD(V)56 b FC(b)-5 b(e)35 b(a)f(morphism)g(of)g FD(V)57 b FC(with)34 b(itself.)45 b(Then)34 b FD(\036)27 b FE(=)h FD(\025I)8 b FC(,)34 b(for)h(some)f FD(\025)27 b FA(2)h Fu(C)20 b FC(.)263 5047 y(3.)48 b(L)-5 b(et)36 b FD(V)58 b FC(and)35 b FD(W)50 b FC(b)-5 b(e)36 b(irr)-5 b(e)g(ducible)35 b(c)-5 b(omplex)35 b(r)-5 b(epr)g(esentations)35 b(of)h(a)f(gr)-5 b(oup)36 b(or)g(Lie)g(algebr)-5 b(a,)391 5167 y(and)44 b(let)g FD(\036)794 5182 y Fz(1)833 5167 y FD(;)17 b(\036)935 5182 y Fz(2)1018 5167 y FE(:)45 b FD(V)67 b FA(!)44 b FD(W)58 b FC(b)-5 b(e)43 b(non-zer)-5 b(o)43 b(morphisms.)71 b(Then)43 b FD(\036)2928 5182 y Fz(1)3012 5167 y FE(=)i FD(\025\036)3248 5182 y Fz(2)3287 5167 y FC(,)h(for)e(some)391 5288 y FD(\025)28 b FA(2)g Fu(C)20 b FC(.)p eop %%Page: 115 115 115 114 bop 147 48 a Fx(Sc)n(h)n(ur's)27 b(Lemma)2915 b FG(115)147 298 y FI(Corollary)36 b(101)49 b FC(L)-5 b(et)31 b FE(\005)f FC(b)-5 b(e)31 b(an)f(irr)-5 b(e)g(ducible)30 b(c)-5 b(omplex)29 b(r)-5 b(epr)g(esentation)30 b(of)g(a)h(matrix)f (Lie)g(gr)-5 b(oup)147 418 y FD(G)p FC(.)42 b(If)25 b FD(A)h FC(is)f(in)g(the)h(c)-5 b(enter)25 b(of)h FD(G)p FC(,)h(then)f FE(\005\()p FD(A)p FE(\))h(=)h FD(\025I)8 b FC(.)41 b(Similarly,)27 b(if)e FD(\031)30 b FC(is)25 b(an)g(irr)-5 b(e)g(ducible)25 b(c)-5 b(omplex)147 539 y(r)g(epr)g(esentation)35 b(of)h(a)g(Lie)g(algebr)-5 b(a)35 b Fk(g)p FC(,)h(and)g(if)g FD(X)43 b FC(is)36 b(in)g(the)g(c)-5 b(enter)36 b(of)f Fk(g)i FC(\(i.e.,)e FE([)p FD(X)r(;)17 b(Y)22 b FE(])30 b(=)f(0)36 b FC(for)147 659 y(al)5 b(l)35 b FD(Y)49 b FA(2)28 b Fk(g)p FC(\),)34 b(then)h FD(\031)t FE(\()p FD(X)8 b FE(\))27 b(=)h FD(\025I)8 b FC(.)147 861 y FI(Corollary)36 b(102)49 b FC(A)n(n)28 b(irr)-5 b(e)g(ducible)28 b(c)-5 b(omplex)28 b(r)-5 b(epr)g(esentation) 28 b(of)g(a)h(c)-5 b(ommutative)28 b(gr)-5 b(oup)29 b(or)f(Lie)147 981 y(algebr)-5 b(a)34 b(is)h(one-dimensionsal.)147 1238 y FI(Pro)s(of)i(of)h(Corollary)e(1)147 1422 y FH(W)-8 b(e)32 b(pro)m(v)m(e)g(the)g(group)e(case;)j(the)f(pro)s(of)e(of)g(the) i(Lie)e(algebra)g(case)i(is)f(the)h(same.)42 b(If)31 b FD(A)h FH(is)e(in)h(the)147 1543 y(cen)m(ter)j(of)e FD(G)p FH(,)h(then)g(for)f(all)e FD(B)j FA(2)28 b FD(G)p FH(,)983 1740 y FE(\005\()p FD(A)p FE(\)\005\()p FD(B)5 b FE(\))28 b(=)f(\005\()p FD(AB)5 b FE(\))28 b(=)g(\005\()p FD(B)5 b(A)p FE(\))28 b(=)f(\005\()p FD(B)5 b FE(\)\005\()p FD(A)p FE(\))p FH(.)147 1936 y(But)41 b(this)g(sa)m(ys)i(exactly)e (that)g FE(\005\()p FD(A)p FE(\))g FH(is)g(a)f(morphism)g(of)g FE(\005)h FH(with)g(itself.)68 b(So)41 b(b)m(y)h(P)m(oin)m(t)f(2)f(of) 147 2057 y(Sc)m(h)m(ur's)35 b(lemma,)30 b FE(\005\()p FD(A)p FE(\))j FH(is)f(a)g(m)m(ultiple)e(of)i(the)h(iden)m(tit)m(y)-8 b(.)43 b Fy(\003)147 2313 y FI(Pro)s(of)37 b(of)h(Corollary)e(2)147 2498 y FH(Again,)30 b(w)m(e)h(pro)m(v)m(e)g(only)e(the)h(group)g(case.) 44 b(If)30 b FD(G)f FH(is)h(comm)m(utativ)m(e,)f(then)i(the)f(cen)m (ter)h(of)f FD(G)g FH(is)f(all)147 2618 y(of)k FD(G)p FH(,)g(so)g(b)m(y)h(the)g(previous)f(corollary)e FE(\005\()p FD(A)p FE(\))i FH(is)g(a)g(m)m(ultiple)d(of)j(the)g(iden)m(tit)m(y)g (for)f(eac)m(h)i FD(A)29 b FA(2)g FD(G)p FH(.)147 2739 y(But)41 b(this)g(means)g(that)g FC(every)50 b FH(subspace)43 b(of)e FD(V)62 b FH(is)41 b(in)m(v)-5 b(arian)m(t!)68 b(Th)m(us)42 b(the)g(only)e(w)m(a)m(y)j(that)e FD(V)147 2859 y FH(can)35 b(fail)e(to)i(ha)m(v)m(e)h(a)f(non-trivial)d(in)m(v)-5 b(arian)m(t)33 b(subspace)k(is)d(for)h(it)f(not)h(to)f(ha)m(v)m(e)i(an) m(y)g(non-trivial)147 2979 y(subspaces.)44 b(This)26 b(means)g(that)g FD(V)48 b FH(m)m(ust)26 b(b)s(e)g(one-dimensional.)38 b(\(Recall)25 b(that)h(w)m(e)h(do)f(not)g(allo)m(w)147 3100 y FD(V)54 b FH(to)32 b(b)s(e)h(zero-dimensional.\))41 b Fy(\003)147 3356 y FI(Pro)s(of)c(of)h(Sc)m(h)m(ur's)g(Lemma)147 3541 y FH(As)31 b(usual,)f(w)m(e)h(will)d(pro)m(v)m(e)j(just)f(the)g (group)g(case;)i(the)e(pro)s(of)f(of)h(the)g(Lie)g(algebra)e(case)j (requires)147 3661 y(only)h(the)h(ob)m(vious)g(notational)d(c)m (hanges.)446 3782 y FC(Pr)-5 b(o)g(of)38 b(of)h(1)p FH(.)55 b(Sa)m(ying)37 b(that)f FD(\036)h FH(is)f(a)g(morphism)f(means)i FD(\036)p FE(\(\005\()p FD(A)p FE(\))p FD(v)t FE(\))d(=)h(\006\()p FD(A)p FE(\))17 b(\()p FD(\036)p FE(\()p FD(v)t FE(\)\))36 b FH(for)147 3902 y(all)31 b FD(v)g FA(2)d FD(V)54 b FH(and)33 b(all)d FD(A)e FA(2)g FD(G)p FH(.)44 b(No)m(w)33 b(supp)s(ose)g(that)g FD(v)e FA(2)d FE(k)m(er)q(\()p FD(\036)p FE(\))p FH(.)43 b(Then)1370 4099 y FD(\036)p FE(\(\005\()p FD(A)p FE(\))p FD(v)t FE(\))27 b(=)h(\006\()p FD(A)p FE(\))p FD(\036)p FE(\()p FD(v)t FE(\))f(=)h(0)p FH(.)147 4296 y(This)33 b(sho)m(ws)i(that)e FE(k)m(er)17 b FD(\036)33 b FH(is)g(an)g(in)m(v)-5 b(arian)m(t)31 b(subspace)k(of)d FD(V)22 b FH(.)45 b(Since)33 b FD(V)54 b FH(is)33 b(irreducible,)f(w)m(e)i(m)m(ust)147 4416 y(ha)m(v)m(e)g FE(k)m(er)18 b FD(\036)27 b FE(=)h(0)k FH(or)g FE(k)m(er)18 b FD(\036)27 b FE(=)h FD(V)21 b FH(.)44 b(Th)m(us)34 b FD(\036)e FH(is)g(either)g(one-to-one)g(or)g (zero.)446 4537 y(Supp)s(ose)39 b FD(\036)e FH(is)g(one-to-one.)57 b(Then)39 b(the)f(image)d(of)i FD(\036)g FH(is)g(a)h(non-zero)f (subspace)i(of)e FD(W)14 b FH(.)147 4657 y(On)35 b(the)g(other)f(hand,) h(the)g(image)e(of)h FD(\036)g FH(is)g(in)m(v)-5 b(arian)m(t,)34 b(for)g(if)f FD(w)g FA(2)f FD(W)48 b FH(is)34 b(of)g(the)h(form)e FD(\036)p FE(\()p FD(v)t FE(\))h FH(for)147 4777 y(some)f FD(v)e FA(2)d FD(V)22 b FH(,)32 b(then)1249 4974 y FE(\006\()p FD(A)p FE(\))p FD(w)e FE(=)e(\006\()p FD(A)p FE(\))p FD(\036)p FE(\()p FD(v)t FE(\))f(=)h FD(\036)p FE(\(\005\()p FD(A)p FE(\))p FD(v)t FE(\))p FH(.)147 5171 y(Since)22 b FD(W)35 b FH(is)22 b(irreducible)e(and)i(image)p FE(\()p FD(V)d FE(\))j FH(is)f(non-zero)h(and)g(in)m(v)-5 b(arian)m(t,)22 b(w)m(e)h(m)m(ust)f(ha)m(v)m(e)h(image)p FE(\()p FD(V)d FE(\))27 b(=)147 5292 y FD(W)14 b FH(.)43 b(Th)m(us)34 b FD(\036)f FH(is)f(either)g(zero)h(or)f(one-to-one)g(and)h(on)m(to.)p eop %%Page: 116 116 116 115 bop 147 48 a FK(116)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)446 298 y FC(Pr)-5 b(o)g(of)33 b(of)f(2)p FH(.)43 b(Supp)s(ose)31 b(no)m(w)g(that)f FD(V)52 b FH(is)30 b(an)g(irreducible)f FC(c)-5 b(omplex)41 b FH(represen)m(tation,)32 b(and)147 418 y(that)k FD(\036)c FE(:)h FD(V)54 b FA(!)33 b FD(V)57 b FH(is)35 b(a)g(morphism)f(of)h FD(V)57 b FH(to)35 b(itself.)51 b(This)36 b(means)g(that)f FD(\036)p FE(\005\()p FD(A)p FE(\))e(=)f(\005\()p FD(A)p FE(\))p FD(\036)k FH(for)147 539 y(all)e FD(A)h FA(2)f FD(G)p FH(,)j(i.e.,)g(that)f FD(\036)g FH(comm)m(utes)g(with)g(all)f(of)g(the) i FE(\005\()p FD(A)p FE(\))p FH('s.)56 b(No)m(w,)37 b(since)g(w)m(e)g (are)g(o)m(v)m(er)g(an)147 659 y(algebraically)25 b(complete)i (\034eld,)h FD(\036)g FH(m)m(ust)g(ha)m(v)m(e)h(at)f(least)f(one)h (eigen)m(v)-5 b(alue)27 b FD(\025)h FA(2)g Fu(C)20 b FH(.)48 b(Let)28 b FD(U)38 b FH(denote)147 779 y(the)i(eigenspace)g (for)e FD(\036)h FH(asso)s(ciated)g(to)g(the)h(eigen)m(v)-5 b(alue)38 b FD(\025)p FH(,)j(and)f(let)e FD(u)h FA(2)g FD(U)10 b FH(.)64 b(Then)40 b(for)f(eac)m(h)147 900 y FD(A)28 b FA(2)g FD(G)1215 1149 y(\036)17 b FE(\(\005\()p FD(A)p FE(\))p FD(u)p FE(\))27 b(=)g(\005\()p FD(A)p FE(\))p FD(\036)p FE(\()p FD(v)t FE(\))h(=)f FD(\025)p FE(\005\()p FD(A)p FE(\))p FD(u)p FH(.)147 1398 y(Th)m(us)40 b(applying)d FE(\005\()p FD(A)p FE(\))h FH(to)g(an)g(eigen)m(v)m(ector) h(of)f FD(\036)g FH(with)g(eigen)m(v)-5 b(alue)38 b FD(\025)g FH(yields)g(another)g(eigen-)147 1518 y(v)m(ector)c(of)e FD(\036)g FH(with)g(eigen)m(v)-5 b(alue)32 b FD(\025)p FH(.)44 b(That)32 b(is,)h FD(U)43 b FH(is)32 b(in)m(v)-5 b(arian)m(t.)446 1648 y(Since)31 b FD(\025)g FH(is)g(an)g(eigen)m(v)-5 b(alue,)31 b FD(U)39 b FA(6)p FE(=)27 b(0)p FH(,)32 b(and)f(so)g(w)m(e) i(m)m(ust)e(ha)m(v)m(e)h FD(U)39 b FE(=)27 b FD(V)22 b FH(.)43 b(But)31 b(this)g(means)147 1769 y(that)i FD(\036)p FE(\()p FD(v)t FE(\))27 b(=)g FD(\025v)36 b FH(for)c(all)f FD(v)g FA(2)d FD(V)22 b FH(,)32 b(i.e.,)h(that)f FD(\036)27 b FE(=)h FD(\025I)8 b FH(.)446 1899 y FC(Pr)-5 b(o)g(of)32 b(of)h(3)p FH(.)42 b(If)30 b FD(\036)1090 1914 y Fz(2)1157 1899 y FA(6)p FE(=)d(0)p FH(,)k(then)f(b)m(y)h(\(1\))f FD(\036)1932 1914 y Fz(2)2001 1899 y FH(is)g(an)g(isomorphism.)40 b(No)m(w)30 b(lo)s(ok)f(at)h FD(\036)3440 1914 y Fz(1)3496 1899 y FA(\016)17 b FD(\036)3621 1858 y Fw(\000)p Fz(1)3621 1923 y(2)3715 1899 y FH(.)147 2019 y(As)32 b(is)e(easily)h(c)m(hec)m(k) m(ed,)j(the)e(comp)s(osition)c(of)j(t)m(w)m(o)g(morphisms)f(is)h(a)f (morphism,)g(so)h FD(\036)3366 2034 y Fz(1)3425 2019 y FA(\016)18 b FD(\036)3551 1978 y Fw(\000)p Fz(1)3551 2044 y(2)3676 2019 y FH(is)147 2140 y(a)32 b(morphism)f(of)h FD(W)46 b FH(with)33 b(itself.)42 b(Th)m(us)34 b(b)m(y)f(\(2\),)f FD(\036)2054 2155 y Fz(1)2116 2140 y FA(\016)22 b FD(\036)2246 2098 y Fw(\000)p Fz(1)2246 2164 y(2)2367 2140 y FE(=)28 b FD(\025I)8 b FH(,)32 b(whence)j FD(\036)3038 2155 y Fz(1)3104 2140 y FE(=)28 b FD(\025\036)3323 2155 y Fz(2)3362 2140 y FH(.)44 b Fy(\003)147 2437 y FI(5.8)112 b(Group)38 b(V)-9 b(ersus)37 b(Lie)g(Algebra)g(Represen)m(tations)147 2614 y FH(W)-8 b(e)24 b(kno)m(w)g(from)e(Chapter)j(3)e(\(Theorem)g (46\))g(that)g(ev)m(ery)i(Lie)e(group)g(homomorphism)d(giv)m(es)k(rise) 147 2734 y(to)i(a)g(Lie)f(algebra)g(homomorphism.)38 b(In)26 b(particular,)g(this)f(sho)m(ws)j(\(Prop)s(osition)c(77\))h (that)h(ev)m(ery)147 2855 y(represen)m(tation)f(of)e(a)g(matrix)g(Lie)g (group)g(giv)m(es)i(rise)e(to)h(a)f(represen)m(tation)i(of)e(the)h (asso)s(ciated)g(Lie)147 2975 y(algebra.)45 b(The)35 b(goal)d(of)h(this)g(section)g(is)g(to)g(in)m(v)m(estigate)h(the)g(rev) m(erse)h(pro)s(cess.)48 b(That)34 b(is,)f(giv)m(en)147 3096 y(a)g(represen)m(tation)h(of)e(the)i(Lie)e(algebra,)g(under)i (what)f(circumstances)h(is)f(there)g(an)g(asso)s(ciated)147 3216 y(represen)m(tation)g(of)f(the)h(Lie)f(group?)446 3346 y(The)26 b(climax)d(of)i(this)f(section)i(is)e(Theorem)i(105,)f (whic)m(h)h(states)g(that)f(if)f FD(G)h FH(is)f(a)h FC(c)-5 b(onne)g(cte)g(d)147 3466 y FH(and)26 b FC(simply)i(c)-5 b(onne)g(cte)g(d)35 b FH(matrix)25 b(Lie)g(group)g(with)h(Lie)f (algebra)g Fk(g)p FH(,)i(and)f(if)f FD(\031)k FH(is)d(a)f(represen)m (tation)147 3587 y(of)35 b Fk(g)p FH(,)h(then)g(there)g(is)f(a)g (unique)h(represen)m(tation)g FE(\005)f FH(of)g FD(G)g FH(suc)m(h)i(that)e FE(\005)g FH(and)h FD(\031)j FH(are)c(related)g(as) 147 3707 y(in)g(Prop)s(osition)f(77.)51 b(Our)36 b(pro)s(of)f(of)g (this)g(theorem)g(will)e(mak)m(e)j(use)g(of)f(the)h(Bak)m(er-Campb)s (ell-)147 3828 y(Hausdor\033)i(form)m(ula)d(from)h(Chapter)h(4.)57 b(Before)38 b(turning)e(to)h(this)g(general)f(theorem,)i(w)m(e)g(will) 147 3948 y(examine)g(t)m(w)m(o)g(sp)s(ecial)f(cases,)k(namely)c FF(SO)p FE(\(3\))h FH(and)g FF(SU)p FE(\(2\))p FH(,)h(for)f(whic)m(h)g (w)m(e)h(can)g(w)m(ork)f(things)147 4068 y(out)33 b(b)m(y)g(hand.)44 b(See)33 b(Br\366c)m(k)m(er)h(and)f(tom)e(Diec)m(k,)j(Chapter)f(I)s(I,) g(Section)f(5.)446 4198 y(W)-8 b(e)27 b(ha)m(v)m(e)h(sho)m(wn)g (\(Theorem)f(82\))f(that)g(ev)m(ery)j(irreducible)c(complex)h(represen) m(tation)h(of)147 4319 y FF(su)q FE(\(2\))32 b FH(is)g(equiv)-5 b(alen)m(t)32 b(to)h(one)f(of)h(the)g(represen)m(tations)g FD(\031)2261 4334 y Ft(m)2361 4319 y FH(describ)s(ed)g(in)f(Section)g (5.3.)44 b(\(Recall)147 4439 y(that)24 b(the)h(irreducible)e(complex)h (represen)m(tations)i(of)e FF(su)p FE(\(2\))g FH(are)h(in)e(one-to-one) h(corresp)s(ondence)147 4560 y(with)40 b(the)g(irreducible)f(represen)m (tations)i(of)e FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(.\))72 b(Eac)m(h)41 b(of)e(the)i(represen)m(tations)g FD(\031)3557 4575 y Ft(m)3663 4560 y FH(of)147 4680 y FF(su)q FE(\(2\))30 b FH(w)m(as)h(constructed)h(from)d(the)h(corresp)s(onding)g(represen)m (tation)h FE(\005)2824 4695 y Ft(m)2921 4680 y FH(of)f(the)h(group)f FF(SU)p FE(\(2\))p FH(.)147 4800 y(Th)m(us)41 b(w)m(e)f(see,)i(b)m(y)d (brute)h(force)f(computation,)g(that)g(ev)m(ery)i(irreducible)c (complex)i(represen-)147 4921 y(tation)44 b(of)g FF(su)q FE(\(2\))g FH(actually)g(comes)h(from)e(a)i(represen)m(tation)g(of)g (the)g(group)g FF(SU)p FE(\(2\))p FH(!)80 b(This)45 b(is)147 5041 y(consisten)m(t)34 b(with)e(the)h(fact)f(that)g FF(SU)q FE(\(2\))g FH(is)g(simply)f(connected)k(\(Chapter)e(2,)f(Prop.) 44 b(24\).)446 5171 y(Let)28 b(us)h(no)m(w)g(consider)f(the)h (situation)d(for)i FF(SO)p FE(\(3\))p FH(.)42 b(\(Whic)m(h)28 b(is)g(not)g(simply)e(connected.\))147 5292 y(W)-8 b(e)42 b(kno)m(w)g(from)d(Exercise)j(10)f(of)f(Chapter)i(3)f(that)f(the)i(Lie) e(algebras)g FF(su)q FE(\(2\))g FH(and)h FF(so)p FE(\(3\))g FH(are)p eop %%Page: 117 117 117 116 bop 147 48 a Fx(Group)27 b(V)-7 b(ersus)27 b(Lie)h(Algebra)e (Represen)n(tations)1919 b FG(117)147 298 y FH(isomorphic.)42 b(In)33 b(particular,)d(if)i(w)m(e)h(tak)m(e)h(the)f(basis)704 572 y FD(E)776 587 y Fz(1)844 572 y FE(=)957 533 y Fz(1)p 957 549 36 4 v 957 606 a(2)1019 431 y Fv(\022)1142 511 y FD(i)122 b FE(0)1134 631 y(0)83 b FA(\000)p FD(i)1418 431 y Fv(\023)1574 572 y FD(E)1646 587 y Fz(2)1714 572 y FE(=)1827 533 y Fz(1)p 1827 549 V 1827 606 a(2)1889 431 y Fv(\022)2043 511 y FE(0)121 b(1)2004 631 y FA(\000)p FE(1)83 b(0)2304 431 y Fv(\023)2460 572 y FD(E)2532 587 y Fz(3)2599 572 y FE(=)2713 533 y Fz(1)p 2713 549 V 2713 606 a(2)2775 431 y Fv(\022)2889 511 y FE(0)91 b FD(i)2897 631 y(i)g FE(0)3111 431 y Fv(\023)147 852 y FH(for)32 b FF(su)q FE(\(2\))g FH(and)g(the)h(basis)540 1191 y FD(F)603 1206 y Fz(1)670 1191 y FE(=)774 991 y Fv(0)774 1171 y(@)903 1070 y FE(0)82 b(0)122 b(0)903 1191 y(0)82 b(0)h FA(\000)p FE(1)903 1311 y(0)f(1)122 b(0)1334 991 y Fv(1)1334 1171 y(A)1504 1191 y FD(F)1567 1206 y Fz(2)1634 1191 y FE(=)1738 991 y Fv(0)1738 1171 y(@)1905 1070 y FE(0)g(0)83 b(1)1905 1191 y(0)122 b(0)83 b(0)1867 1311 y FA(\000)p FE(1)g(0)g(0)2298 991 y Fv(1)2298 1171 y(A)2468 1191 y FD(F)2531 1206 y Fz(3)2598 1191 y FE(=)2702 991 y Fv(0)2702 1171 y(@)2831 1070 y FE(0)f FA(\000)p FE(1)i(0)2831 1191 y(1)121 b(0)h(0)2831 1311 y(0)f(0)h(0)3262 991 y Fv(1)3262 1171 y(A)147 1531 y FH(then)34 b(direct)f(computation)e(sho)m (ws)j(that)f FE([)p FD(E)1807 1546 y Fz(1)1847 1531 y FD(;)17 b(E)1963 1546 y Fz(2)2002 1531 y FE(])29 b(=)f FD(E)2234 1546 y Fz(3)2274 1531 y FH(,)33 b FE([)p FD(E)2433 1546 y Fz(2)2472 1531 y FD(;)17 b(E)2588 1546 y Fz(3)2628 1531 y FE(])28 b(=)h FD(E)2860 1546 y Fz(1)2899 1531 y FH(,)k FE([)p FD(E)3058 1546 y Fz(3)3098 1531 y FD(;)17 b(E)3214 1546 y Fz(1)3253 1531 y FE(])29 b(=)f FD(E)3485 1546 y Fz(2)3525 1531 y FH(,)33 b(and)147 1652 y(similarly)h(with)j (the)h FD(E)6 b FH('s)37 b(replaced)h(b)m(y)g(the)g FD(F)14 b FH('s.)58 b(Th)m(us)39 b(the)f(map)e FD(\036)g FE(:)g FF(so)p FE(\(3\))g FA(!)f FF(su)q FE(\(2\))i FH(whic)m(h)147 1772 y(tak)m(es)d FD(F)460 1787 y Ft(i)521 1772 y FH(to)e FD(E)712 1787 y Ft(i)773 1772 y FH(will)e(b)s(e)j(a)f(Lie)g(algebra)f (isomorphism.)446 1893 y(Since)f FF(su)q FE(\(2\))g FH(and)g FF(so)p FE(\(3\))g FH(are)h(isomorphic)d(Lie)i(algebras,)g(they)h(m)m (ust)f(ha)m(v)m(e)i(\020the)e(same\021)147 2013 y(represen)m(tations.) 70 b(Sp)s(eci\034cally)-8 b(,)42 b(if)e FD(\031)k FH(is)d(a)g(represen) m(tation)g(of)g FF(su)p FE(\(2\))p FH(,)i(then)e FD(\031)32 b FA(\016)c FD(\036)41 b FH(will)d(b)s(e)j(a)147 2134 y(represen)m(tation)25 b(of)g FF(so)p FE(\(3\))p FH(,)h(and)e(ev)m(ery) j(represen)m(tation)e(of)f FF(so)p FE(\(3\))h FH(is)f(of)g(this)g (form.)40 b(In)25 b(particular,)147 2254 y(the)39 b(irreducible)f (represen)m(tations)i(of)e FF(so)p FE(\(3\))h FH(are)g(precisely)g(of)f (the)h(form)f FD(\033)3016 2269 y Ft(m)3121 2254 y FE(=)g FD(\031)3290 2269 y Ft(m)3383 2254 y FA(\016)27 b FD(\036)p FH(.)62 b(W)-8 b(e)147 2375 y(wish)39 b(to)g(determine,)h(for)e(a)g (particular)f FD(m)p FH(,)k(whether)f(there)f(is)g(a)f(represen)m (tation)i FE(\006)3384 2390 y Ft(m)3489 2375 y FH(of)f(the)147 2495 y(group)33 b FF(SO)p FE(\(3\))f FH(suc)m(h)i(that)e FD(\033)1193 2510 y Ft(m)1293 2495 y FH(and)g FE(\006)1552 2510 y Ft(m)1652 2495 y FH(are)g(related)g(as)h(in)f(Prop)s(osition)f (77.)147 2727 y FI(Prop)s(osition)36 b(103)48 b FC(L)-5 b(et)36 b FD(\033)1193 2742 y Ft(m)1288 2727 y FE(=)28 b FD(\031)1447 2742 y Ft(m)1536 2727 y FA(\016)22 b FD(\036)35 b FC(b)-5 b(e)35 b(the)g(irr)-5 b(e)g(ducible)34 b(c)-5 b(omplex)34 b(r)-5 b(epr)g(esentations)34 b(of)h(the)147 2848 y(Lie)42 b(algebr)-5 b(a)41 b FF(so)p FE(\(3\))h FC(\()p FD(m)f FA(\025)g FE(0)p FC(\).)66 b(If)41 b FD(m)i FC(is)e(even,)i(then)f(ther)-5 b(e)42 b(is)g(a)f(r)-5 b(epr)g(esentation)41 b FE(\006)3384 2863 y Ft(m)3493 2848 y FC(of)h(the)147 2968 y(gr)-5 b(oup)33 b FF(SO)p FE(\(3\))g FC(such)g(that)g FD(\033)1169 2983 y Ft(m)1269 2968 y FC(and)f FE(\006)1526 2983 y Ft(m)1626 2968 y FC(ar)-5 b(e)33 b(r)-5 b(elate)g(d)33 b(as)g(in)f(Pr)-5 b(op)g(osition)32 b(77.)44 b(If)33 b FD(m)g FC(is)g(o)-5 b(dd,)32 b(then)147 3089 y(ther)-5 b(e)35 b(is)g(no)f(such)h(r)-5 b(epr)g(esentation)34 b(of)g FF(SO)p FE(\(3\))p FC(.)147 3352 y FI(Remarks)147 3537 y FH(Note)d(that)f(the)i(condition)d(that)h FD(m)h FH(b)s(e)g(ev)m(en)h(is)e(equiv)-5 b(alen)m(t)31 b(to)f(the)h(condition)e(that)i FE(dim)15 b FD(V)3572 3552 y Ft(m)3666 3537 y FE(=)147 3658 y FD(m)f FE(+)g(1)29 b FH(b)s(e)f(o)s(dd.)42 b(Th)m(us)30 b(it)e(is)g(the)h(o)s (dd-dimensional)c(represen)m(tations)30 b(of)e(the)h(Lie)f(algebra)f FF(so)p FE(\(3\))147 3778 y FH(whic)m(h)33 b(come)f(from)g(group)g (represen)m(tations.)446 3899 y(In)g(the)g(ph)m(ysics)h(literature,)d (the)i(represen)m(tations)g(of)f FF(su)q FE(\(2\))p FD(=)p FF(so)o FE(\(3\))h FH(are)f(lab)s(eled)f(b)m(y)j(the)147 4019 y(parameter)i FD(l)f FE(=)d FD(m=)p FE(2)p FH(.)51 b(In)35 b(terms)g(of)g(this)f(notation,)h(a)f(represen)m(tation)i(of)e FF(so)q FE(\(3\))g FH(comes)h(from)147 4140 y(a)c(represen)m(tation)h (of)e FF(SO)p FE(\(3\))h FH(if)f(and)h(only)g(if)e FD(l)34 b FH(is)c(an)h(in)m(teger.)43 b(The)32 b(represen)m(tations)g(with)f FD(l)i FH(an)147 4260 y(in)m(teger)g(are)f(called)f(\020in)m(teger)i (spin\021;)f(the)h(others)g(are)g(called)e(\020half-in)m(teger)f (spin.\021)147 4523 y FI(Pro)s(of)147 4709 y FC(Case)40 b(1:)57 b(m)40 b(o)-5 b(dd)p FH(.)63 b(In)39 b(this)g(case,)j(w)m(e)e (w)m(an)m(t)g(to)f(pro)m(v)m(e)h(that)f(there)h(is)e(no)h(represen)m (tation)h FE(\006)3675 4724 y Ft(m)147 4829 y FH(suc)m(h)35 b(that)f FD(\033)636 4844 y Ft(m)736 4829 y FH(and)g FE(\006)997 4844 y Ft(m)1097 4829 y FH(are)g(related)f(as)h(in)f(Prop)s (osition)e(77.)46 b(\(W)-8 b(e)34 b(ha)m(v)m(e)h(already)e(considered) 147 4950 y(the)40 b(case)g FD(m)f FE(=)g(1)g FH(in)f(Exercise)j(7.\))62 b(Supp)s(ose,)42 b(to)d(the)h(con)m(trary)-8 b(,)41 b(that)e(there)h (is)e(suc)m(h)j(a)e FE(\006)3648 4965 y Ft(m)3715 4950 y FH(.)147 5070 y(Then)34 b(Prop)s(osition)d(77)h(sa)m(ys)i(that)1583 5292 y FE(\006)1653 5307 y Ft(m)1720 5292 y FE(\()p FD(e)1803 5250 y Ft(X)1871 5292 y FE(\))27 b(=)h FD(e)2085 5250 y Ft(\033)2125 5258 y Fn(m)2184 5250 y Fz(\()p Ft(X)5 b Fz(\))p eop %%Page: 118 118 118 117 bop 147 48 a FK(118)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FH(for)35 b(all)d FD(X)40 b FA(2)32 b FF(so)p FE(\(3\))p FH(.)50 b(In)36 b(particular,)d(tak)m(e)j FD(X)k FE(=)31 b(2)p FD(\031)t(F)2160 313 y Fz(1)2199 298 y FH(.)51 b(Then,)37 b(computing)c(as)i(in)g(Chapter)g(3,)147 418 y(Section)e(3.2)f(w)m(e)h(see)h(that)1138 712 y FD(e)1183 671 y Fz(2)p Ft(\031)r(F)1306 680 y Fr(1)1373 712 y FE(=)1476 511 y Fv(0)1476 691 y(@)1605 590 y FE(1)186 b(0)331 b(0)1605 711 y(0)83 b(cos)17 b(2)p FD(\031)87 b FA(\000)17 b FE(sin)f(2)p FD(\031)1605 831 y FE(0)88 b(sin)17 b(2)p FD(\031)133 b FE(cos)18 b(2)p FD(\031)2454 511 y Fv(1)2454 691 y(A)2569 712 y FE(=)28 b FD(I)8 b FH(.)147 1040 y(Th)m(us)27 b(on)e(the)h(one)f (hand)h FE(\006)1154 1055 y Ft(m)1238 959 y Fv(\000)1283 1040 y FD(e)1328 1004 y Fz(2)p Ft(\031)r(F)1451 1013 y Fr(1)1490 959 y Fv(\001)1563 1040 y FE(=)i(\006)1737 1055 y Ft(m)1804 1040 y FE(\()p FD(I)8 b FE(\))27 b(=)h FD(I)8 b FH(,)27 b(while)d(on)h(the)h(other)f(hand)h FE(\006)3257 1055 y Ft(m)3340 959 y Fv(\000)3386 1040 y FD(e)3431 1004 y Fz(2)p Ft(\031)r(F)3554 1013 y Fr(1)3593 959 y Fv(\001)3666 1040 y FE(=)147 1169 y FD(e)192 1133 y Fz(2)p Ft(\031)r(\033)310 1141 y Fn(m)370 1133 y Fz(\()p Ft(F)442 1142 y Fr(1)476 1133 y Fz(\))508 1169 y FH(.)446 1290 y(Let)31 b(us)h(compute)e FD(e)1181 1253 y Fz(2)p Ft(\031)r(\033)1299 1261 y Fn(m)1359 1253 y Fz(\()p Ft(F)1431 1262 y Fr(1)1466 1253 y Fz(\))1497 1290 y FH(.)43 b(By)31 b(de\034nition,)g FD(\033)2233 1305 y Ft(m)2300 1290 y FE(\()p FD(F)2401 1305 y Fz(1)2440 1290 y FE(\))d(=)f FD(\031)2664 1305 y Ft(m)2731 1290 y FE(\()p FD(\036)p FE(\()p FD(F)2928 1305 y Fz(1)2967 1290 y FE(\)\))h(=)f FD(\031)3229 1305 y Ft(m)3296 1290 y FE(\()p FD(E)3406 1305 y Fz(1)3446 1290 y FE(\))p FH(.)43 b(But,)147 1410 y FD(E)219 1425 y Fz(1)287 1410 y FE(=)406 1371 y Ft(i)p 400 1387 36 4 v 400 1444 a Fz(2)445 1410 y FD(H)8 b FH(,)32 b(where)i(as)f(usual)1569 1678 y FD(H)i FE(=)1789 1537 y Fv(\022)1904 1617 y FE(1)121 b(0)1904 1737 y(0)82 b FA(\000)p FE(1)2203 1537 y Fv(\023)2293 1678 y FH(.)147 1930 y(W)-8 b(e)39 b(kno)m(w)h(that)f(there)g(is)f(a)h(basis)g FD(u)1549 1945 y Fz(0)1588 1930 y FD(;)17 b(u)1688 1945 y Fz(1)1726 1930 y FD(;)g FA(\001)g(\001)g(\001)31 b FD(;)17 b(u)2019 1945 y Ft(m)2124 1930 y FH(for)38 b FD(V)2336 1945 y Ft(m)2441 1930 y FH(suc)m(h)j(that)d FD(u)2941 1945 y Ft(k)3022 1930 y FH(is)g(an)h(eigen)m(v)m(ector)147 2050 y(for)i FD(\031)360 2065 y Ft(m)427 2050 y FE(\()p FD(H)8 b FE(\))41 b FH(with)f(eigen)m(v)-5 b(alue)41 b FD(m)29 b FA(\000)f FE(2)p FD(k)s FH(.)70 b(This)41 b(means)g(that)g FD(u)2571 2065 y Ft(k)2655 2050 y FH(is)g(also)f(an)h (eigen)m(v)m(ector)i(for)147 2171 y FD(\033)202 2186 y Ft(m)269 2171 y FE(\()p FD(F)370 2186 y Fz(1)410 2171 y FE(\))27 b(=)594 2131 y Ft(i)p 589 2148 V 589 2205 a Fz(2)634 2171 y FD(\031)689 2186 y Ft(m)756 2171 y FE(\()p FD(H)8 b FE(\))p FH(,)32 b(with)g(eigen)m(v)-5 b(alue)1684 2131 y Ft(i)p 1678 2148 V 1678 2205 a Fz(2)1723 2171 y FE(\()p FD(m)23 b FA(\000)f FE(2)p FD(k)s FE(\))p FH(.)44 b(Th)m(us)34 b(in)d(the)i(basis)g FA(f)p FD(u)3054 2186 y Ft(k)3096 2171 y FA(g)f FH(w)m(e)i(ha)m(v)m(e)986 2587 y FD(\033)1041 2602 y Ft(m)1108 2587 y FE(\()p FD(F)1209 2602 y Fz(1)1248 2587 y FE(\))28 b(=)1417 2297 y Fv(0)1417 2472 y(B)1417 2532 y(B)1417 2592 y(B)1417 2656 y(@)1562 2345 y Ft(i)p 1556 2362 V 1556 2419 a Fz(2)1601 2385 y FD(m)1785 2466 y Ft(i)p 1780 2482 V 1780 2539 a Fz(2)1825 2505 y FE(\()p FD(m)22 b FA(\000)h FE(2\))2245 2609 y FH(.)2283 2634 y(.)2322 2659 y(.)2453 2748 y Ft(i)p 2447 2764 V 2447 2822 a Fz(2)2492 2787 y FE(\()p FA(\000)p FD(m)p FE(\))2773 2297 y Fv(1)2773 2472 y(C)2773 2532 y(C)2773 2592 y(C)2773 2656 y(A)2876 2587 y FH(.)446 2988 y(But)30 b(w)m(e)g(are)f(assuming)g(the)g FD(m)h FH(is)f(o)s(dd!)42 b(This)29 b(means)h(that)f FD(m)16 b FA(\000)g FE(2)p FD(k)32 b FH(is)d(an)g(o)s(dd)g(in)m(teger.)147 3109 y(Th)m(us)34 b FD(e)439 3072 y Fz(2)p Ft(\031)532 3045 y Fn(i)p 528 3057 31 3 v 528 3098 a Fr(2)568 3072 y Fz(\()p Ft(m)p Fw(\000)p Fz(2)p Ft(k)r Fz(\))846 3109 y FE(=)27 b FA(\000)p FE(1)p FH(,)33 b(and)g(in)f(the)h(basis)f FA(f)p FD(u)1952 3124 y Ft(k)1994 3109 y FA(g)757 3527 y FD(e)802 3486 y Fz(2)p Ft(\031)r(\033)920 3494 y Fn(m)979 3486 y Fz(\()p Ft(F)1051 3495 y Fr(1)1086 3486 y Fz(\))1145 3527 y FE(=)1249 3237 y Fv(0)1249 3412 y(B)1249 3472 y(B)1249 3532 y(B)1249 3596 y(@)1378 3318 y FD(e)1423 3282 y Fz(2)p 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Fz(2)p Ft(\031)r(\033)310 4056 y Fn(m)370 4048 y Fz(\()p Ft(F)442 4057 y Fr(1)476 4048 y Fz(\))536 4084 y FE(=)k FA(\000)p FD(I)8 b FH(.)43 b(This)29 b(is)g(a)g(con)m(tradiction,)g(so)h(there)g(can)g(b)s(e)f(no) g(suc)m(h)i(group)e(represen)m(tation)147 4205 y FE(\006)217 4220 y Ft(m)284 4205 y FH(.)446 4325 y FC(Case)34 b(2:)45 b(m)34 b(is)h(even)p FH(.)42 b(W)-8 b(e)33 b(will)e(use)i(the)g(follo)m (wing:)147 4531 y FI(Lemma)g(104)49 b FC(Ther)-5 b(e)34 b(exists)h(a)f(Lie)h(gr)-5 b(oup)35 b(homomorphism)d FE(\010)c(:)g FF(SU)q FE(\(2\))f FA(!)g FF(SO)p FE(\(3\))35 b FC(such)g(that)263 4736 y(1.)48 b FE(\010)35 b FC(maps)f FF(SU)q FE(\(2\))g FC(onto)h FF(SO)p FE(\(3\))263 4934 y FC(2.)48 b FE(k)m(er)18 b(\010)28 b(=)f FA(f)p FD(I)8 b(;)17 b FA(\000)p FD(I)8 b FA(g)263 5142 y FC(3.)48 b(The)30 b(asso)-5 b(ciate)g(d)29 b(Lie)h(algebr)-5 b(a)29 b(homomorphism)2215 5117 y Fv(e)2207 5142 y FE(\010)f(:)g FF(su)p FE(\(2\))g FA(!)f FF(so)p FE(\(3\))j FC(is)g(an)g(isomorphism) 391 5272 y(which)k(takes)g FD(E)983 5287 y Ft(i)1047 5272 y FC(to)h FD(F)1227 5287 y Ft(i)1255 5272 y FC(.)45 b(That)34 b(is,)1708 5247 y Fv(e)1701 5272 y FE(\010)28 b(=)f FD(\036)1960 5236 y Fw(\000)p Fz(1)2055 5272 y FC(.)p eop %%Page: 119 119 119 118 bop 147 48 a Fx(Group)27 b(V)-7 b(ersus)27 b(Lie)h(Algebra)e (Represen)n(tations)1919 b FG(119)147 298 y FI(Pro)s(of)37 b(of)h(the)f(Lemma)147 482 y FH(Exercise)d(17.)43 b Fy(\003)446 603 y FH(No)m(w)31 b(consider)f(the)h(represen)m(tations)g FE(\005)1955 618 y Ft(m)2052 603 y FH(of)e FF(SU)q FE(\(2\))p FH(.)42 b(I)31 b(claim)c(that)j(if)f FD(m)h FH(is)g(ev)m(en,)i(then)147 723 y FE(\005)220 738 y Ft(m)287 723 y FE(\()p FA(\000)p FD(I)8 b FE(\))28 b(=)f FD(I)8 b FH(.)44 b(T)-8 b(o)32 b(see)i(this,)e(note)h(that)1259 981 y FD(e)1304 940 y Fz(2)p Ft(\031)r(E)1434 949 y Fr(1)1500 981 y FE(=)28 b(exp)1769 841 y Fv(\022)1884 920 y FD(\031)t(i)144 b FE(0)1906 1040 y(0)104 b FA(\000)p FD(\031)t(i)2271 841 y Fv(\023)2372 981 y FE(=)27 b FA(\000)p FD(I)8 b FH(.)147 1250 y(Th)m(us)34 b FE(\005)467 1265 y Ft(m)534 1250 y FE(\()p FA(\000)p FD(I)8 b FE(\))28 b(=)f(\005)942 1265 y Ft(m)1009 1250 y FE(\()p FD(e)1092 1214 y Fz(2)p Ft(\031)r(E)1222 1223 y Fr(1)1261 1250 y FE(\))h(=)f FD(e)1475 1214 y Ft(\031)1516 1222 y Fn(m)1574 1214 y Fz(\(2)p Ft(\031)r(E)1731 1223 y Fr(1)1766 1214 y Fz(\))1798 1250 y FH(.)43 b(But)33 b(as)g(in)f(Case)h(1,)875 1668 y FD(e)920 1627 y Ft(\031)961 1635 y Fn(m)1019 1627 y Fz(\(2)p Ft(\031)r(E)1176 1636 y Fr(1)1211 1627 y Fz(\))1270 1668 y FE(=)1374 1378 y Fv(0)1374 1554 y(B)1374 1613 y(B)1374 1673 y(B)1374 1737 y(@)1503 1460 y FD(e)1548 1423 y Fz(2)p Ft(\031)1640 1396 y Fn(i)p 1636 1408 31 3 v 1636 1449 a Fr(2)1676 1423 y Ft(m)1826 1593 y FD(e)1871 1556 y Fz(2)p Ft(\031)1963 1529 y Fn(i)p 1959 1541 V 1959 1582 a Fr(2)2000 1556 y Fz(\()p Ft(m)p Fw(\000)p Fz(2\))2300 1697 y FH(.)2338 1721 y(.)2376 1747 y(.)2492 1888 y FD(e)2537 1851 y Fz(2)p Ft(\031)2629 1824 y Fn(i)p 2625 1836 V 2625 1877 a Fr(2)2666 1851 y Fz(\()p Fw(\000)p Ft(m)p Fz(\))2884 1378 y Fv(1)2884 1554 y(C)2884 1613 y(C)2884 1673 y(C)2884 1737 y(A)2987 1668 y FH(.)147 2095 y(Only)-8 b(,)52 b(this)c(time,)j FD(m)e FH(is)f(ev)m(en,)54 b(and)48 b(so)1775 2055 y Ft(i)p 1770 2072 36 4 v 1770 2129 a Fz(2)1815 2095 y FE(\()p FD(m)33 b FA(\000)g FE(2)p FD(k)s FE(\))49 b FH(is)e(an)i(in)m(teger,)j(so)d(that)f FE(\005)3341 2110 y Ft(m)3407 2095 y FE(\()p FA(\000)p FD(I)8 b FE(\))55 b(=)147 2215 y FD(e)192 2179 y Ft(\031)233 2187 y Fn(m)292 2179 y Fz(\(2)p Ft(\031)r(E)449 2188 y Fr(1)484 2179 y Fz(\))543 2215 y FE(=)27 b FD(I)8 b FH(.)446 2335 y(Since)47 b FE(\005)788 2350 y Ft(m)855 2335 y FE(\()p FA(\000)p FD(I)8 b FE(\))52 b(=)h FD(I)8 b FH(,)50 b FE(\005)1441 2350 y Ft(m)1508 2335 y FE(\()p FA(\000)p FD(U)10 b FE(\))53 b(=)f(\005)1991 2350 y Ft(m)2058 2335 y FE(\()p FD(U)10 b FE(\))48 b FH(for)e(all)f FD(U)63 b FA(2)53 b FF(SU)p FE(\(2\))p FH(.)87 b(According)46 b(to)147 2456 y(Lemma)39 b(104,)i(for)e(eac)m(h)i FD(R)h FA(2)f FF(SO)p FE(\(3\))p FH(,)g(there)g(is)e(a)h(unique)h(pair)e(of)g (elemen)m(ts)h FA(f)p FD(U;)17 b FA(\000)p FD(U)10 b FA(g)41 b FH(suc)m(h)147 2576 y(that)33 b FE(\010\()p FD(U)10 b FE(\))28 b(=)g(\010\()p FA(\000)p FD(U)10 b FE(\))29 b(=)e FD(R)q FH(.)44 b(Since)33 b FE(\005)1618 2591 y Ft(m)1684 2576 y FE(\()p FD(U)10 b FE(\))28 b(=)g(\005)2041 2591 y Ft(m)2108 2576 y FE(\()p FA(\000)p FD(U)10 b FE(\))p FH(,)33 b(it)f(mak)m(es)h(sense)h(to)e(de\034ne)1575 2776 y FE(\006)1645 2791 y Ft(m)1712 2776 y FE(\()p FD(R)q FE(\))c(=)g(\005)2068 2791 y Ft(m)2134 2776 y FE(\()p FD(U)10 b FE(\))p FH(.)147 2975 y(It)30 b(is)f(easy)i(to)e(see)i(that)f FE(\006)1107 2990 y Ft(m)1203 2975 y FH(is)g(a)f(Lie)g(group)h (homomorphism)c(\(hence,)32 b(a)d(represen)m(tation\).)44 b(By)147 3096 y(construction,)33 b(w)m(e)g(ha)m(v)m(e)1645 3296 y FE(\005)1718 3311 y Ft(m)1812 3296 y FE(=)28 b(\006)1986 3311 y Ft(m)2075 3296 y FA(\016)22 b FE(\010)p FH(.)1249 b(\(5.12\))446 3516 y(No)m(w,)39 b(if)803 3491 y Fv(e)795 3516 y FE(\006)865 3531 y Ft(m)969 3516 y FH(denotes)f(the)g(Lie)e (algebra)g(represen)m(tation)i(asso)s(ciated)f(to)g FE(\006)3318 3531 y Ft(m)3385 3516 y FH(,)h(then)g(it)147 3637 y(follo)m(ws)31 b(from)h(\(5.12\))f(that)1654 3846 y FD(\031)1709 3861 y Ft(m)1803 3846 y FE(=)1914 3821 y Fv(e)1907 3846 y FE(\006)1977 3861 y Ft(m)2066 3846 y FA(\016)2146 3821 y Fv(e)2138 3846 y FE(\010)p FH(.)147 4046 y(But)40 b(the)h(Lie)e (algebra)g(homomorphism)1748 4021 y Fv(e)1740 4046 y FE(\010)i FH(tak)m(es)g FD(E)2180 4061 y Ft(i)2248 4046 y FH(to)f FD(F)2438 4061 y Ft(i)2466 4046 y FH(,)i(that)e(is,)2896 4021 y Fv(e)2889 4046 y FE(\010)h(=)f FD(\036)3174 4010 y Fw(\000)p Fz(1)3268 4046 y FH(.)66 b(So)40 b FD(\031)3559 4061 y Ft(m)3666 4046 y FE(=)155 4141 y Fv(e)147 4167 y FE(\006)217 4182 y Ft(m)306 4167 y FA(\016)22 b FD(\036)436 4130 y Fw(\000)p Fz(1)530 4167 y FH(,)33 b(or)717 4141 y Fv(e)709 4167 y FE(\006)779 4182 y Ft(m)874 4167 y FE(=)27 b FD(\031)1032 4182 y Ft(m)1121 4167 y FA(\016)22 b FD(\036)p FH(.)44 b(Th)m(us)1576 4141 y Fv(e)1569 4167 y FE(\006)1639 4182 y Ft(m)1733 4167 y FE(=)28 b FD(\033)1892 4182 y Ft(m)1959 4167 y FH(,)33 b(whic)m(h)g(is)f(what)g(w)m(e)i(w)m (an)m(t)f(to)g(sho)m(w.)44 b Fy(\003)446 4287 y FH(It)33 b(is)f(no)m(w)h(time)e(to)h(state)h(the)g(main)e(theorem.)147 4492 y FI(Theorem)k(105)164 b FC(1.)49 b(L)-5 b(et)44 b FD(G;)17 b(H)51 b FC(b)-5 b(e)43 b(a)g(matrix)g(Lie)h(gr)-5 b(oups,)45 b(let)f FD(\036)2787 4507 y Fz(1)2826 4492 y FD(;)17 b(\036)2928 4507 y Fz(2)3011 4492 y FE(:)44 b FD(G)f FA(!)h FD(H)51 b FC(b)-5 b(e)43 b(Lie)391 4612 y(gr)-5 b(oup)46 b(homomorphisms,)g(and)g(let)1796 4586 y Fv(e)1786 4612 y FD(\036)1844 4627 y Fz(1)1883 4612 y FD(;)1937 4586 y Fv(e)1927 4612 y FD(\036)1985 4627 y Fz(2)2073 4612 y FE(:)j Fk(g)g FA(!)f Fk(h)e FC(b)-5 b(e)46 b(the)g(asso)-5 b(ciate)g(d)45 b(Lie)h(algebr)-5 b(a)391 4733 y(homomorphisms.)42 b(If)35 b FD(G)f FC(is)h(c)-5 b(onne)g(cte)g(d)34 b(and)2112 4706 y Fv(e)2103 4733 y FD(\036)2161 4748 y Fz(1)2228 4733 y FE(=)2341 4706 y Fv(e)2331 4733 y FD(\036)2389 4748 y Fz(2)2428 4733 y FC(,)h(then)g FD(\036)2768 4748 y Fz(1)2835 4733 y FE(=)27 b FD(\036)2996 4748 y Fz(2)3035 4733 y FC(.)263 4930 y(2.)48 b(L)-5 b(et)36 b FD(G;)17 b(H)43 b FC(b)-5 b(e)36 b(a)f(matrix)h(Lie)g(gr)-5 b(oups)35 b(with)h(Lie)f(algebr)-5 b(as)35 b Fk(g)h FC(and)f Fk(h)p FC(.)48 b(L)-5 b(et)3144 4904 y Fv(e)3135 4930 y FD(\036)29 b FE(:)h Fk(g)e FA(!)f Fk(h)36 b FC(b)-5 b(e)35 b(a)391 5051 y(Lie)d(algebr)-5 b(a)32 b(homomorphism.)42 b(If)32 b FD(G)g FC(is)h(c)-5 b(onne)g(cte)g(d)31 b(and)h(simply)g(c)-5 b(onne)g(cte)g(d,)32 b(then)g(ther)-5 b(e)391 5171 y(exists)42 b(a)g(unique)g(Lie)g(gr)-5 b(oup)42 b(homomorphism)e FD(\036)h FE(:)g FD(G)h FA(!)f FD(H)49 b FC(such)42 b(that)h FD(\036)f FC(and)3521 5145 y Fv(e)3511 5171 y FD(\036)g FC(ar)-5 b(e)391 5292 y(r)g(elate)g(d)35 b(as)f(in)h(The)-5 b(or)g(em)33 b(46)i(of)f(Chapter)h(3.)p eop %%Page: 120 120 120 119 bop 147 48 a FK(120)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)446 298 y FH(This)33 b(has)g(the)g(follo)m(wing)c(corollaries.) 147 487 y FI(Corollary)36 b(106)49 b FC(Supp)-5 b(ose)28 b FD(G)i FC(and)e FD(H)37 b FC(ar)-5 b(e)29 b(c)-5 b(onne)g(cte)g(d,)29 b(simply)g(c)-5 b(onne)g(cte)g(d)28 b(matrix)h(Lie)g(gr)-5 b(oups)147 607 y(with)35 b(Lie)g(algebr)-5 b(as)33 b Fk(g)i FC(and)g Fk(h)p FC(.)44 b(If)34 b Fk(g)1480 579 y FA(\030)1481 611 y FE(=)1585 607 y Fk(h)h FC(then)g FD(G)1994 579 y FA(\030)1994 611 y FE(=)2099 607 y FD(H)8 b FC(.)147 862 y FI(Pro)s(of)147 1046 y FH(Let)344 1020 y Fv(e)335 1046 y FD(\036)49 b FE(:)h Fk(g)g FA(!)f Fk(h)c FH(b)s(e)h(a)f(Lie)g(algebra)f(isomorphism.)79 b(By)46 b(Theorem)g(105,)i(there)e(exists)g(an)147 1167 y(asso)s(ciated)40 b(Lie)f(group)g(homomorphism)e FD(\036)j FE(:)g FD(G)g FA(!)g FD(H)8 b FH(.)64 b(Since)2643 1141 y Fv(e)2633 1167 y FD(\036)2691 1131 y Fw(\000)p Fz(1)2825 1167 y FE(:)41 b Fk(h)e FA(!)h Fk(g)g FH(is)f(also)g(a)h(Lie)147 1287 y(algebra)30 b(homomorphism,)f(there)j(is)f(a)g(corresp)s(onding)g (Lie)f(group)h(homomorphism)e FD( )i FE(:)d FD(H)35 b FA(!)147 1408 y FD(G)p FH(.)44 b(W)-8 b(e)33 b(w)m(an)m(t)g(to)f(sho)m (w)i(that)e FD(\036)g FH(and)h FD( )j FH(are)d(in)m(v)m(erses)h(of)e (eac)m(h)i(other.)446 1541 y(W)-8 b(ell,)723 1517 y Fu(])708 1541 y FD(\036)21 b FA(\016)h FD( )49 b FE(=)1100 1514 y Fv(e)1091 1541 y FD(\036)29 b FA(\016)1273 1514 y Fv(e)1256 1541 y FD( )49 b FE(=)44 b FD(I)1531 1556 y Fd(h)1571 1541 y FH(,)h(so)e(b)m(y)g(the)g(P)m(oin)m(t)f(1)g(of)g(the)h(Theorem,) i FD(\036)28 b FA(\016)h FD( )48 b FE(=)d FD(I)3648 1556 y Ft(H)3715 1541 y FH(.)147 1661 y(Similarly)-8 b(,)29 b FD( )d FA(\016)c FD(\036)27 b FE(=)h FD(I)974 1676 y Ft(G)1033 1661 y FH(.)43 b Fy(\003)147 1850 y FI(Corollary)36 b(107)164 b FC(1.)49 b(L)-5 b(et)30 b FD(G)h FC(b)-5 b(e)30 b(a)g(c)-5 b(onne)g(cte)g(d)30 b(matrix)g(Lie)g(gr)-5 b(oup,)31 b(let)g FE(\005)2983 1865 y Fz(1)3053 1850 y FC(and)e FE(\005)3310 1865 y Fz(2)3380 1850 y FC(b)-5 b(e)31 b(r)-5 b(epr)g(e-)391 1970 y(sentations)30 b(of)h FD(G)p FC(,)g(and)g(let)g FD(\031)1473 1985 y Fz(1)1543 1970 y FC(and)g FD(\031)1784 1985 y Fz(2)1854 1970 y FC(b)-5 b(e)31 b(the)g(asso)-5 b(ciate)g(d)30 b(Lie)g(algebr)-5 b(a)30 b(r)-5 b(epr)g(esentations.)391 2091 y(If)34 b FD(\031)548 2106 y Fz(1)623 2091 y FC(and)g FD(\031)867 2106 y Fz(2)942 2091 y FC(ar)-5 b(e)34 b(e)-5 b(quivalent,)35 b(then)f FE(\005)1880 2106 y Fz(1)1955 2091 y FC(and)g FE(\005)2217 2106 y Fz(2)2291 2091 y FC(ar)-5 b(e)35 b(e)-5 b(quivalent.)263 2284 y(2.)48 b(L)-5 b(et)42 b FD(G)f FC(b)-5 b(e)41 b(c)-5 b(onne)g(cte)g(d)40 b(and)g(simply)h(c)-5 b(onne)g(cte)g(d.)63 b(If)40 b FD(\031)45 b FC(is)c(a)g(r)-5 b(epr)g(esentation)41 b(of)f Fk(g)p FC(,)j(then)391 2405 y(ther)-5 b(e)31 b(exists)g(a)h(r)-5 b(epr)g(esentation)30 b FE(\005)i FC(of)f FD(G)p FC(,)g(acting)g(on)g(the)h(same)e(sp)-5 b(ac)g(e,)31 b(such)g(that)h FE(\005)g FC(and)391 2525 y FD(\031)39 b FC(ar)-5 b(e)34 b(r)-5 b(elate)g(d)35 b(as)f(in)h(Pr)-5 b(op)g(osition)34 b(77.)147 2780 y FI(Pro)s(of)j(of)h(the)f(Corollary)147 2964 y FH(F)-8 b(or)36 b(\(1\),)h(let)e FE(\005)732 2979 y Fz(1)808 2964 y FH(act)i(on)f FD(V)58 b FH(and)36 b FE(\005)1495 2979 y Fz(2)1571 2964 y FH(on)g FD(W)14 b FH(.)55 b(W)-8 b(e)37 b(assume)f(that)h(the)f(asso)s(ciated)h(Lie)e(algebra)147 3085 y(represen)m(tations)29 b(are)f(equiv)-5 b(alen)m(t,)29 b(i.e.,)g(that)f(there)g(exists)h(an)f(in)m(v)m(ertible)f(linear)g(map) g FD(\036)g FE(:)h FD(V)49 b FA(!)147 3205 y FD(W)d FH(suc)m(h)34 b(that)1427 3391 y FD(\036)17 b FE(\()o FD(\031)1594 3406 y Fz(1)1634 3391 y FE(\()p FD(X)8 b FE(\))p FD(v)t FE(\))27 b(=)h FD(\031)2074 3406 y Fz(2)2113 3391 y FE(\()p FD(X)8 b FE(\))p FD(\036)p FE(\()p FD(v)t FE(\))147 3576 y FH(for)36 b(all)e FD(X)42 b FA(2)34 b Fk(g)i FH(and)g(all)e FD(v)k FA(2)c FD(V)22 b FH(.)54 b(This)37 b(is)e(the)i(same)f(as)g(sa)m (ying)g(that)g FD(\036\031)2926 3591 y Fz(1)2966 3576 y FE(\()p FD(X)8 b FE(\))33 b(=)h FD(\031)3329 3591 y Fz(2)3369 3576 y FE(\()p FD(X)8 b FE(\))p FD(\036)p FH(,)36 b(or)147 3697 y(equiv)-5 b(alen)m(tly)32 b(that)g FD(\036\031)1013 3712 y Fz(1)1053 3697 y FE(\()p FD(X)8 b FE(\))p FD(\036)1276 3660 y Fw(\000)p Fz(1)1397 3697 y FE(=)28 b FD(\031)1556 3712 y Fz(2)1596 3697 y FE(\()p FD(X)8 b FE(\))32 b FH(\(for)g(all)e FD(X)36 b FA(2)28 b Fk(g)p FH(\).)446 3817 y(No)m(w)33 b(de\034ne)h(a)e(map)g FE(\006)1318 3832 y Fz(2)1385 3817 y FE(:)c FD(G)g FA(!)f FF(GL)p FE(\()p FD(W)14 b FE(\))32 b FH(b)m(y)h(the)g(form)m(ula)1500 4003 y FE(\006)1570 4018 y Fz(2)1610 4003 y FE(\()p FD(A)p FE(\))28 b(=)f FD(\036)p FE(\005)2021 4018 y Fz(1)2061 4003 y FE(\()p FD(A)p FE(\))p FD(\036)2268 3961 y Fw(\000)p Fz(1)2362 4003 y FH(.)147 4188 y(It)32 b(is)f(trivial)f(to)h(c)m(hec)m(k)j(that)e FE(\006)1300 4203 y Fz(2)1371 4188 y FH(is)g(a)f(homomorphism.)40 b(F)-8 b(urthermore,)32 b(di\033eren)m(tiation)d(sho)m(ws)147 4309 y(that)k(the)f(asso)s(ciated)h(Lie)f(algebra)f(homomorphism)f(is) 1319 4494 y FD(\033)1374 4509 y Fz(2)1414 4494 y FE(\()p FD(X)8 b FE(\))28 b(=)f FD(\036\031)1823 4509 y Fz(1)1863 4494 y FE(\()p FD(X)8 b FE(\))p FD(\036)2086 4453 y Fw(\000)p Fz(1)2207 4494 y FE(=)28 b FD(\031)2366 4509 y Fz(2)2405 4494 y FE(\()p FD(X)8 b FE(\))147 4680 y FH(for)32 b(all)f FD(X)8 b FH(.)43 b(Then)34 b(b)m(y)f(\(1\))f(in)g(the)h(Theorem,)g(w)m (e)g(m)m(ust)g(also)f(ha)m(v)m(e)h FE(\006)2734 4695 y Fz(2)2802 4680 y FE(=)28 b(\005)2979 4695 y Fz(2)3018 4680 y FH(,)33 b(i.e.,)1513 4865 y FD(\036)p FE(\005)1644 4880 y Fz(1)1683 4865 y FE(\()p FD(A)p FE(\))p FD(\036)1890 4824 y Fw(\000)p Fz(1)2012 4865 y FE(=)27 b(\005)2188 4880 y Fz(2)2228 4865 y FE(\()p FD(A)p FE(\))147 5051 y FH(for)32 b(all)f FD(A)c FA(2)i FD(G)p FH(.)43 b(But)33 b(this)f(sho)m(ws)i(that)e FE(\005)1722 5066 y Fz(1)1794 5051 y FH(and)h FE(\005)2057 5066 y Fz(2)2129 5051 y FH(are)g(equiv)-5 b(alen)m(t.)446 5171 y(P)m(oin)m(t)29 b(\(2\))g(of)f(the)i(Corollary)d(follo)m(ws)h(immediately)d(from)j(P)m (oin)m(t)h(\(2\))g(of)f(the)i(Theorem,)147 5292 y(b)m(y)k(taking)d FD(H)36 b FE(=)27 b FF(GL)p FE(\()p FD(V)21 b FE(\))p FH(.)44 b Fy(\003)p eop %%Page: 121 121 121 120 bop 147 48 a Fx(Group)27 b(V)-7 b(ersus)27 b(Lie)h(Algebra)e (Represen)n(tations)1919 b FG(121)147 298 y FI(Pro)s(of)37 b(of)h(the)f(Theorem)446 603 y FC(Step)e(1:)44 b(V)-7 b(erify)35 b(Point)g(\(1\))f(of)h(the)f(The)-5 b(or)g(em.)446 723 y FH(Since)32 b FD(G)f FH(is)g(connected,)i(Corollary)d(54)h(of)g (Chapter)h(3)g(tells)e(us)i(that)f(ev)m(ery)j(elemen)m(t)d FD(A)147 844 y FH(of)j FD(G)h FH(is)f(a)g(\034nite)g(pro)s(duct)g(of)g (the)h(form)f FD(A)d FE(=)f(exp)18 b FD(X)2146 859 y Fz(1)2202 844 y FE(exp)g FD(X)2449 859 y Fz(2)2505 844 y FA(\001)f(\001)g(\001)e FE(exp)i FD(X)2884 859 y Ft(n)2931 844 y FH(,)35 b(with)f FD(X)3298 859 y Ft(i)3357 844 y FA(2)e Fk(g)p FH(.)49 b(But)147 964 y(then)33 b(if)468 938 y Fv(e)459 964 y FD(\036)517 979 y Fz(1)584 964 y FE(=)696 938 y Fv(e)687 964 y FD(\036)745 979 y Fz(2)784 964 y FH(,)g(w)m(e)g(ha)m(v)m(e)375 1184 y FD(\036)433 1199 y Fz(1)489 1103 y Fv(\000)535 1184 y FD(e)580 1143 y Ft(X)638 1152 y Fr(1)693 1184 y FA(\001)17 b(\001)g(\001)e FD(e)871 1143 y Ft(X)929 1151 y Fn(n)976 1103 y Fv(\001)1049 1184 y FE(=)28 b FD(e)1205 1125 y Fl(e)1198 1143 y Ft(\036)1240 1152 y Fr(1)1275 1143 y Fz(\()p Ft(X)1360 1152 y Fr(1)1395 1143 y Fz(\))1443 1184 y FA(\001)17 b(\001)g(\001)e FD(e)1628 1125 y Fl(e)1621 1143 y Ft(\036)1663 1152 y Fr(1)1698 1143 y Fz(\()p Ft(X)1783 1151 y Fn(n)1826 1143 y Fz(\))1885 1184 y FE(=)27 b FD(e)2040 1125 y Fl(e)2033 1143 y Ft(\036)2075 1152 y Fr(2)2110 1143 y Fz(\()p Ft(X)2195 1152 y Fr(1)2230 1143 y Fz(\))2278 1184 y FA(\001)17 b(\001)g(\001)e FD(e)2463 1125 y Fl(e)2456 1143 y Ft(\036)2498 1152 y Fr(2)2533 1143 y Fz(\()p Ft(X)2618 1151 y Fn(n)2661 1143 y Fz(\))2720 1184 y FE(=)28 b FD(\036)2882 1199 y Fz(2)2938 1103 y Fv(\000)2983 1184 y FD(e)3028 1143 y Ft(X)3086 1152 y Fr(1)3142 1184 y FA(\001)17 b(\001)g(\001)e FD(e)3320 1143 y Ft(X)3378 1151 y Fn(n)3425 1103 y Fv(\001)3487 1184 y FH(.)446 1389 y(So)33 b(w)m(e)g(no)m(w)g(need)h(only)e(pro)m(v)m (e)i(P)m(oin)m(t)e(\(2\).)446 1630 y FC(Step)j(2:)44 b(De\034ne)d FD(\036)33 b FC(in)h(a)h(neighb)-5 b(orho)g(o)g(d)33 b(of)h(the)h(identity.)446 1751 y FH(Prop)s(osition)i(51)h(of)g (Chapter)h(3)g(sa)m(ys)h(that)e(the)h(exp)s(onen)m(tial)f(mapping)f (for)h FD(G)g FH(has)h(a)147 1871 y(lo)s(cal)26 b(in)m(v)m(erse)k(whic) m(h)f(maps)f(a)h(neigh)m(b)s(orho)s(o)s(d)e FD(V)50 b FH(of)28 b(the)h(iden)m(tit)m(y)f(in)m(to)g(the)h(Lie)e(algebra)h Fk(g)p FH(.)42 b(On)147 1991 y(this)32 b(neigh)m(b)s(orho)s(o)s(d)g FD(V)54 b FH(w)m(e)33 b(can)g FC(de\034ne)39 b FD(\036)27 b FE(:)h FD(V)49 b FA(!)28 b FD(H)40 b FH(b)m(y)1430 2219 y FD(\036)p FE(\()p FD(A)p FE(\))27 b(=)h(exp)1933 2108 y Fv(n)2009 2193 y(e)2000 2219 y FD(\036)p FE(\(log)16 b FD(A)p FE(\))2349 2108 y Fv(o)2432 2219 y FH(.)147 2446 y(That)33 b(is)1582 2652 y FD(\036)27 b FE(=)h(exp)17 b FA(\016)1995 2625 y Fv(e)1986 2652 y FD(\036)22 b FA(\016)g FE(log)16 b FH(.)147 2857 y(\(Note)33 b(that)f(if)f(there)i(is)f(to)g (b)s(e)h(a)f(homomorphism)d FD(\036)j FH(as)h(in)f(Theorem)g(46)g(of)g (Chapter)h(3,)f(then)147 2989 y(on)h FD(V)21 b FH(,)33 b FD(\036)f FH(m)m(ust)h(b)s(e)f FE(exp)18 b FA(\016)1110 2963 y Fv(e)1101 2989 y FD(\036)k FA(\016)g FE(log)p FH(.\))446 3109 y(It)43 b(follo)m(ws)e(from)g(Corollary)f(68)i(to)g (the)h(Bak)m(er-Campb)s(ell-Hausdor\033)e(form)m(ula)f(that)147 3230 y(this)30 b FD(\036)g FH(is)g(a)g(\020lo)s(cal)e (homomorphism.\021)40 b(That)30 b(is,)h(if)e FD(A)i FH(and)f FD(B)35 b FH(are)c(in)e FD(V)22 b FH(,)31 b(and)f FC(if)51 b FD(AB)36 b FH(happ)s(ens)147 3350 y(to)c(b)s(e)g(in)g FD(V)54 b FH(as)32 b(w)m(ell,)g(then)g FD(\036)p FE(\()p FD(AB)5 b FE(\))28 b(=)g FD(\036)p FE(\()p FD(A)p FE(\))p FD(\036)p FE(\()p FD(B)5 b FE(\))p FH(.)42 b(\(See)33 b(the)g(discussion)f(at)g(the)h(b)s(eginning)d(of)147 3470 y(Chapter)j(4.\))446 3711 y FC(Step)i(3:)44 b(De\034ne)d FD(\036)33 b FC(along)g(a)i(p)-5 b(ath.)446 3832 y FH(Recall)46 b(that)i(when)g(w)m(e)h(sa)m(y)g FD(G)e FH(is)h(connected,)53 b(w)m(e)c(really)d(mean)h(that)h FD(G)f FH(is)g(path-)147 3952 y(connected.)64 b(Th)m(us)41 b(for)d(an)m(y)i FD(A)e FA(2)h FD(G)p FH(,)i(there)e(exists)h(a)f(path)g FD(A)p FE(\()p FD(t)p FE(\))f FA(2)h FD(G)g FH(with)g FD(A)p FE(\(0\))f(=)g FD(I)47 b FH(and)147 4072 y FD(A)p FE(\(1\))31 b(=)g FD(A)p FH(.)50 b(A)35 b(compactness)h(argumen)m(t)e(sho)m(ws)i (that)f(there)g(exists)g(n)m(um)m(b)s(ers)h FE(0)31 b(=)g FD(t)3382 4087 y Fz(0)3453 4072 y FD(<)g(t)3595 4087 y Fz(1)3666 4072 y FD(<)147 4193 y(t)182 4208 y Fz(2)238 4193 y FA(\001)17 b(\001)g(\001)26 b FD(<)i(t)521 4208 y Ft(n)596 4193 y FE(=)f(1)33 b FH(suc)m(h)h(that)1594 4399 y FD(A)p FE(\()p FD(s)p FE(\))p FD(A)p FE(\()p FD(t)1935 4414 y Ft(i)1963 4399 y FE(\))2001 4357 y Fw(\000)p Fz(1)2123 4399 y FA(2)28 b FD(V)1219 b FH(\(5.13\))147 4604 y(for)32 b(all)f FD(s)h FH(b)s(et)m(w)m(een)j FD(t)922 4619 y Ft(i)983 4604 y FH(and)d FD(t)1207 4619 y Ft(i)p Fz(+1)1326 4604 y FH(.)446 4725 y(In)44 b(particular,)g(for)e FD(i)k FE(=)g(0)p FH(,)g(w)m(e)e(ha)m(v)m(e)g FD(A)p FE(\()p FD(s)p FE(\))i FA(2)g FD(V)65 b FH(for)42 b FE(0)k FA(\024)g FD(s)g FA(\024)g FD(t)3042 4740 y Fz(1)3082 4725 y FH(.)75 b(Th)m(us)45 b(w)m(e)f(can)147 4845 y(de\034ne)50 b FD(\036)17 b FE(\()o FD(A)p FE(\()p FD(s)p FE(\)\))49 b FH(b)m(y)h(Step)f(2)f(for) h FD(s)55 b FA(2)g FE([0)p FD(;)17 b(t)1869 4860 y Fz(1)1909 4845 y FE(])p FH(.)92 b(No)m(w,)53 b(for)48 b FD(s)55 b FA(2)h FE([)p FD(t)2774 4860 y Fz(1)2814 4845 y FD(;)17 b(t)2893 4860 y Fz(2)2932 4845 y FE(])49 b FH(w)m(e)h(ha)m(v)m(e)g(b)m (y)f(\()p FI(??)p FH(\))147 4965 y FD(A)p FE(\()p FD(s)p FE(\))p FD(A)p FE(\()p FD(t)488 4980 y Fz(1)528 4965 y FE(\))566 4929 y Fw(\000)p Fz(1)689 4965 y FA(2)28 b FD(V)22 b FH(.)45 b(Mo)m(ving)33 b(the)g FD(A)p FE(\()p FD(t)1598 4980 y Fz(1)1638 4965 y FE(\))g FH(to)f(the)i(other)f(side,)g (this)f(means)h(that)g(for)g FD(s)28 b FA(2)h FE([)p FD(t)3557 4980 y Fz(1)3597 4965 y FD(;)17 b(t)3676 4980 y Fz(2)3715 4965 y FE(])147 5086 y FH(w)m(e)34 b(can)f(write)1350 5292 y FD(A)p FE(\()p FD(s)p FE(\))28 b(=)1676 5211 y Fv(\002)1718 5292 y FD(A)p FE(\()p FD(s)p FE(\))p FD(A)p FE(\()p FD(t)2059 5307 y Fz(1)2098 5292 y FE(\))2136 5250 y Fw(\000)p Fz(1)2230 5211 y Fv(\003)2289 5292 y FD(A)p FE(\()p FD(t)2435 5307 y Fz(1)2474 5292 y FE(\))p FH(.)p eop %%Page: 122 122 122 121 bop 147 48 a FK(122)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)147 298 y FH(with)32 b FD(A)p FE(\()p FD(s)p FE(\))p FD(A)p FE(\()p FD(t)710 313 y Fz(1)750 298 y FE(\))788 262 y Fw(\000)p Fz(1)910 298 y FA(2)c FD(V)22 b FH(.)43 b(If)33 b FD(\036)f FH(is)g(to)g(b)s(e)h(a)f(homomorphism,)e (w)m(e)j(m)m(ust)g(ha)m(v)m(e)459 503 y FD(\036)17 b FE(\()o FD(A)p FE(\()p FD(s)p FE(\)\))28 b(=)f FD(\036)1010 422 y Fv(\000)o(\002)1097 503 y FD(A)p FE(\()p FD(s)p FE(\))p FD(A)p FE(\()p FD(t)1438 518 y Fz(1)1477 503 y FE(\))1515 462 y Fw(\000)p Fz(1)1610 422 y 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b(of)d(simple)g (connectedness,)k(an)m(y)e(t)m(w)m(o)g(paths)g FD(A)2292 5066 y Fz(1)2331 5051 y FE(\()p FD(t)p FE(\))g FH(and)f FD(A)2739 5066 y Fz(2)2778 5051 y FE(\()p FD(t)p FE(\))h FH(joining)d(the)i(iden)m(tit)m(y)147 5171 y(to)k FD(A)g FH(will)e(b)s(e)i(homotopic)f(with)g(endp)s(oin)m(ts)i(\034xed.)58 b(\(This)37 b(is)g(a)f(standard)i(top)s(ological)33 b(fact.\))147 5292 y(Using)f(this,)h(w)m(e)g(w)m(an)m(t)h(to)e(pro)m(v)m(e)i(that)e (Step)h(3)f(giv)m(es)h(the)g(same)g(answ)m(er)g(for)f FD(A)3113 5307 y Fz(1)3186 5292 y FH(and)g FD(A)3448 5307 y Fz(2)3488 5292 y FH(.)p eop %%Page: 123 123 123 122 bop 147 48 a Fx(Group)27 b(V)-7 b(ersus)27 b(Lie)h(Algebra)e (Represen)n(tations)1919 b FG(123)446 298 y FH(Our)37 b(strategy)h(is)e(to)h(deform)f FD(A)1664 313 y Fz(1)1741 298 y FH(in)m(to)g FD(A)2016 313 y Fz(2)2093 298 y FH(in)g(a)h(series)g (of)g(steps,)j(where)e(during)e(eac)m(h)147 418 y(step)j(w)m(e)g(only)f (c)m(hange)h(the)f(path)g(in)f(a)h(small)e(time)g(in)m(terv)-5 b(al)37 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FH(.)391 1413 y(V)-8 b(erify)26 b(\(still)d(formally\))h(that)h (these)j(op)s(erators)e(ha)m(v)m(e)h(the)f(righ)m(t)g(comm)m(utation)d (relations)391 1533 y(to)41 b(generate)h(a)e(represen)m(tation)i(of)f (the)g(Lie)g(algebra)f(of)g(the)i(real)e(Heisen)m(b)s(erg)i(group.)391 1654 y(\(That)33 b(is,)f(v)m(erify)h(that)f(on)h(this)f(basis,)g FD(\031)1927 1669 y Fi(~)1970 1654 y FE([)p FD(X)r(;)17 b(Y)22 b FE(])27 b(=)h([)p FD(\031)2443 1669 y Fi(~)2486 1654 y FE(\()p FD(X)8 b FE(\))p FD(;)17 b(\031)2750 1669 y Fi(~)2792 1654 y FE(\()p FD(Y)k FE(\)])p FH(.\))391 1828 y(Wh)m(y)34 b(is)e(this)g(computation)f(not)h(rigorous?)218 2056 y(11.)48 b(Consider)26 b(the)g(Heisen)m(b)s(erg)g(group)f(o)m(v)m (er)i(the)f(\034eld)f Fu(Z)2346 2071 y Ft(p)2409 2056 y FH(of)g(in)m(tegers)g(mo)s(d)g FD(p)p FH(,)i(with)e FD(p)g FH(prime,)391 2176 y(namely)1292 2524 y FD(H)1373 2539 y Ft(p)1440 2524 y FE(=)1544 2319 y Fv(8)1544 2409 y(<)1544 2588 y(:)1632 2323 y(0)1632 2503 y(@)1761 2402 y FE(1)83 b FD(a)g(b)1761 2523 y FE(0)g(1)i FD(c)1761 2643 y FE(0)e(0)i(1)2117 2323 y Fv(1)2117 2503 y(A)2221 2524 y FA(j)p FD(a;)17 b(b;)g(c)28 b FA(2)g Fu(Z)2662 2539 y Ft(p)2709 2319 y Fv(9)2709 2409 y(=)2709 2588 y(;)2814 2524 y FH(.)391 2881 y(This)33 b(is)f(a)g(subgroup)h(of)f(the) h(group)g FF(GL)16 b FE(\(3;)h Fu(Z)2106 2896 y Ft(p)2142 2881 y FE(\))p FH(,)33 b(and)g(has)g FD(p)2653 2845 y Fz(3)2724 2881 y FH(elemen)m(ts.)391 3055 y(Let)51 b FD(V)641 3070 y Ft(p)732 3055 y FH(denote)g(the)h(space)g(of)e (complex-v)-5 b(alued)50 b(functions)g(on)h Fu(Z)3027 3070 y Ft(p)3064 3055 y FH(,)56 b(whic)m(h)51 b(is)g(a)g FD(p)p FH(-)391 3175 y(dimensional)41 b(complex)i(v)m(ector)i(space.)78 b(F)-8 b(or)43 b(eac)m(h)h(non-zero)g FD(n)j FA(2)g Fu(Z)3078 3190 y Ft(p)3115 3175 y FH(,)g(de\034ne)e(a)e(rep-)391 3296 y(resen)m(tation)33 b(of)f FD(H)1085 3311 y Ft(p)1157 3296 y FH(b)m(y)h(the)g(form)m(ula)1087 3531 y FE(\(\005)1198 3546 y Ft(n)1245 3531 y FD(f)11 b FE(\))17 b(\()o FD(x)p FE(\))28 b(=)g FD(e)1666 3489 y Fw(\000)p Ft(i)p Fz(2)p Ft(\031)r(nb=p)1971 3531 y FD(e)2016 3489 y Ft(i)p Fz(2)p Ft(\031)r(ncx=p)2307 3531 y FD(f)f FE(\()p FD(x)c FA(\000)f FD(a)p FE(\))50 b FD(x)28 b FA(2)g Fu(Z)2982 3546 y Ft(p)3019 3531 y FH(.)391 3765 y(\(These)34 b(represen)m(tations)g(are)e (analogous)f(to)i(the)f(unitary)g(represen)m(tations)i(of)e(the)h(real) 391 3886 y(Heisen)m(b)s(erg)g(group,)g(with)f(the)h(quan)m(tit)m(y)g FE(2)p FD(\031)t(n=p)f FH(pla)m(ying)f(the)i(role)f(of)g Fu(~)p FH(.\))391 4060 y(a\))37 b(Sho)m(w)h(that)g(for)f(eac)m(h)h FD(n)p FH(,)h FE(\005)1569 4075 y Ft(n)1654 4060 y FH(is)e(actually)f (a)h(represen)m(tation)h(of)f FD(H)3058 4075 y Ft(p)3097 4060 y FH(,)i(and)f(that)f(it)f(is)391 4180 y(irreducible.)391 4354 y(b\))27 b(Determine)g(\(up)h(to)f(equiv)-5 b(alence\))27 b(all)e(the)j(one-dimensional)c(represen)m(tations)29 b(of)d FD(H)3675 4369 y Ft(p)3715 4354 y FH(.)391 4528 y(c\))i(Sho)m(w)h(that)f(ev)m(ery)i(irreducible)d(represen)m(tation)h (of)g FD(H)2511 4543 y Ft(p)2579 4528 y FH(is)f(either)h (one-dimensional)d(or)391 4649 y(equiv)-5 b(alen)m(t)32 b(to)h(one)f(of)g(the)h FE(\005)1504 4664 y Ft(n)1551 4649 y FH('s.)218 4877 y(12.)48 b(Pro)m(v)m(e)34 b(Theorem)f(92.)391 5051 y FC(Hints)8 b FH(:)52 b(F)-8 b(or)35 b(existence,)k(c)m(ho)s(ose) f(bases)f FA(f)p FD(e)2000 5066 y Ft(i)2029 5051 y FA(g)f FH(and)g FA(f)p FD(f)2406 5066 y Ft(j)2443 5051 y FA(g)g FH(for)g FD(U)47 b FH(and)36 b FD(V)22 b FH(.)55 b(Then)38 b(de\034ne)f(a)391 5171 y(space)23 b FD(W)35 b FH(whic)m(h)22 b(has)h(as)f(a)f(basis)h FA(f)o FD(w)1727 5186 y Ft(ij)1804 5171 y FA(j)p FE(0)27 b FA(\024)i FD(i)f FA(\024)g FD(n;)17 b FE(0)27 b FA(\024)h FD(j)34 b FA(\024)28 b FD(m)10 b FA(g)p FH(.)40 b(De\034ne)23 b FD(\036)p FE(\()p FD(e)3286 5186 y Ft(i)3314 5171 y FD(;)17 b(f)3406 5186 y Ft(j)3443 5171 y FE(\))27 b(=)h FD(w)3682 5186 y Ft(ij)391 5292 y FH(and)33 b(extend)h(b)m(y)f(bilinearit)m(y)-8 b(.)41 b(F)-8 b(or)32 b(uniqueness,)i(use)g(the)f(univ)m(ersal)f(prop)s(ert)m (y)-8 b(.)p eop %%Page: 129 129 129 128 bop 147 48 a Fx(Exercises)3138 b FG(129)218 298 y FH(13.)48 b(Let)38 b Fk(g)g FH(and)g Fk(h)g FH(b)s(e)g(Lie)g (algebras,)g(and)g(consider)g(the)h(v)m(ector)g(space)g Fk(g)26 b FA(\010)g Fk(h)p FH(.)60 b(Sho)m(w)38 b(that)391 418 y(the)33 b(follo)m(wing)d(op)s(eration)h(mak)m(es)i Fk(g)22 b FA(\010)h Fk(h)32 b FH(in)m(to)g(a)h(Lie)e(algebra)1220 647 y FE([\()p FD(X)1366 662 y Fz(1)1405 647 y FD(;)17 b(Y)1506 662 y Fz(1)1545 647 y FE(\))p FD(;)g FE(\()p FD(X)1746 662 y Fz(2)1785 647 y FD(;)g(Y)1886 662 y Fz(2)1925 647 y FE(\)])28 b(=)f(\([)p FD(X)2267 662 y Fz(1)2307 647 y FD(;)17 b(X)2432 662 y Fz(2)2471 647 y FE(])p FD(;)g FE([)p FD(Y)2626 662 y Fz(1)2665 647 y FD(;)g(Y)2766 662 y Fz(2)2804 647 y FE(]\))g FH(.)391 924 y(No)m(w)34 b(let)f FD(G)g FH(and)h FD(H)40 b FH(b)s(e)34 b(matrix)e(Lie)h(groups,) h(with)f(Lie)f(algebras)h Fk(g)h FH(and)f Fk(h)p FH(.)46 b(Sho)m(w)34 b(that)391 1045 y FD(G)22 b FA(\002)h FD(H)41 b FH(can)33 b(b)s(e)g(regarded)g(as)g(a)g(matrix)e(Lie)h(group)h(in)f (an)h(ob)m(vious)g(w)m(a)m(y)-8 b(,)34 b(and)f(that)g(the)391 1165 y(Lie)f(algebra)f(of)h FD(G)22 b FA(\002)h FD(H)40 b FH(is)32 b(isomorphic)f(to)h Fk(g)22 b FA(\010)h Fk(h)p FH(.)218 1383 y(14.)48 b(Supp)s(ose)23 b(that)e FD(\031)26 b FH(is)21 b(a)g(represen)m(tation)i(of)e(a)g(Lie)h(algebra)e Fk(g)i FH(acting)f(on)g(a)h(\034nite-dimensional)391 1504 y(v)m(ector)34 b(space)g FD(V)22 b FH(.)44 b(Let)33 b FD(V)1349 1467 y Fw(\003)1421 1504 y FH(denote)h(as)f(usual)g(the)g (dual)f(space)i(of)f FD(V)21 b FH(,)33 b(that)g(is,)g(the)g(space)391 1624 y(of)38 b(linear)f(functionals)g(on)i FD(V)21 b FH(.)62 b(If)38 b FD(A)h FH(is)f(a)g(linear)f(op)s(erator)g(on)i FD(V)21 b FH(,)41 b(let)c FD(A)3184 1588 y Ft(tr)3286 1624 y FH(denote)j(the)391 1744 y(dual)32 b(or)g(transp)s(ose)h(op)s (erator)f(on)h FD(V)1771 1708 y Fw(\003)1834 1744 y FH(,)1649 1892 y Fv(\000)1695 1973 y FD(A)1768 1932 y Ft(tr)1832 1973 y FD(\036)1890 1892 y Fv(\001)1952 1973 y FE(\()p FD(v)t FE(\))27 b(=)h FD(\036)17 b FE(\()o FD(Av)t FE(\))391 2202 y FH(for)32 b FD(\036)27 b FA(2)i FD(V)798 2166 y Fw(\003)838 2202 y FH(,)j FD(v)g FA(2)c FD(V)21 b FH(.)44 b(De\034ne)34 b(a)e(represen)m(tation)h FD(\031)2298 2166 y Fw(\003)2370 2202 y FH(of)f Fk(g)h FH(on)f FD(V)2778 2166 y Fw(\003)2850 2202 y FH(b)m(y)h(the)g(form)m(ula)1641 2431 y FD(\031)1700 2390 y Fw(\003)1756 2431 y FE(\()p FD(X)8 b FE(\))27 b(=)h FA(\000)p FD(\031)2205 2350 y Fv(\000)2251 2431 y FD(X)2340 2390 y Ft(tr)2403 2350 y Fv(\001)2465 2431 y FH(.)391 2708 y(a\))k(Sho)m(w)i(that)e FD(\031)1038 2672 y Fw(\003)1110 2708 y FH(is)g(really)f(a)h(represen)m (tation)i(of)e Fk(g)p FH(.)391 2877 y(b\))h(Sho)m(w)g(that)f FE(\()p FD(\031)1081 2841 y Fw(\003)1121 2877 y FE(\))1159 2829 y Fw(\003)1231 2877 y FH(is)g(isomorphic)e(to)i FD(\031)t FH(.)391 3047 y(c\))h(Sho)m(w)g(that)g FD(\031)1033 3010 y Fw(\003)1104 3047 y FH(is)f(irreducible)g(if)f(and)i(only)f(if)f FD(\031)37 b FH(is.)391 3216 y(d\))d(What)f(is)g(the)h(analogous)e (construction)i(of)f(the)h(dual)f(represen)m(tation)h(for)f(represen-) 391 3336 y(tations)f(of)g(groups?)218 3554 y(15.)48 b(Recall)26 b(the)j(spaces)g FD(V)1195 3569 y Ft(m)1290 3554 y FH(in)m(tro)s(duced) f(in)f(Section)g(5.3,)i(view)m(ed)g(as)f(represen)m(tations)h(of)f(the) 391 3675 y(Lie)33 b(algebra)g FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(.)54 b(In)34 b(particular,)f(consider)h(the)g(space)h FD(V)2746 3690 y Fz(1)2820 3675 y FH(\(whic)m(h)f(has)g(dimension)391 3795 y(2\).)391 3964 y(a\))29 b(Regard)g FD(V)898 3979 y Fz(1)953 3964 y FA(\012)15 b FD(V)1102 3979 y Fz(1)1171 3964 y FH(as)30 b(a)f(represen)m(tation)g(of)g FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(,)36 b(as)29 b(in)g(De\034nition)f(99.) 42 b(Sho)m(w)30 b(that)391 4084 y(this)i(represen)m(tation)h(is)f(not)h (irreducible.)391 4254 y(b\))j(No)m(w)g(view)g FD(V)1029 4269 y Fz(1)1093 4254 y FA(\012)25 b FD(V)1252 4269 y Fz(1)1327 4254 y FH(as)36 b(a)g(represen)m(tation)g(of)g FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))31 b FA(\010)24 b FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(,)43 b(as)36 b(in)f(De\034nition)391 4374 y(96.)43 b(Sho)m(w)33 b(that)g(this)f(represen)m(tation)h(is)f (irreducible.)391 4543 y(c\))50 b(More)h(generally)-8 b(,)53 b(sho)m(w)e(that)f FD(V)1792 4558 y Ft(m)1892 4543 y FA(\012)35 b FD(V)2061 4558 y Ft(n)2158 4543 y FH(is)49 b(irreducible)g(as)h(a)g(represen)m(tation)g(of)391 4664 y FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))33 b FA(\010)c FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))p FH(,)49 b(but)41 b(reducible)g(\(except)h(if)e(one)h(of)f FD(n)h FH(or)g FD(m)g FH(is)f(zero\))h(as)g(a)g(repre-)391 4784 y(sen)m(tation)33 b(of)f FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))p FH(.)218 5002 y(16.)48 b(Sho)m(w)33 b(explicitly)e(that)h FE(exp)d(:)f FF(so)p FE(\(3\))f FA(!)h FF(SO)p FE(\(3\))k FH(is)g(on)m(to.)391 5171 y FC(Hint)9 b FH(:)61 b(Using)40 b(the)h(fact)g(that)f FF(SO)p FE(\(3\))i FA(\032)g FF(SU)p FE(\(3\))p FH(,)h(sho)m(w)f(that)e (the)h(eigen)m(v)-5 b(alues)41 b(of)f FD(R)j FA(2)391 5292 y FF(SO)p FE(\(3\))i FH(m)m(ust)h(b)s(e)g(of)f(one)h(of)f(the)h (three)h(follo)m(wing)42 b(forms:)69 b FE(\(1)p FD(;)17 b FE(1)p FD(;)g FE(1\))p FH(,)48 b FE(\(1)p FD(;)17 b FA(\000)p FE(1)p FD(;)g FA(\000)p FE(1\))p FH(,)48 b(or)p eop %%Page: 130 130 130 129 bop 147 48 a FK(130)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)391 298 y FE(\(1)p FD(;)17 b(e)567 262 y Ft(i\022)630 298 y FD(;)g(e)719 262 y Fw(\000)p Ft(i\022)837 298 y FE(\))p FH(.)43 b(In)33 b(particular,)d FD(R)k FH(m)m(ust)e(ha)m(v)m(e) i(an)e(eigen)m(v)-5 b(alue)32 b(equal)g(to)g(one.)44 b(No)m(w)33 b(sho)m(w)391 418 y(that)f(in)g(a)g(suitable)g(orthonormal) e(basis,)j FD(R)g FH(is)f(of)g(the)h(form)1468 733 y FD(R)c FE(=)1675 533 y Fv(0)1675 712 y(@)1803 612 y FE(1)198 b(0)271 b(0)1803 732 y(0)125 b(cos)17 b FD(\022)133 b FE(sin)17 b FD(\022)1803 853 y FE(0)83 b FA(\000)17 b FE(sin)g FD(\022)86 b FE(cos)17 b FD(\022)2534 533 y Fv(1)2534 712 y(A)2638 733 y FH(.)218 1087 y(17.)48 b FC(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emma)34 b(104)p FH(.)391 1246 y(Let)h FA(f)p FD(E)690 1261 y Fz(1)730 1246 y FD(;)17 b(E)846 1261 y Fz(2)885 1246 y FD(;)g(E)1001 1261 y Fz(3)1040 1246 y FA(g)35 b FH(b)s(e)g(the)h(usual)e(basis)h(for)g FF(su)p FE(\(2\))p FH(,)g(and)g FA(f)p FD(F)2661 1261 y Fz(1)2701 1246 y FD(;)17 b(F)2808 1261 y Fz(2)2847 1246 y FD(;)g(F)2954 1261 y Fz(3)2993 1246 y FA(g)35 b FH(b)s(e)g(the)h(basis)f(for)391 1367 y FF(so)p FE(\(3\))d FH(in)m(tro)s(duced)h(in)f(Section)g(5.8.)44 b(Iden)m(tify)33 b FF(su)p FE(\(2\))f FH(with)h Fu(R)2669 1330 y Fz(3)2747 1367 y FH(b)m(y)g(iden)m(tifying)e(the)i(basis)391 1487 y FA(f)p FD(E)513 1502 y Fz(1)553 1487 y FD(;)17 b(E)669 1502 y Fz(2)708 1487 y FD(;)g(E)824 1502 y Fz(3)863 1487 y FA(g)38 b FH(with)f(the)h(standard)g(basis)g(for)f Fu(R)2228 1451 y Fz(3)2273 1487 y FH(.)59 b(Consider)38 b FE(ad)p FD(E)2946 1502 y Fz(1)2986 1487 y FH(,)h FE(ad)p FD(E)3227 1502 y Fz(2)3267 1487 y FH(,)g(and)e FE(ad)p FD(E)3702 1502 y Fz(3)391 1607 y FH(as)i(op)s(erators)f(on)h FF(su)p FE(\(2\))p FH(,)h(hence)g(on)f Fu(R)1860 1571 y Fz(3)1905 1607 y FH(.)62 b(Sho)m(w)40 b(that)e FE(ad)p FD(E)2650 1622 y Ft(i)2717 1607 y FE(=)g FD(F)2894 1622 y Ft(i)2922 1607 y FH(,)i(for)f FD(i)f FE(=)g(1)p FD(;)17 b FE(2)p FD(;)g FE(3)p FH(.)61 b(In)391 1728 y(particular,)31 b FC(ad)42 b FH(is)32 b(a)h(Lie)f(algebra)f(isomorphism)f(of)i FF(su)p FE(\(2\))h FH(on)m(to)f FF(so)p FE(\(3\))p FH(.)391 1887 y(No)m(w)39 b(consider)f FE(Ad)g(:)g FF(SU)p FE(\(2\))g FA(!)f FF(GL)16 b FE(\()p FF(SU)p FE(\(2\)\))38 b(=)f FF(GL)16 b FE(\(3;)h Fu(R)5 b FE(\))g FH(.)61 b(Sho)m(w)39 b(that)f(the)h(image)d(of)391 2007 y FC(A)-5 b(d)43 b FH(is)32 b(precisely)h FF(SO)p FE(\(3\))p FH(.)43 b(Sho)m(w)34 b(that)e(the)h(k)m(ernel)g(of)f FC(A)-5 b(d)43 b FH(is)32 b FA(f)p FD(I)8 b(;)17 b FA(\000)p FD(I)8 b FA(g)p FH(.)391 2167 y(Sho)m(w)40 b(that)e FE(Ad)h(:)f FF(SU)q FE(\(2\))g FA(!)g FF(SO)p FE(\(3\))g FH(is)g(the)i(homomorphism)35 b FE(\010)40 b FH(required)f(b)m(y)g(Lemma)391 2287 y(104.)218 2486 y(18.)48 b FC(Pr)-5 b(o)g(of)35 b(of)f(Pr)-5 b(op)g(osition)34 b(110.)391 2645 y FH(Supp)s(ose)40 b(that)e FD(G)g FH(and)1327 2620 y Fv(e)1308 2645 y FD(G)g FH(are)h(matrix)e(Lie)h(groups.)61 b(Supp)s(ose)40 b(that)e FD(\036)g FE(:)3240 2620 y Fv(e)3222 2645 y FD(G)f FA(!)h FD(G)g FH(is)g(a)391 2765 y(Lie)h(group)h (homomorphism)d(suc)m(h)k(that)f FD(\036)f FH(maps)h(some)f(neigh)m(b)s (orho)s(o)s(d)g FD(U)50 b FH(of)39 b FD(I)48 b FH(in)3684 2740 y Fv(e)3665 2765 y FD(G)391 2886 y FH(homeomorphically)27 b(on)m(to)j(a)g(neigh)m(b)s(orho)s(o)s(d)f FD(V)52 b FH(of)30 b FD(I)38 b FH(in)30 b FD(G)p FH(.)43 b(Pro)m(v)m(e)31 b(that)f(the)h(asso)s(ciated)391 3006 y(Lie)h(algebra)f(map)1122 2980 y Fv(e)1113 3006 y FD(\036)c FE(:)1251 3002 y Fv(e)1253 3006 y Fk(g)h FA(!)f Fk(g)33 b FH(is)f(an)h(isomorphism.)391 3165 y FC(Hints)8 b FH(:)61 b(Supp)s(ose)42 b(that)1337 3139 y Fv(e)1327 3165 y FD(\036)f FH(w)m(ere)i(not)e(one-to-one.)68 b(Sho)m(w,)45 b(then,)f(that)d(there)g(exists)h(a)391 3286 y(sequence)i(of)d(p)s(oin)m(ts)g FD(A)1299 3301 y Ft(n)1387 3286 y FH(in)1529 3261 y Fv(e)1510 3286 y FD(G)g FH(with)g FD(A)1932 3301 y Ft(n)2022 3286 y FA(6)p FE(=)i FD(I)8 b FH(,)43 b FD(A)2335 3301 y Ft(n)2425 3286 y FA(!)f FD(I)50 b FH(and)41 b FD(\036)p FE(\()p FD(A)3027 3301 y Ft(n)3074 3286 y FE(\))h(=)h FD(I)8 b FH(,)44 b(giving)39 b(a)391 3406 y(con)m(tradiction.)391 3566 y(T)-8 b(o)37 b(sho)m(w)h(that)1010 3539 y Fv(e)1000 3566 y FD(\036)f FH(is)f(on)m(to,)i(use)g(Step)f(1)f(of)h(the)g(pro)s (of)f(of)g(Theorem)h(105)f(to)g(sho)m(w)i(that)391 3686 y(on)33 b(a)f(su\036cien)m(tly)h(small)d(neigh)m(b)s(orho)s(o)s(d)h(of) h(zero)h(in)2392 3682 y Fv(e)2394 3686 y Fk(g)p FH(,)1713 3862 y Fv(e)1704 3888 y FD(\036)27 b FE(=)h(log)16 b FA(\016)p FD(\036)22 b FA(\016)g FE(exp)17 b FH(.)391 4091 y(Use)29 b(this)f(to)g(sho)m(w)h(that)f(the)h(image)d(of)1876 4064 y Fv(e)1867 4091 y FD(\036)i FH(con)m(tains)g(a)g(neigh)m(b)s (orho)s(o)s(d)f(of)g(zero)i(in)e Fk(g)p FH(.)42 b(No)m(w)391 4211 y(use)34 b(linearit)m(y)c(to)i(sho)m(w)i(that)e(the)h(image)e(of) 2087 4185 y Fv(e)2077 4211 y FD(\036)h FH(is)h(all)d(of)i Fk(g)p FH(.)218 4409 y(19.)48 b FC(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(113.)391 4569 y FH(First)d(supp)s(ose)j (that)d FE(k)m(er)1361 4544 y Fv(e)1352 4569 y FE(\005)d FA(\033)g FE(k)m(er)18 b FD(\036)p FH(.)43 b(Then)33 b(construct)g FE(\005)f FH(as)h(in)e(the)h(pro)s(of)g(of)f(Prop)s(osi-) 391 4689 y(tion)h(103.)391 4848 y(No)m(w)f(supp)s(ose)g(that)f(there)h (is)f(a)g(represen)m(tation)h FE(\005)f FH(of)g FD(G)g FH(for)f(whic)m(h)i(the)g(asso)s(ciated)f(Lie)391 4969 y(algebra)k(represen)m(tation)h(is)g FD(\031)t FH(.)51 b(W)-8 b(e)35 b(w)m(an)m(t)h(to)e(sho)m(w,)j(then,)g(that)d FE(k)m(er)3038 4944 y Fv(e)3029 4969 y FE(\005)e FA(\033)g FE(k)m(er)18 b FD(\036)p FH(.)50 b(W)-8 b(ell,)391 5089 y(de\034ne)34 b(a)e(new)i(represen)m(tation)f FE(\006)g FH(of)1825 5064 y Fv(e)1806 5089 y FD(G)f FH(b)m(y)1840 5292 y FE(\006)c(=)f(\005)c FA(\016)f FD(\036)p FH(.)p eop %%Page: 131 131 131 130 bop 147 48 a Fx(Exercises)3138 b FG(131)391 298 y FH(Sho)m(w)37 b(that)f(the)g(asso)s(ciated)g(Lie)f(algebra)g (homomorphism)e FD(\033)40 b FH(is)c(equal)f(to)h FD(\031)t FH(,)h(so)f(that,)391 418 y(b)m(y)30 b(P)m(oin)m(t)g(\(1\))f(of)g (Theorem)g(105,)1665 393 y Fv(e)1656 418 y FE(\005)f(=)f(\006)p FH(.)43 b(What)29 b(can)h(y)m(ou)g(sa)m(y)h(ab)s(out)e(the)h(k)m(ernel) g(of)f FE(\006)p FH(?)218 622 y(20.)48 b(Fix)32 b(an)g(in)m(teger)h FD(n)28 b FA(\025)g FE(2)p FH(.)391 783 y(a\))i(Sho)m(w)i(that)e(ev)m (ery)j(\(\034nite-dimensional)27 b(complex\))j(represen)m(tation)h(of)g (the)g(Lie)f(alge-)391 904 y(bra)37 b FF(sl)17 b FE(\()p FD(n)p FE(;)g Fu(R)5 b FE(\))42 b FH(giv)m(es)c(rise)f(to)f(a)h (represen)m(tation)h(of)e(the)h(group)g FF(SL)17 b FE(\()p FD(n)p FE(;)g Fu(R)t FE(\))6 b FH(,)38 b(ev)m(en)h(though)391 1024 y FF(SL)17 b FE(\()p FD(n)p FE(;)g Fu(R)t FE(\))30 b FH(is)24 b(not)g(simply)e(connected.)43 b(\(Y)-8 b(ou)24 b(ma)m(y)f(use)i(the)g(fact)f(that)g FF(SL)16 b FE(\()p FD(n)p FE(;)h Fu(C)j FE(\))30 b FH(is)23 b(simply)391 1145 y(connected.\))391 1306 y(b\))37 b(Sho)m(w)h(that)e(the)h(univ)m (ersal)g(co)m(v)m(er)h(of)f FF(SL)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))43 b FH(is)36 b(not)h(isomorphic)e(to)h(an)m(y)i (matrix)391 1427 y(Lie)32 b(group.)43 b(\(Y)-8 b(ou)33 b(ma)m(y)f(use)h(the)g(fact)g(that)f FF(SL)16 b FE(\()p FD(n)p FE(;)h Fu(R)5 b FE(\))39 b FH(is)32 b(not)g(simply)f (connected.\))218 1630 y(21.)48 b(Let)30 b FD(G)g FH(b)s(e)g(a)g (matrix)e(Lie)h(group)h(with)f(Lie)g(algebra)g Fk(g)p FH(,)i(let)e Fk(h)h FH(b)s(e)g(a)f(subalgebra)h(of)f Fk(g)p FH(,)i(and)391 1751 y(let)i FD(H)40 b FH(b)s(e)33 b(the)h(unique)f(connected)i(Lie)e(subgroup)g(of)g FD(G)g FH(with)g(Lie)f(algebra)g Fk(h)p FH(.)45 b(Supp)s(ose)391 1871 y(that)27 b(there)i(exists)f(a)f(compact)h(simply)e(connected)j (matrix)d(Lie)h(group)g FD(K)35 b FH(suc)m(h)29 b(that)f(the)391 1991 y(Lie)36 b(algebra)g(of)h FD(K)44 b FH(is)36 b(isomorphic)f(to)i Fk(h)p FH(.)57 b(Sho)m(w)38 b(that)e FD(H)45 b FH(is)36 b(closed.)57 b(Is)38 b FD(H)45 b FH(necessarily)391 2112 y(isomorphic)31 b(to)h FD(K)7 b FH(?)p eop %%Page: 132 132 132 131 bop 147 48 a FK(132)2407 b FG(Basic)27 b(Represen)n(tation)f (Theory)p eop %%Page: 133 133 133 132 bop 1688 1047 a FL(Chapter)37 b(6)499 1222 y(THE)h(REPRESENT) -10 b(A)g(TIONS)37 b(OF)i Fc(SU)p Fb(\(3\))p FL(,)e(AND)i(BEYOND)147 1491 y FI(6.1)112 b(Preliminaries)147 1659 y FH(There)47 b(is)f(a)f(theory)i(of)e(the)h(represen)m(tations)h(of)f(semisimple)d (groups/Lie)i(algebras)h(whic)m(h)147 1779 y(includes)34 b(as)f(a)h(sp)s(ecial)e(case)j(the)f(represen)m(tation)g(theory)g(of)g FF(SU)p FE(\(3\))p FH(.)46 b(Ho)m(w)m(ev)m(er,)37 b(I)c(feel)h(that)f (it)147 1900 y(is)f(w)m(orth)m(while)f(to)h(examine)f(the)h(case)h(of)e FF(SU)q FE(\(3\))g FH(separately)-8 b(.)44 b(I)32 b(feel)f(this)h(w)m (a)m(y)h(partly)e(b)s(ecause)147 2020 y FF(SU)q FE(\(3\))37 b FH(is)g(an)g(imp)s(ortan)m(t)f(group)h(in)g(ph)m(ysics,)j(but)e(c)m (hie\035y)g(b)s(ecause)h(the)f(general)e(semisimple)147 2141 y(theory)c(is)e(di\036cult)g(to)h(digest.)42 b(Considering)31 b(a)f(non-trivial)e(example)i(mak)m(es)i(it)e(m)m(uc)m(h)h(clearer)147 2261 y(what)d(is)g(going)e(on.)42 b(In)29 b(fact,)f(all)e(of)i(the)g (elemen)m(ts)g(of)g(the)g(general)f(theory)i(are)f(presen)m(t)i (already)147 2381 y(in)f(the)h(case)h(of)e FF(SU)p FE(\(3\))p FH(,)h(so)g(w)m(e)h(do)e(not)h(lose)f(to)s(o)g(m)m(uc)m(h)h(b)m(y)g (considering)f(at)h(\034rst)g(just)g(this)f(case.)446 2502 y(The)51 b(main)d(result)i(of)f(this)h(c)m(hapter)g(is)g(Theorem)g (1,)k(whic)m(h)c(states)h(that)e(an)h(irre-)147 2622 y(ducible)44 b(\034nite-dimensional)d(represen)m(tation)j(of)g FF(SU)q FE(\(3\))f FH(can)i(b)s(e)f(classi\034ed)g(in)g(terms)g(of)g (its)147 2742 y(\020highest)32 b(w)m(eigh)m(t.\021)44 b(This)32 b(is)g(analogous)f(to)h(lab)s(eling)e(the)i(irreducible)f (represen)m(tations)j FD(V)3565 2757 y Ft(m)3663 2742 y FH(of)147 2863 y FF(SU)q FE(\(2\))p FD(=)p FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))34 b FH(b)m(y)29 b(the)g(highest)f(eigen)m(v) -5 b(alue)28 b(of)g FD(\031)2021 2878 y Ft(m)2088 2863 y FE(\()p FD(H)8 b FE(\))p FH(.)41 b(\(The)29 b(highest)f(eigen)m(v)-5 b(alue)28 b(of)g FD(\031)3511 2878 y Ft(m)3578 2863 y FE(\()p FD(H)8 b FE(\))147 2983 y FH(in)33 b FD(V)319 2998 y Ft(m)419 2983 y FH(is)g(precisely)h FD(m)p FH(.\))47 b(W)-8 b(e)35 b(will)c(then)j(discuss,)h(without)f(pro)s(ofs,)f(what)h (the)g(corresp)s(onding)147 3104 y(results)f(are)g(for)f(general)g (semisimple)e(Lie)i(algebras.)446 3224 y(The)c(group)f FF(SU)q FE(\(3\))g FH(is)g(connected)i(and)e(simply)f(connected)j (\(Br\366c)m(k)m(er)f(and)g(tom)e(Diec)m(k\),)147 3344 y(so)39 b(b)m(y)h(Corollary)e(1)h(of)f(Chapter)i(5,)h(the)e (\034nite-dimensional)d(represen)m(tations)k(of)f FF(SU)p FE(\(3\))g FH(are)147 3465 y(in)g(one-to-one)f(corresp)s(ondence)k (with)d(the)h(\034nite-dimensional)c(represen)m(tations)k(of)f(the)h (Lie)147 3585 y(algebra)28 b FF(su)p FE(\(3\))p FH(.)42 b(Mean)m(while,)29 b(the)g(complex)f(represen)m(tations)i(of)e FF(su)p FE(\(3\))g FH(are)h(in)f(one-to-one)g(cor-)147 3705 y(resp)s(ondence)36 b(with)e(the)g(complex-linear)e(represen)m (tations)j(of)e(the)i(complexi\034ed)e(Lie)g(algebra)147 3826 y FF(su)q FE(\(3\))360 3841 y Fi(C)411 3826 y FH(.)72 b(But)42 b FF(su)q FE(\(3\))926 3841 y Fi(C)1021 3798 y FA(\030)1022 3830 y FE(=)1142 3826 y FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))6 b FH(,)44 b(as)e(is)g(easily)f(v)m(eri\034ed.)72 b(Moreo)m(v)m(er,)46 b(since)d FF(SU)p FE(\(3\))f FH(is)f(con-)147 3946 y(nected,)g(it)d(follo)m(ws)f(that)h(a)g(subspace)i FD(W)51 b FA(\032)38 b FD(V)60 b FH(is)38 b(in)m(v)-5 b(arian)m(t)37 b(under)i(the)f(action)g(of)g FF(SU)p FE(\(3\))g FH(if)147 4067 y(and)33 b(only)f(if)f(it)h(is)g(in)m(v)-5 b(arian)m(t)31 b(under)i(the)g(action)f(of)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(.)44 b(Th)m(us)34 b(w)m(e)f(ha)m(v)m(e) h(the)f(follo)m(wing:)147 4278 y FI(Prop)s(osition)j(114)48 b FC(Ther)-5 b(e)25 b(is)f(a)h(one-to-one)f(c)-5 b(orr)g(esp)g(ondenc)g (e)22 b(b)-5 b(etwe)g(en)25 b(the)f(\034nite-dimensional)147 4399 y(c)-5 b(omplex)35 b(r)-5 b(epr)g(esentations)35 b FE(\005)i FC(of)f FF(SU)q FE(\(3\))g FC(and)f(the)i (\034nite-dimensional)c(c)-5 b(omplex-line)g(ar)35 b(r)-5 b(epr)g(e-)147 4519 y(sentations)34 b FD(\031)39 b FC(of)c FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FC(.)45 b(This)34 b(c)-5 b(orr)g(esp)g(ondenc)g(e)33 b(is)h(determine)-5 b(d)34 b(by)h(the)g(pr)-5 b(op)g(erty)35 b(that)1627 4725 y FE(\005)1717 4644 y Fv(\000)1762 4725 y FD(e)1807 4684 y Ft(X)1875 4644 y Fv(\001)1948 4725 y FE(=)28 b FD(e)2097 4684 y Ft(\031)r Fz(\()p Ft(X)5 b Fz(\))147 4930 y FC(for)35 b(al)5 b(l)34 b FD(X)i FA(2)28 b FF(su)p FE(\(3\))g FA(\032)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FC(.)446 5051 y(The)43 b(r)-5 b(epr)g(esentation)42 b FE(\005)h FC(is)g(irr)-5 b(e)g(ducible)42 b(if)h(and)f(only)h(the)g (r)-5 b(epr)g(esentation)42 b FD(\031)47 b FC(is)c(irr)-5 b(e-)147 5171 y(ducible.)44 b(Mor)-5 b(e)g(over,)32 b(a)h(subsp)-5 b(ac)g(e)32 b FD(W)42 b FA(\032)28 b FD(V)54 b FC(is)33 b(invariant)f(for)h FE(\005)g FC(if)g(and)f(only)h(if)g(it)g(is)g (invariant)147 5292 y(for)i FD(\031)t FC(.)p eop %%Page: 134 134 134 133 bop 147 48 a FK(134)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)446 298 y FH(Since)g FF(SU)q FE(\(3\))g FH(is)g(compact,)g(Prop)s(osition)f(90)h (of)f(Chapter)i(5)f(tells)f(us)i(that)f(all)e(the)j(\034nite-)147 418 y(dimensional)22 b(represen)m(tations)j(of)e FF(SU)q FE(\(3\))g FH(are)h(completely)f(reducible.)40 b(The)25 b(ab)s(o)m(v)m(e)g(prop)s(osition)147 539 y(then)32 b(implies)c(that)j (all)e(the)i(\034nite-dimensional)d(represen)m(tations)k(of)e FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))37 b FH(are)31 b(completely)147 659 y(reducible.)446 786 y(Moreo)m(v)m(er,)f(w)m(e)d(can)g(apply)g(the) 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FD(H)r(;)17 b(X)r(;)g(Y)22 b FA(g)o FC(.)45 b(Then)34 b(al)5 b(l)35 b(the)f(eigenvalues)g(of)g FD(\031)t FE(\()p FD(H)8 b FE(\))35 b FC(ar)-5 b(e)34 b(inte)-5 b(gers.)p eop %%Page: 137 137 137 136 bop 147 48 a Fx(W)-7 b(eigh)n(ts)28 b(and)f(Ro)r(ots)2781 b FG(137)147 298 y FI(Pro)s(of)147 483 y FH(By)31 b(Prop)s(osition)d (115,)i FD(V)52 b FH(decomp)s(oses)31 b(as)f(a)g(direct)g(sum)g(of)f (irreducible)g(in)m(v)-5 b(arian)m(t)29 b(subspaces)147 603 y FD(V)204 618 y Ft(i)232 603 y FH(.)44 b(Eac)m(h)34 b FD(V)603 618 y Ft(i)664 603 y FH(m)m(ust)e(b)s(e)h(one)g(of)g(the)g (irreducible)e(represen)m(tations)j(of)e FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(,)39 b(whic)m(h)33 b(w)m(e)h(ha)m(v)m(e)147 723 y(classi\034ed.)43 b(In)31 b(particular,)f(in)g(eac)m(h)h FD(V)1574 738 y Ft(i)1602 723 y FH(,)h FD(\031)t FE(\()p FD(H)8 b FE(\))30 b FH(can)h(b)s(e)g(diagonalized,)e(and)i(the)g(eigen) m(v)-5 b(alues)30 b(of)147 844 y FD(\031)t FE(\()p FD(H)8 b FE(\))33 b FH(are)h(in)m(tegers.)48 b(Th)m(us)35 b FD(\031)t FE(\()p FD(H)8 b FE(\))34 b FH(can)g(b)s(e)g(diagonalized)d (on)j(the)g(whole)g(space)h FD(V)21 b FH(,)35 b(and)f(all)d(of)147 964 y(the)i(eigen)m(v)-5 b(alues)33 b(are)f(in)m(tegers.)44 b Fy(\003)147 1193 y FI(Corollary)36 b(119)49 b FC(If)38 b FD(\031)k FC(is)c(a)g(r)-5 b(epr)g(esentation)38 b(of)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FC(,)39 b(then)g(al)5 b(l)38 b(of)g(the)g(weights)g(of)g FD(\031)k FC(ar)-5 b(e)147 1313 y(of)35 b(the)g(form)1665 1533 y FD(\026)27 b FE(=)h(\()p FD(m)1978 1548 y Fz(1)2018 1533 y FD(;)17 b(m)2147 1548 y Fz(2)2186 1533 y FE(\))147 1753 y FC(with)35 b FD(m)444 1768 y Fz(1)518 1753 y FC(and)g FD(m)793 1768 y Fz(2)867 1753 y FC(inte)-5 b(gers.)147 2013 y FI(Pro)s(of)147 2198 y FH(Apply)35 b(Prop)s(osition)e(118)h(to)g(the)h(restriction)f (of)g FD(\031)39 b FH(to)c FA(f)o FD(H)2369 2213 y Fz(1)2409 2198 y FD(;)17 b(X)2534 2213 y Fz(1)2573 2198 y FD(;)g(Y)2674 2213 y Fz(1)2712 2198 y FA(g)p FH(,)36 b(and)e(to)h(the)g(restriction) 147 2318 y(of)d FD(\031)37 b FH(to)32 b FA(f)p FD(H)600 2333 y Fz(2)639 2318 y FD(;)17 b(X)764 2333 y Fz(2)803 2318 y FD(;)g(Y)904 2333 y Fz(2)943 2318 y FA(g)p FH(.)43 b Fy(\003)446 2439 y FH(Our)28 b(strategy)h(no)m(w)g(is)f(to)g(b)s (egin)g(with)g(one)h(sim)m(ultaneous)e(eigen)m(v)m(ector)j(for)e FD(\031)t FE(\()p FD(H)3480 2454 y Fz(1)3519 2439 y FE(\))g FH(and)147 2559 y FD(\031)t FE(\()p FD(H)325 2574 y Fz(2)364 2559 y FE(\))p FH(,)38 b(and)f(then)g(to)g(apply)f FD(\031)t FE(\()p FD(X)1461 2574 y Ft(i)1489 2559 y FE(\))h FH(or)f FD(\031)t FE(\()p FD(Y)1841 2574 y Ft(i)1869 2559 y FE(\))p FH(,)i(and)f(see)h(what)f(the)g(e\033ect)g(is.)56 b(The)38 b(follo)m(wing)147 2679 y(de\034nition)32 b(is)g(relev)-5 b(an)m(t)32 b(in)g(this)g(con)m(text.)45 b(\(See)33 b(Lemma)e(121)h(b)s (elo)m(w.\))147 2908 y FI(De\034nition)k(120)49 b FC(A)n(n)35 b(or)-5 b(der)g(e)g(d)34 b(p)-5 b(air)34 b FD(\013)29 b FE(=)e(\()p FD(\013)1882 2923 y Fz(1)1921 2908 y FD(;)17 b(\013)2027 2923 y Fz(2)2067 2908 y FE(\))27 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b(com-)147 4316 y(m)m(utation)c(relations)g(\(6.1\))h(tell)f(us)i(what)g(the)g(ro) s(ots)f(for)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))39 b FH(are.)k(There)34 b(are)f(six)f(ro)s(ots.)1812 4533 y FD(\013)264 b Fs(Z)1696 4654 y FE(\(2)p FD(;)17 b FA(\000)p FE(1\))121 b FD(X)2193 4669 y Fz(1)1696 4774 y FE(\()p FA(\000)p FE(1)p FD(;)17 b FE(2\))121 b FD(X)2193 4789 y Fz(2)1734 4894 y FE(\(1)p FD(;)17 b FE(1\))160 b FD(X)2193 4909 y Fz(3)1696 5015 y FE(\()p FA(\000)p FE(2)p FD(;)17 b FE(1\))133 b FD(Y)2181 5030 y Fz(1)1696 5135 y FE(\(1)p FD(;)17 b FA(\000)p FE(2\))133 b FD(Y)2181 5150 y Fz(2)1657 5255 y FE(\()p FA(\000)p FE(1)p FD(;)17 b FA(\000)p FE(1\))95 b FD(Y)2181 5270 y Fz(3)3542 4895 y FH(\(6.3\))p eop %%Page: 138 138 138 137 bop 147 48 a FK(138)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)446 298 y FH(It)36 b(is)g(con)m(v)m(enien)m(t)h(to)f(single)f(out)h(the)g(t)m(w)m (o)h(ro)s(ots)f(corresp)s(onding)f(to)h FD(X)3160 313 y Fz(1)3235 298 y FH(and)h FD(X)3510 313 y Fz(2)3585 298 y FH(and)147 418 y(giv)m(e)c(them)f(sp)s(ecial)f(names:)1640 620 y FD(\013)1703 578 y Fz(\(1\))1824 620 y FE(=)d(\(2)p FD(;)17 b FA(\000)p FE(1\))1640 780 y FD(\013)1703 739 y Fz(\(2\))1824 780 y FE(=)28 b(\()p FA(\000)p FE(1)p FD(;)17 b FE(2\))p FH(.)1292 b(\(6.4\))147 991 y(The)39 b(ro)s(ots)e FD(\013)668 955 y Fz(\(1\))799 991 y FH(and)h FD(\013)1057 955 y Fz(\(2\))1188 991 y FH(are)g(called)e(the)i FI(simple)h(ro)s(ots)p FH(.)58 b(They)39 b(ha)m(v)m(e)g(the)f(prop)s (ert)m(y)g(that)147 1112 y(all)31 b(of)i(the)h(ro)s(ots)f(can)g(b)s(e)h (expressed)h(as)f(linear)e(com)m(binations)f(of)i FD(\013)2724 1075 y Fz(\(1\))2851 1112 y FH(and)g FD(\013)3104 1075 y Fz(\(2\))3232 1112 y FH(with)f FC(inte)-5 b(ger)147 1232 y FH(co)s(e\036cien)m(ts,)41 b(and)e(these)h(co)s(e\036cien)m(ts)f (are)g(either)f(all)e(greater)j(than)f(or)h(equal)f(to)g(zero)h(or)f (all)147 1352 y(less)33 b(than)g(or)f(equal)g(to)g(zero.)44 b(This)33 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b(its)g(pro)s(of)g(is)h(v)m(ery)h (easy)-8 b(,)41 b(this)d(Lemma)g(pla)m(ys)147 2598 y(a)h(crucial)f (role)g(in)h(the)g(classi\034cation)f(of)h(the)g(represen)m(tations)i (of)d FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))6 b FH(.)63 b(Note)39 b(that)g(this)147 2719 y(Lemma)22 b(is)g(the)i(analog)d(of)h(Lemma)g (83)g(of)h(Chapter)h(5,)g(whic)m(h)g(w)m(as)g(the)f(k)m(ey)i(to)d(the)i (classi\034cation)147 2839 y(of)32 b(the)h(represen)m(tations)h(of)e FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(.)147 3046 y FI(Lemma)33 b(121)49 b FC(L)-5 b(et)44 b FD(\013)h FE(=)f(\()p FD(\013)1265 3061 y Fz(1)1305 3046 y FD(;)17 b(\013)1411 3061 y Fz(2)1450 3046 y FE(\))44 b FC(b)-5 b(e)43 b(a)h(r)-5 b(o)g(ot,)46 b(and)d FD(Z)2258 3061 y Ft(\013)2352 3046 y FA(6)p FE(=)h(0)g FC(a)f(c)-5 b(orr)g(esp)g(onding)42 b(r)-5 b(o)g(ot)44 b(ve)-5 b(ctor)147 3166 y(in)38 b FF(sl)17 b FE(\(3;)g Fs(C)p FE(\))o FC(.)54 b(L)-5 b(et)38 b FD(\031)k FC(b)-5 b(e)38 b(a)g(r)-5 b(epr)g(esentation)37 b(of)h FF(sl)17 b FE(\()o(3;)g Fu(C)j FE(\))6 b FC(,)39 b FD(\026)33 b FE(=)g(\()p FD(m)2625 3181 y Fz(1)2665 3166 y FD(;)17 b(m)2794 3181 y Fz(2)2833 3166 y FE(\))38 b FC(a)g(weight)f(for)h FD(\031)t FC(,)h(and)147 3287 y FD(v)32 b FA(6)p FE(=)27 b(0)35 b FC(a)f(c)-5 b(orr)g(esp)g(onding)33 b(weight)i(ve)-5 b(ctor.)44 b(Then)1224 3488 y FD(\031)t FE(\()p FD(H)1402 3503 y Fz(1)1441 3488 y FE(\))p FD(\031)t FE(\()p FD(Z)1643 3503 y Ft(\013)1692 3488 y FE(\))p FD(v)31 b FE(=)d(\()p FD(m)2035 3503 y Fz(1)2097 3488 y FE(+)22 b FD(\013)2257 3503 y Fz(1)2296 3488 y FE(\))p FD(\031)t FE(\()p FD(Z)2498 3503 y Ft(\013)2547 3488 y FE(\))p FD(v)1224 3633 y(\031)t FE(\()p FD(H)1402 3648 y Fz(2)1441 3633 y FE(\))p FD(\031)t FE(\()p FD(Z)1643 3648 y Ft(\013)1692 3633 y FE(\))p FD(v)31 b FE(=)d(\()p FD(m)2035 3648 y Fz(2)2097 3633 y FE(+)22 b FD(\013)2257 3648 y Fz(2)2296 3633 y FE(\))p FD(\031)t FE(\()p FD(Z)2498 3648 y Ft(\013)2547 3633 y FE(\))p FD(v)t FC(.)147 3835 y(Thus)35 b(either)f FD(\031)t FE(\()p FD(Z)835 3850 y Ft(\013)884 3835 y FE(\))p FD(v)e FE(=)27 b(0)35 b FC(or)g(else)f FD(\031)t FE(\()p FD(Z)1667 3850 y Ft(\013)1716 3835 y FE(\))p FD(v)39 b FC(is)34 b(a)h(new)f(weight)g(ve)-5 b(ctor)35 b(with)g(weight)1337 4036 y FD(\026)22 b FE(+)g FD(\013)28 b FE(=)g(\()p FD(m)1833 4051 y Fz(1)1895 4036 y FE(+)22 b FD(\013)2055 4051 y Fz(1)2094 4036 y FD(;)17 b(m)2223 4051 y Fz(2)2285 4036 y FE(+)22 b FD(\013)2445 4051 y Fz(2)2484 4036 y FE(\))p FC(.)147 4293 y FI(Pro)s(of)147 4478 y FH(The)h(de\034nition)e(of)h(a)f (ro)s(ot)g(tells)g(us)i(that)f(w)m(e)h(ha)m(v)m(e)g(the)f(comm)m (utation)e(relation)g FE([)p FD(H)3183 4493 y Fz(1)3223 4478 y FD(Z)3290 4493 y Ft(\013)3339 4478 y FE(])28 b(=)f FD(\013)3559 4493 y Fz(1)3599 4478 y FD(Z)3666 4493 y Ft(\013)3715 4478 y FH(.)147 4598 y(Th)m(us)1044 4800 y FD(\031)t FE(\()p FD(H)1222 4815 y Fz(1)1261 4800 y FE(\))p FD(\031)t FE(\()p FD(Z)1463 4815 y Ft(\013)1512 4800 y FE(\))p FD(v)k FE(=)d(\()p FD(\031)t FE(\()p FD(Z)1934 4815 y Ft(\013)1983 4800 y FE(\))p FD(\031)t FE(\()p FD(H)2199 4815 y Fz(1)2238 4800 y FE(\))22 b(+)g FD(\013)2458 4815 y Fz(1)2497 4800 y FD(\031)t FE(\()p FD(Z)2661 4815 y Ft(a)2702 4800 y FE(\)\))17 b FD(v)1628 4945 y FE(=)28 b FD(\031)t FE(\()p FD(Z)1896 4960 y Ft(\013)1945 4945 y FE(\)\()p FD(m)2106 4960 y Fz(1)2145 4945 y FD(v)t FE(\))22 b(+)g FD(\013)2416 4960 y Fz(1)2456 4945 y FD(\031)t FE(\()p FD(Z)2620 4960 y Ft(\013)2669 4945 y FE(\))p FD(v)1628 5090 y FE(=)28 b(\()p FD(m)1855 5105 y Fz(1)1916 5090 y FE(+)22 b FD(\013)2076 5105 y Fz(1)2116 5090 y FE(\))p FD(\031)t FE(\()p FD(Z)2318 5105 y Ft(\013)2367 5090 y FE(\))p FD(v)t FH(.)147 5292 y(A)33 b(similar)c(argumen)m(t)j (allo)m(ws)g(us)h(to)f(compute)g FD(\031)t FE(\()p FD(H)2117 5307 y Fz(2)2157 5292 y FE(\))p FD(\031)t FE(\()p FD(Z)2359 5307 y Ft(\013)2408 5292 y FE(\))p FD(v)t FH(.)43 b Fy(\003)p eop %%Page: 139 139 139 138 bop 147 48 a Fx(Highest)28 b(W)-7 b(eigh)n(ts)27 b(and)g(the)h(Classi\034cation)f(Theorem)1715 b FG(139)147 298 y FI(6.3)112 b(Highest)37 b(W)-9 b(eigh)m(ts)36 b(and)i(the)f (Classi\034cation)f(Theorem)147 465 y FH(W)-8 b(e)39 b(see)h(then)f(that)f(if)f(w)m(e)i(ha)m(v)m(e)h(a)e(represen)m(tation)h (with)f(a)g(w)m(eigh)m(t)h FD(\026)f FE(=)f(\()p FD(m)3098 480 y Fz(1)3138 465 y FD(;)17 b(m)3267 480 y Fz(2)3306 465 y FE(\))p FH(,)40 b(then)f(b)m(y)147 586 y(applying)e(the)h(ro)s (ot)f(v)m(ectors)j FD(X)1356 601 y Fz(1)1395 586 y FD(;)17 b(X)1520 601 y Fz(2)1559 586 y FD(;)g(X)1684 601 y Fz(3)1723 586 y FD(;)g(Y)1824 601 y Fz(1)1863 586 y FD(;)g(Y)1964 601 y Fz(2)2003 586 y FD(;)g(Y)2104 601 y Fz(3)2180 586 y FH(w)m(e)39 b(can)g(get)f(some)f(new)i(w)m(eigh)m(ts)g(of)f(the)147 706 y(form)24 b FD(\026)7 b FE(+)g FD(\013)q FH(,)25 b(where)h FD(\013)g FH(is)e(the)h(ro)s(ot.)40 b(Of)25 b(course,)j(some)c(of)h(the)g(w)m(eigh)m(t)g(v)m(ectors)i(ma)m(y)e (simply)e(giv)m(e)147 827 y(zero.)43 b(In)31 b(fact,)g(since)g(our)g (represen)m(tation)g(is)f(\034nite-dimensional,)e(there)j(can)g(b)s(e)g (only)f(\034nitely)147 947 y(man)m(y)39 b(w)m(eigh)m(ts,)i(so)e(w)m(e)h (m)m(ust)f(get)g(zero)g(quite)g(often.)62 b(By)39 b(analogy)f(to)g(the) h(classi\034cation)e(of)147 1067 y(the)i(represen)m(tations)h(of)e FF(sl)q FE(\(2;)17 b Fs(C)p FE(\))p FH(,)39 b(w)m(e)h(w)m(ould)e(lik)m (e)g(to)h(single)e(out)i(in)e(eac)m(h)j(represen)m(tation)f(a)147 1188 y(\020highest\021)24 b(w)m(eigh)m(t,)i(and)e(then)g(w)m(ork)h (from)e(there.)41 b(The)25 b(follo)m(wing)c(de\034nition)i(giv)m(es)h (the)g(\020righ)m(t\021)147 1308 y(notion)32 b(of)g(highest.)147 1537 y FI(De\034nition)k(122)49 b FC(L)-5 b(et)48 b FD(\013)1131 1501 y Fz(\(1\))1276 1537 y FE(=)j(\(2)p FD(;)17 b FA(\000)p FE(1\))47 b FC(and)g FD(\013)2010 1501 y Fz(\(2\))2155 1537 y FE(=)j(\()p FA(\000)p FE(1)p FD(;)17 b FE(2\))47 b FC(b)-5 b(e)48 b(the)f(r)-5 b(o)g(ots)47 b(intr)-5 b(o)g(duc)g(e)g(d)47 b(in)147 1657 y(\(6.4\).)e(L)-5 b(et)35 b FD(\026)659 1672 y Fz(1)733 1657 y FC(and)g FD(\026)982 1672 y Fz(2)1056 1657 y FC(b)-5 b(e)35 b(two)g(weights.)44 b(Then)34 b FD(\026)2052 1672 y Fz(1)2127 1657 y FC(is)g FB(higher)i FC(than)e FD(\026)2848 1672 y Fz(2)2923 1657 y FC(\(or)g(e)-5 b(quivalently,)35 b FD(\026)3703 1672 y Fz(2)147 1778 y FC(is)g FB(lower)g FC(than)f FD(\026)828 1793 y Fz(1)867 1778 y FC(\))h(if)g FD(\026)1096 1793 y Fz(1)1157 1778 y FA(\000)23 b FD(\026)1316 1793 y Fz(2)1390 1778 y FC(c)-5 b(an)34 b(b)-5 b(e)34 b(written)h(in)g(the)g(form)1457 1998 y FD(\026)1516 2013 y Fz(1)1577 1998 y FA(\000)22 b FD(\026)1735 2013 y Fz(2)1802 1998 y FE(=)28 b FD(a\013)2020 1957 y Fz(\(1\))2136 1998 y FE(+)22 b FD(b\013)2338 1957 y Fz(\(2\))147 2218 y FC(with)35 b FD(a)28 b FA(\025)g FE(0)35 b FC(and)f FD(b)28 b FA(\025)g FE(0)p FC(.)45 b(This)34 b(r)-5 b(elationship)33 b(is)i(written)g(as)f FD(\026)2487 2233 y Fz(1)2554 2218 y FA(\027)28 b FD(\026)2718 2233 y Fz(2)2792 2218 y FC(or)35 b FD(\026)2977 2233 y Fz(2)3044 2218 y FA(\026)28 b FD(\026)3208 2233 y Fz(1)3247 2218 y FC(.)446 2338 y(If)46 b FD(\031)k FC(is)c(a)h(r)-5 b(epr)g(esentation)45 b(of)h FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FC(,)50 b(then)c(a)g(weight)g FD(\026)2743 2353 y Fz(0)2828 2338 y FC(for)g FD(\031)51 b FC(is)46 b(said)g(to)g(b)-5 b(e)46 b(a)147 2459 y FB(highest)41 b(weight)34 b FC(if)h(for)f(al)5 b(l)35 b(weights)f FD(\026)h FC(of)f FD(\031)t FC(,)h FD(\026)27 b FA(\026)h FD(\026)2180 2474 y Fz(0)2219 2459 y FC(.)446 2687 y FH(Note)k(that)g(the)g (relation)e(of)h(\020higher\021)g(is)h(only)f(a)h FC(p)-5 b(artial)41 b FH(ordering.)i(That)32 b(is,)g(one)g(can)147 2808 y(easily)f(ha)m(v)m(e)i FD(\026)698 2823 y Fz(1)769 2808 y FH(and)f FD(\026)1017 2823 y Fz(2)1087 2808 y FH(suc)m(h)h(that)f FD(\026)1576 2823 y Fz(1)1647 2808 y FH(is)f(neither)g(higher)g(nor)h(lo)m(w)m(er)g(than)f FD(\026)3082 2823 y Fz(2)3122 2808 y FH(.)43 b(F)-8 b(or)30 b(example,)147 2928 y FD(\013)210 2892 y Fz(\(1\))328 2928 y FA(\000)23 b FD(\013)491 2892 y Fz(\(2\))620 2928 y FH(is)34 b(neither)g(higher)g(nor)g(lo)m(w)m(er)g(than)h FE(0)p FH(.)48 b(This)35 b(in)e(particular)g(means)h(that)g(a)g (\034nite)147 3048 y(set)g(of)e(w)m(eigh)m(ts)h(need)h(not)e(ha)m(v)m (e)i(a)f(highest)f(elemen)m(t.)44 b(\(E.g.,)33 b(the)g(set)2788 2968 y Fv(\010)2846 3048 y FE(0)p FD(;)17 b(\013)3002 3012 y Fz(\(1\))3118 3048 y FA(\000)22 b FD(\013)3280 3012 y Fz(\(2\))3374 2968 y Fv(\011)3465 3048 y FH(has)33 b(no)147 3169 y(highest)g(elemen)m(t.\))446 3289 y(W)-8 b(e)34 b(are)g(no)m(w)g(ready)g(to)f(state)h(the)g(main)e(theorem)h (regarding)g(the)h(irreducible)e(repre-)147 3410 y(sen)m(tations)h(of)f FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(.)147 3638 y FI(Theorem)35 b(123)164 b FC(1.)49 b(Every)35 b(irr)-5 b(e)g(ducible)35 b(r)-5 b(epr)g(esentation)35 b FD(\031)k FC(of)c FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))41 b FC(is)35 b(the)h(dir)-5 b(e)g(ct)35 b(sum)391 3759 y(of)j(its)h(weight)f(sp)-5 b(ac)g(es.)54 b(That)39 b(is,)g FD(\031)t FE(\()p FD(H)1857 3774 y Fz(1)1896 3759 y FE(\))f FC(and)g FD(\031)t FE(\()p FD(H)2343 3774 y Fz(2)2382 3759 y FE(\))h FC(ar)-5 b(e)38 b(simultane)-5 b(ously)38 b(diagonaliz-)391 3879 y(able.)263 4083 y(2.)48 b(Every)33 b(irr)-5 b(e)g(ducible)32 b(r)-5 b(epr)g(esentation)33 b(of)f FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))38 b FC(has)33 b(a)g(unique)g(highest)f(weight)g FD(\026)3485 4098 y Fz(0)3524 4083 y FC(,)i(and)391 4203 y(two)h(e)-5 b(quivalent)34 b(irr)-5 b(e)g(ducible)34 b(r)-5 b(epr)g(esentations)34 b(have)g(the)h(same)f(highest)g(weight.) 263 4407 y(3.)48 b(Two)g(irr)-5 b(e)g(ducible)48 b(r)-5 b(epr)g(esentations)47 b(of)h FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))54 b FC(with)48 b(the)h(same)f(highest)f(weight)h(ar)-5 b(e)391 4527 y(e)g(quivalent.)263 4731 y(4.)48 b(If)34 b FD(\031)k FC(is)c(an)g(irr)-5 b(e)g(ducible)34 b(r)-5 b(epr)g(esentation)34 b(of)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FC(,)34 b(then)g(the)h(highest)f(weight)f FD(\026)3495 4746 y Fz(0)3569 4731 y FC(of)h FD(\031)391 4851 y FC(is)h(of)f(the)h (form)1767 5071 y FD(\026)1826 5086 y Fz(0)1893 5071 y FE(=)28 b(\()p FD(m)2120 5086 y Fz(1)2159 5071 y FD(;)17 b(m)2288 5086 y Fz(2)2328 5071 y FE(\))391 5292 y FC(with)35 b FD(m)688 5307 y Fz(1)762 5292 y FC(and)g FD(m)1037 5307 y Fz(2)1111 5292 y FC(non-ne)-5 b(gative)33 b(inte)-5 b(gers.)p eop %%Page: 140 140 140 139 bop 147 48 a FK(140)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)263 298 y FC(5.)48 b(Conversely,)34 b(if)h FD(m)1098 313 y Fz(1)1173 298 y FC(and)g FD(m)1448 313 y Fz(2)1522 298 y FC(ar)-5 b(e)35 b(non-ne)-5 b(gative)34 b(inte)-5 b(gers,)34 b(then)h(ther)-5 b(e)35 b(exists)g(a)g(unique)391 418 y(irr)-5 b(e)g(ducible)34 b(r)-5 b(epr)g(esentation)34 b FD(\031)39 b FC(of)34 b FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))41 b FC(with)34 b(highest)g(weight) h FD(\026)2958 433 y Fz(0)3025 418 y FE(=)27 b(\()p FD(m)3251 433 y Fz(1)3291 418 y FD(;)17 b(m)3420 433 y Fz(2)3459 418 y FE(\))p FC(.)446 650 y FH(Note)34 b(the)g(parallels)d(b)s(et)m(w) m(een)36 b(this)d(result)h(and)g(the)g(classi\034cation)e(of)h(the)h (irreducible)147 771 y(represen)m(tations)28 b(of)e FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(:)46 b(In)27 b(eac)m(h)h(irreducible)d (represen)m(tation)j(of)e FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(,)34 b FD(\031)t FE(\()p FD(H)8 b FE(\))26 b FH(is)g(diag-)147 891 y(onalizable,)i(and)h(there)g(is)g(a)g(largest) f(eigen)m(v)-5 b(alue)28 b(of)h FD(\031)t FE(\()p FD(H)8 b FE(\))p FH(.)41 b(Tw)m(o)30 b(irreducible)e(represen)m(tations)147 1012 y(of)33 b FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))39 b FH(with)32 b(the)h(same)g(largest)f(eigen)m(v)-5 b(alue)32 b(are)h(equiv)-5 b(alen)m(t.)44 b(The)34 b(highest)e(eigen)m(v)-5 b(alue)32 b(is)147 1132 y(alw)m(a)m(ys)27 b(a)g(non-negativ)m(e)f(in)m (teger,)i(and)e(con)m(v)m(ersely)-8 b(,)30 b(for)c(ev)m(ery)i (non-negativ)m(e)f(in)m(teger)f FD(m)p FH(,)i(there)147 1252 y(is)k(an)h(irreducible)e(represen)m(tation)i(with)f(highest)h (eigen)m(v)-5 b(alue)32 b FD(m)p FH(.)446 1373 y(Ho)m(w)m(ev)m(er,)40 b(note)d(that)f(in)g(the)h(classi\034cation)e(of)h(the)h(represen)m (tations)h(of)e FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))43 b FH(the)147 1494 y(notion)31 b(of)g(\020highest\021)g(do)s(es)h(not)g (mean)f(what)h(w)m(e)h(migh)m(t)d(ha)m(v)m(e)j(though)m(t)f(it)f (should)g(mean.)43 b(F)-8 b(or)147 1614 y(example,)40 b(the)f(w)m(eigh)m(t)g FE(\(1)p FD(;)17 b FE(1\))37 b FH(is)h(higher)g(than)h(the)g(w)m(eigh)m(ts)g FE(\()p FA(\000)p FE(1)p FD(;)17 b FE(2\))39 b FH(and)f FE(\(2)p FD(;)17 b FA(\000)p FE(1\))p FH(.)61 b(\(In)39 b(fact,)147 1734 y FE(\(1)p FD(;)17 b FE(1\))32 b FH(is)g(the)h(highest)f(w)m(eigh) m(t)h(for)f(the)h(adjoin)m(t)f(represen)m(tation,)h(whic)m(h)g(is)f (irreducible.\))446 1855 y(It)d(is)f(p)s(ossible)g(to)g(obtain)f(m)m (uc)m(h)i(more)f(information)e(ab)s(out)i(the)h(irreducible)e (represen-)147 1976 y(tations)37 b(b)s(esides)i(the)f(highest)g(w)m (eigh)m(t.)60 b(F)-8 b(or)38 b(example,)h(w)m(e)g(ha)m(v)m(e)g(the)f (follo)m(wing)e(form)m(ula)g(for)147 2096 y(the)d(dimension)e(of)h(the) h(represen)m(tation)g(with)g(highest)f(w)m(eigh)m(t)h FE(\()p FD(m)2683 2111 y Fz(1)2723 2096 y FD(;)17 b(m)2852 2111 y Fz(2)2891 2096 y FE(\))p FH(.)147 2329 y FI(Theorem)35 b(124)49 b FC(The)40 b(dimension)f(of)h(the)h(irr)-5 b(e)g(ducible)40 b(r)-5 b(epr)g(esentation)40 b(with)h(highest)f (weight)147 2449 y FE(\()p FD(m)270 2464 y Fz(1)310 2449 y FD(;)17 b(m)439 2464 y Fz(2)478 2449 y FE(\))35 b FC(is)1229 2648 y FE(1)p 1229 2693 49 4 v 1229 2784 a(2)1287 2716 y(\()p FD(m)1410 2731 y Fz(1)1472 2716 y FE(+)22 b(1\)\()p FD(m)1780 2731 y Fz(2)1842 2716 y FE(+)g(1\)\()p FD(m)2150 2731 y Fz(1)2211 2716 y FE(+)g FD(m)2394 2731 y Fz(2)2456 2716 y FE(+)g(2\))p FC(.)446 2973 y FH(W)-8 b(e)41 b(will)d(not)i(pro)m (v)m(e)i(this)e(form)m(ula.)66 b(It)40 b(is)g(a)h(consequence)i(of)d (the)h(\020W)-8 b(eyl)40 b(c)m(haracter)147 3093 y(form)m(ula.\021)h (See)34 b(Humphreys,)g(Section)e(24.3.)43 b(Humphreys)33 b(refers)h(to)e FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))39 b FH(as)32 b FD(A)3343 3108 y Fz(2)3383 3093 y FH(.)147 3357 y FI(6.4)112 b(Pro)s(of)37 b(of)h(the)f(Classi\034cation)f (Theorem)147 3526 y FH(It)i(will)d(tak)m(e)k(us)f(some)f(time)g(to)g (pro)m(v)m(e)i(Theorem)f(1.)58 b(The)39 b(pro)s(of)e(will)e(consist)j (of)f(a)g(series)h(of)147 3646 y(Prop)s(ositions.)147 3879 y FI(Prop)s(osition)e(125)48 b FC(In)40 b(every)h(irr)-5 b(e)g(ducible)40 b(r)-5 b(epr)g(esentation)39 b FE(\()p FD(\031)t(;)17 b(V)k FE(\))41 b FC(of)f FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FC(,)42 b FD(\031)t FE(\()p FD(H)3470 3894 y Fz(1)3509 3879 y FE(\))f FC(and)147 3999 y FD(\031)t FE(\()p FD(H)325 4014 y Fz(2)364 3999 y FE(\))c FC(c)-5 b(an)35 b(b)-5 b(e)36 b(simultane)-5 b(ously)36 b(diagonalize)-5 b(d.)47 b(That)36 b(is,)g FD(V)58 b FC(is)36 b(the)g(dir)-5 b(e)g(ct)36 b(sum)g(of)g(its)h(weight)147 4120 y(sp)-5 b(ac)g(es.)147 4383 y FI(Pro)s(of)147 4569 y FH(Let)34 b FD(W)47 b FH(b)s(e)34 b(the)g(direct)g(sum)f(of)g(the)i(w)m(eigh)m(t) f(spaces)h(in)e FD(V)21 b FH(.)47 b(Equiv)-5 b(alen)m(tly)d(,)33 b FD(W)47 b FH(is)34 b(the)g(space)g(of)147 4689 y(all)27 b(v)m(ectors)j FD(w)g FA(2)e FD(V)50 b FH(suc)m(h)30 b(that)e FD(w)j FH(can)e(b)s(e)g(written)f(as)h(a)g(linear)d(com)m (bination)h(of)h(sim)m(ultaneous)147 4809 y(eigen)m(v)m(ectors)34 b(for)d FD(\031)t FE(\()p FD(H)1018 4824 y Fz(1)1057 4809 y FE(\))h FH(and)g FD(\031)t FE(\()p FD(H)1494 4824 y Fz(2)1533 4809 y FE(\))p FH(.)43 b(Since)33 b(\(Prop)s(osition)d (117\))h FD(\031)36 b FH(alw)m(a)m(ys)c(has)h(at)e(least)h(one)147 4930 y(w)m(eigh)m(t,)h FD(W)41 b FA(6)p FE(=)28 b FA(f)p FE(0)p FA(g)o FH(.)446 5051 y(On)23 b(the)g(other)g(hand,)j(Lemma)21 b(121)h(tells)g(us)i(that)e(if)g FD(Z)2434 5066 y Ft(\013)2506 5051 y FH(is)h(a)f(ro)s(ot)g(v)m(ector)i(corresp)s(onding)147 5171 y(to)32 b(the)h(ro)s(ot)f FD(\013)q FH(,)g(then)h FD(\031)t FE(\()p FD(Z)1151 5186 y Ft(\013)1200 5171 y FE(\))g FH(maps)f(the)h(w)m(eigh)m(t)g(space)g(corresp)s(onding)g(to) f FD(\026)g FH(in)m(to)g(the)h(w)m(eigh)m(t)147 5292 y(space)e(corresp)s(onding)e(to)h FD(\026)16 b FE(+)g FD(\013)q FH(.)42 b(Th)m(us)31 b FD(W)43 b FH(is)29 b(in)m(v)-5 b(arian)m(t)28 b(under)i(the)h(action)d(of)h(all)f(of)h(the)h(ro)s(ot)p eop %%Page: 141 141 141 140 bop 147 48 a Fx(Pro)r(of)28 b(of)f(the)h(Classi\034cation)e (Theorem)2174 b FG(141)147 298 y FH(v)m(ectors,)47 b(namely)-8 b(,)44 b(under)g(the)f(action)e FD(X)1736 313 y Fz(1)1776 298 y FD(;)17 b(X)1901 313 y Fz(2)1940 298 y FD(;)g(X)2065 313 y Fz(3)2104 298 y FD(;)g(Y)2205 313 y Fz(1)2244 298 y FD(;)g(Y)2345 313 y Fz(2)2383 298 y FD(;)43 b FH(and)g FD(Y)2710 313 y Fz(3)2749 298 y FH(.)74 b(Since)42 b FD(W)57 b FH(is)42 b(certainly)147 418 y(in)m(v)-5 b(arian)m(t)43 b(under)j(the)f(action)e(of)h FD(H)1545 433 y Fz(1)1629 418 y FH(and)h FD(H)1912 433 y Fz(2)1951 418 y FH(,)j FD(W)58 b FH(is)44 b(in)m(v)-5 b(arian)m(t.)78 b(Th)m(us)46 b(b)m(y)g(irreducibilit)m(y)-8 b(,)147 539 y FD(W)42 b FE(=)27 b FD(V)22 b FH(.)43 b Fy(\003)147 775 y FI(De\034nition)36 b(126)49 b FC(A)39 b(r)-5 b(epr)g(esentation)37 b FE(\()p FD(\031)t(;)17 b(V)k FE(\))39 b FC(of)f FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))44 b FC(is)38 b(said)g(to)h(b)-5 b(e)38 b(a)g FB(highest)44 b(weight)147 895 y(cyclic)d(r)-6 b(epr)g(esentation)44 b(with)e(weight)g FD(\026)1855 910 y Fz(0)1925 895 y FE(=)31 b(\()p FD(m)2155 910 y Fz(1)2194 895 y FD(;)17 b(m)2323 910 y Fz(2)2363 895 y FE(\))36 b FC(if)g(ther)-5 b(e)37 b(exists)f FD(v)e FA(6)p FE(=)d(0)36 b FC(in)g FD(V)58 b FC(such)147 1016 y(that)263 1251 y(1.)48 b FD(v)39 b FC(is)34 b(a)h(weight)f(ve)-5 b(ctor)35 b(with)f(weight)h FD(\026)1827 1266 y Fz(0)1866 1251 y FC(.)263 1460 y(2.)48 b FD(\031)t FE(\()p FD(X)569 1475 y Fz(1)608 1460 y FE(\))p FD(v)32 b FE(=)27 b FD(\031)t FE(\()p FD(X)1006 1475 y Fz(2)1045 1460 y FE(\))p FD(v)32 b FE(=)27 b(0)p FC(.)263 1668 y(3.)48 b(The)34 b(smal)5 b(lest)34 b(invariant)g(subsp)-5 b(ac)g(e)34 b(of)h FD(V)56 b FC(c)-5 b(ontaining)34 b FD(v)k FC(is)d(al)5 b(l)34 b(of)h FD(V)22 b FC(.)147 1904 y(The)34 b(ve)-5 b(ctor)35 b FD(v)k FC(is)34 b(c)-5 b(al)5 b(le)-5 b(d)34 b(a)h FB(cyclic)j(ve)-6 b(ctor)36 b FC(for)e FD(\031)t FC(.)147 2140 y FI(Prop)s(osition)i(127)48 b FC(L)-5 b(et)29 b FE(\()p FD(\031)t(;)17 b(V)k FE(\))28 b FC(b)-5 b(e)27 b(a)h(highest)f(weight)h(cyclic)f(r)-5 b(epr)g(esentation)27 b(of)h FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))34 b FC(with)147 2260 y(weight)g FD(\026)507 2275 y Fz(0)547 2260 y FC(.)44 b(Then)263 2495 y(1.)k FD(\031)39 b FC(has)34 b(highest)g(weight)h FD(\026)1346 2510 y Fz(0)1385 2495 y FC(.)263 2704 y(2.)48 b(The)34 b(weight)h(sp)-5 b(ac)g(e)34 b(c)-5 b(orr)g(esp)g(onding)33 b(to)i(the)f(highest)h(weight)f FD(\026)2726 2719 y Fz(0)2800 2704 y FC(is)h(one-dimensional.)147 2970 y FI(Pro)s(of)147 3157 y FH(Let)e FD(v)j FH(b)s(e)c(as)h(in)e(the)i(de\034nition.)42 b(Consider)33 b(the)g(subspace)h FD(W)46 b FH(of)31 b FD(V)54 b FH(spanned)34 b(b)m(y)f(elemen)m(ts)f(of)147 3277 y(the)h(form)1358 3500 y FD(w)d FE(=)d FD(\031)t FE(\()p FD(Y)1715 3515 y Ft(i)1739 3524 y Fr(1)1777 3500 y FE(\))p FD(\031)t FE(\()p FD(Y)1969 3515 y Ft(i)1993 3524 y Fr(2)2032 3500 y FE(\))17 b FA(\001)g(\001)g(\001)d FD(\031)t FE(\()p FD(Y)2373 3515 y Ft(i)2397 3523 y Fn(n)2443 3500 y FE(\))p FD(v)1014 b FH(\(6.5\))147 3723 y(with)33 b(eac)m(h)h FD(i)623 3738 y Ft(l)678 3723 y FE(=)29 b(1)p FD(;)17 b FE(2)p FH(,)32 b(and)i FD(n)29 b FA(\025)g FE(0)p FH(.)45 b(\(If)33 b FD(n)c FE(=)g(0)p FH(,)k(it)f(is)h(understo) s(o)s(d)g(that)g FD(w)j FH(in)c(\(6.5\))h(is)f(equal)h(to)147 3844 y FD(v)t FH(.\))49 b(I)35 b(assert)g(that)g FD(W)48 b FH(is)34 b(in)m(v)-5 b(arian)m(t.)48 b(T)-8 b(o)34 b(see)i(this,)f(it)e(su\036ces)k(to)d(c)m(hec)m(k)j(that)d FD(W)48 b FH(is)34 b(in)m(v)-5 b(arian)m(t)147 3964 y(under)34 b(eac)m(h)f(of)f(the)h(basis)f(elemen)m(ts.)446 4086 y(By)i(de\034nition,)f FD(W)47 b FH(is)33 b(in)m(v)-5 b(arian)m(t)33 b(under)h FD(\031)t FE(\()p FD(Y)2142 4101 y Fz(1)2181 4086 y FE(\))f FH(and)h FD(\031)t FE(\()p FD(Y)2597 4101 y Fz(2)2636 4086 y FE(\))p FH(.)46 b(It)34 b(is)f(th)m(us)i(also)d(in)m(v)-5 b(arian)m(t)147 4206 y(under)34 b FD(\031)t FE(\()p FD(Y)578 4221 y Fz(3)616 4206 y FE(\))28 b(=)g FA(\000)17 b FE([)p FD(\031)t FE(\()p FD(Y)1061 4221 y Fz(1)1100 4206 y FE(\))p FD(;)g(\031)t FE(\()p FD(Y)1336 4221 y Fz(2)1374 4206 y FE(\)])p FH(.)446 4327 y(No)m(w,)29 b(Lemma)d(121)g(tells)g(us)i(that)f(applying)f(a)h (ro)s(ot)f(v)m(ector)i FD(Z)2760 4342 y Ft(\013)2837 4327 y FA(2)h FF(sl)16 b FE(\(3;)h Fu(C)j FE(\))33 b FH(to)27 b(a)g(w)m(eigh)m(t)147 4448 y(v)m(ector)g FD(v)i FH(with)c(w)m(eigh)m(t)h FD(\026)f FH(giv)m(es)h(either)g(zero,)h(or)f (else)f(a)h(new)g(w)m(eigh)m(t)g(v)m(ector)h(with)e(w)m(eigh)m(t)h FD(\026)8 b FE(+)g FD(\013)q FH(.)147 4568 y(No)m(w,)41 b(b)m(y)e(assumption,)g FD(v)j FH(is)c(a)g(w)m(eigh)m(t)h(v)m(ector)g (with)f(w)m(eigh)m(t)h FD(\026)2590 4583 y Fz(0)2629 4568 y FH(.)61 b(F)-8 b(urthermore,)39 b FD(Y)3373 4583 y Fz(1)3451 4568 y FH(and)f FD(Y)3703 4583 y Fz(2)147 4689 y FH(are)f(ro)s(ot)f(v)m(ectors)j(with)d(ro)s(ots)h FA(\000)p FD(\013)1481 4652 y Fz(\(1\))1611 4689 y FE(=)e(\()p FA(\000)p FE(2)p FD(;)17 b FE(1\))37 b FH(and)g FA(\000)p FD(\013)2388 4652 y Fz(\(2\))2517 4689 y FE(=)f(\(1)p FD(;)17 b FA(\000)p FE(2\))p FH(,)37 b(resp)s(ectiv)m(ely)-8 b(.)58 b(\(See)147 4809 y(Equation)34 b(\(6.3\).\))46 b(Th)m(us)35 b(eac)m(h)g(application)c(of)i FD(\031)t FE(\()p FD(Y)2122 4824 y Fz(1)2161 4809 y FE(\))h FH(or)f FD(\031)t FE(\()p FD(Y)2507 4824 y Fz(2)2546 4809 y FE(\))h FH(subtracts)h FD(\013)3106 4773 y Fz(\(1\))3233 4809 y FH(or)f FD(\013)3417 4773 y Fz(\(2\))3544 4809 y FH(from)147 4929 y(the)h(w)m(eigh)m(t.)51 b(In)35 b(particular,)e(eac)m(h)j (non-zero)f(elemen)m(t)f(of)g(the)i(form)d(\(6.5\))h(is)h(a)f(sim)m (ultaneous)147 5050 y(eigen)m(v)m(ector)g(for)e FD(\031)t FE(\()p FD(H)981 5065 y Fz(1)1020 5050 y FE(\))g FH(and)h FD(\031)t FE(\()p FD(H)1458 5065 y Fz(2)1497 5050 y FE(\))p FH(.)43 b(Th)m(us)35 b FD(W)46 b FH(is)32 b(in)m(v)-5 b(arian)m(t)31 b(under)i FD(\031)t FE(\()p FD(H)2952 5065 y Fz(1)2991 5050 y FE(\))g FH(and)f FD(\031)t FE(\()p FD(H)3429 5065 y Fz(2)3468 5050 y FE(\))p FH(.)446 5171 y(T)-8 b(o)36 b(sho)m(w)g(that)g FD(W)49 b FH(is)35 b(in)m(v)-5 b(arian)m(t)34 b(under)i FD(\031)t FE(\()p FD(X)2163 5186 y Fz(1)2203 5171 y FE(\))f FH(and)h FD(\031)t FE(\()p FD(X)2647 5186 y Fz(2)2686 5171 y FE(\))p FH(,)g(w)m(e)h(argue)f(b)m(y) g(induction)147 5292 y(on)d FD(n)p FH(.)45 b(F)-8 b(or)32 b FD(n)d FE(=)g(0)p FH(,)k(w)m(e)h(ha)m(v)m(e)g FD(\031)t FE(\()p FD(X)1436 5307 y Fz(1)1475 5292 y FE(\))p FD(v)e FE(=)d FD(\031)t FE(\()p FD(X)1875 5307 y Fz(2)1914 5292 y FE(\))p FD(v)j FE(=)d(0)f FA(2)h FD(W)14 b FH(.)44 b(No)m(w)34 b(consider)f(applying)f FD(\031)t FE(\()p FD(X)3665 5307 y Fz(1)3704 5292 y FE(\))p eop %%Page: 142 142 142 141 bop 147 48 a FK(142)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)147 298 y FH(or)38 b FD(\031)t FE(\()p FD(X)450 313 y Fz(2)489 298 y FE(\))g FH(to)f(a)h(v)m(ector)g(of)g(the)g(form)e(\(6.5\).)59 b(Recall)36 b(the)j(comm)m(utation)c(relations)h(in)m(v)m(olving)147 418 y(an)d FD(X)364 433 y Fz(1)436 418 y FH(or)f FD(X)636 433 y Fz(2)708 418 y FH(and)g(a)h FD(Y)1036 433 y Fz(1)1107 418 y FH(or)f FD(Y)1283 433 y Fz(2)1322 418 y FH(:)1172 579 y FE([)p FD(X)1280 594 y Fz(1)1319 579 y FD(;)17 b(Y)1420 594 y Fz(1)1459 579 y FE(])83 b(=)g FD(H)1809 594 y Fz(1)2014 579 y FE([)p FD(X)2122 594 y Fz(1)2162 579 y FD(;)17 b(Y)2263 594 y Fz(2)2301 579 y FE(])83 b(=)133 b(0)1172 699 y([)p FD(X)1280 714 y Fz(2)1319 699 y FD(;)17 b(Y)1420 714 y Fz(1)1459 699 y FE(])83 b(=)119 b(0)201 b([)p FD(X)2122 714 y Fz(2)2162 699 y FD(;)17 b(Y)2263 714 y Fz(2)2301 699 y FE(])83 b(=)g FD(H)2651 714 y Fz(2)2691 699 y FH(.)147 871 y(Th)m(us)39 b(\(for)d FD(i)h FH(and)g FD(j)43 b FH(equal)37 b(to)f(1)h(or)f(2\))h FD(\031)t FE(\()p FD(X)1833 886 y Ft(i)1861 871 y FE(\))p FD(\031)t FE(\()p FD(Y)2053 886 y Ft(j)2089 871 y FE(\))e(=)g FD(\031)t FE(\()p FD(Y)2427 886 y Ft(j)2463 871 y FE(\))p FD(\031)t FE(\()p FD(X)2679 886 y Ft(i)2707 871 y FE(\))25 b(+)g FD(\031)t FE(\()p FD(H)3049 886 y Ft(ij)3109 871 y FE(\))p FH(,)38 b(where)g FD(H)3579 886 y Ft(ij)3676 871 y FH(is)147 992 y(either)33 b FD(H)505 1007 y Fz(1)576 992 y FH(or)f FD(H)776 1007 y Fz(2)848 992 y FH(or)g(zero.)44 b(Hence)34 b(\(for)e FD(i)h FH(equal)f(to)g(1)h(or)f(2\))702 1165 y FD(\031)t FE(\()p FD(X)880 1180 y Ft(i)908 1165 y FE(\))p FD(\031)t FE(\()p FD(Y)1100 1180 y Ft(i)1124 1189 y Fr(1)1162 1165 y FE(\))p FD(\031)t FE(\()p FD(Y)1354 1180 y Ft(i)1378 1189 y Fr(2)1416 1165 y FE(\))17 b FA(\001)g(\001)g (\001)e FD(\031)t FE(\()p FD(Y)1758 1180 y Ft(i)1782 1188 y Fn(n)1828 1165 y FE(\))p FD(v)730 1310 y FE(=)27 b FD(\031)t FE(\()p FD(Y)987 1325 y Ft(i)1011 1334 y Fr(1)1049 1310 y FE(\))p FD(\031)t FE(\()p FD(X)1265 1325 y Ft(i)1293 1310 y FE(\))p FD(\031)t FE(\()p FD(Y)1485 1325 y Ft(i)1509 1334 y Fr(2)1547 1310 y FE(\))17 b FA(\001)g(\001)g (\001)e FD(\031)t FE(\()p FD(Y)1889 1325 y Ft(i)1913 1333 y Fn(n)1959 1310 y FE(\))p FD(v)26 b FE(+)c FD(\031)t FE(\()p FD(H)2346 1325 y Ft(ij)2406 1310 y FE(\))p FD(\031)t FE(\()p FD(Y)2598 1325 y Ft(i)2622 1334 y Fr(2)2660 1310 y FE(\))17 b FA(\001)g(\001)g(\001)d FD(\031)t FE(\()p FD(Y)3001 1325 y Ft(i)3025 1333 y Fn(n)3072 1310 y FE(\))p FD(v)t FH(.)147 1484 y(But)40 b FD(\031)t FE(\()p FD(X)526 1499 y Ft(i)554 1484 y FE(\))p FD(\031)t FE(\()p FD(Y)746 1499 y Ft(i)770 1508 y Fr(2)808 1484 y FE(\))17 b FA(\001)g(\001)g (\001)d FD(\031)t FE(\()p FD(Y)1149 1499 y Ft(i)1173 1507 y Fn(n)1219 1484 y FE(\))p FD(v)43 b FH(is)c(in)f FD(W)53 b FH(b)m(y)40 b(induction,)g(and)f FD(\031)t FE(\()p FD(H)2702 1499 y Ft(ij)2763 1484 y FE(\))p FD(\031)t FE(\()p FD(Y)2955 1499 y Ft(i)2979 1508 y Fr(2)3017 1484 y FE(\))17 b FA(\001)g(\001)g(\001)d FD(\031)t FE(\()p FD(Y)3358 1499 y Ft(i)3382 1507 y Fn(n)3428 1484 y FE(\))p FD(v)43 b FH(is)c(in)147 1604 y FD(W)46 b FH(since)33 b FD(W)46 b FH(is)33 b(in)m(v)-5 b(arian)m(t)31 b(under)i FD(\031)t FE(\()p FD(H)1624 1619 y Fz(1)1663 1604 y FE(\))f FH(and)h FD(\031)t FE(\()p FD(H)2101 1619 y Fz(2)2140 1604 y FE(\))p FH(.)446 1724 y(Finally)-8 b(,)28 b FD(W)44 b FH(is)30 b(in)m(v)-5 b(arian)m(t)29 b(under)j FD(\031)t FE(\()p FD(X)1887 1739 y Fz(3)1926 1724 y FE(\))e FH(since)h FD(\031)t FE(\()p FD(X)2409 1739 y Fz(3)2448 1724 y FE(\))d(=)f([)q FD(\031)t FE(\()p FD(X)2823 1739 y Fz(1)2862 1724 y FE(\))p FD(;)17 b(\031)t FE(\()p FD(X)3122 1739 y Fz(2)3161 1724 y FE(\)])p FH(.)42 b(Th)m(us)32 b FD(W)44 b FH(is)147 1845 y(in)m(v)-5 b(arian)m(t.)42 b(Since)33 b(b)m(y)g(de\034nition)f FD(W)46 b FH(con)m(tains)32 b FD(v)t FH(,)h(w)m(e)g(m)m(ust)g(ha)m(v)m (e)h FD(W)41 b FE(=)28 b FD(V)21 b FH(.)446 1965 y(Since)41 b FD(Y)766 1980 y Fz(1)846 1965 y FH(is)g(a)g(ro)s(ot)f(v)m(ector)i (with)f(ro)s(ot)f FA(\000)p FD(\013)2149 1929 y Fz(\(1\))2285 1965 y FH(and)h FD(Y)2540 1980 y Fz(2)2620 1965 y FH(is)g(a)g(ro)s(ot)f (v)m(ector)i(with)f(ro)s(ot)147 2086 y FA(\000)p FD(\013)287 2049 y Fz(\(2\))382 2086 y FH(,)k(Lemma)c(121)g(tells)h(us)h(that)f (eac)m(h)h(elemen)m(t)f(of)g(the)h(form)e(\(6.5\))h(is)g(either)g(zero) h(or)f(a)147 2206 y(w)m(eigh)m(t)h(v)m(ector)g(with)e(w)m(eigh)m(t)i FD(\026)1377 2221 y Fz(0)1444 2206 y FA(\000)29 b FD(\013)1613 2170 y Fz(\()p Ft(i)1664 2179 y Fr(1)1699 2170 y Fz(\))1759 2206 y FA(\000)g(\001)17 b(\001)g(\001)27 b(\000)i FD(\013)2179 2170 y Fz(\()p Ft(i)2230 2178 y Fn(n)2273 2170 y Fz(\))2304 2206 y FH(.)72 b(Th)m(us)44 b FD(V)65 b FE(=)44 b FD(W)56 b FH(is)41 b(spanned)i(b)m(y)g FD(v)147 2326 y FH(together)i(with)f(w)m (eigh)m(t)h(v)m(ectors)h(with)e(w)m(eigh)m(ts)h(lo)m(w)m(er)g(than)g FD(\026)2601 2341 y Fz(0)2640 2326 y FH(.)79 b(Th)m(us)46 b FD(\026)3064 2341 y Fz(0)3148 2326 y FH(is)e(the)h(highest)147 2447 y(w)m(eigh)m(t)33 b(for)f FD(V)22 b FH(.)446 2567 y(F)-8 b(urthermore,ev)m(ery)48 b(elemen)m(t)f(of)f FD(W)60 b FH(can)47 b(b)s(e)g(written)f(as)h(a)g(m)m(ultiple)d(of)i FD(v)51 b FH(plus)46 b(a)147 2688 y(linear)39 b(com)m(bination)f(of)h (w)m(eigh)m(t)i(v)m(ectors)g(with)f(w)m(eigh)m(ts)h(lo)m(w)m(er)f(than) g FD(\026)2903 2703 y Fz(0)2942 2688 y FH(.)67 b(Th)m(us)41 b(the)g(w)m(eigh)m(t)147 2808 y(space)e(corresp)s(onding)e(to)g FD(\026)1227 2823 y Fz(0)1303 2808 y FH(is)g(spanned)i(b)m(y)f FD(v)t FH(;)i(that)d(is,)h(the)g(w)m(eigh)m(t)g(space)g(corresp)s (onding)147 2928 y(to)32 b FD(\026)325 2943 y Fz(0)397 2928 y FH(is)g(one-dimensional.)41 b Fy(\003)147 3103 y FI(Prop)s(osition)36 b(128)48 b FC(Every)25 b(irr)-5 b(e)g(ducible)25 b(r)-5 b(epr)g(esentation)24 b(of)g FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))30 b FC(is)25 b(a)g(highest)f(weight) g(cylic)147 3224 y(r)-5 b(epr)g(esentation,)34 b(with)h(a)f(unique)h (highest)f(weight)h FD(\026)2109 3239 y Fz(0)2148 3224 y FC(.)147 3477 y FI(Pro)s(of)147 3661 y FH(Uniqueness)c(is)e (immediate,)f(since)i(b)m(y)g(the)g(previous)g(Prop)s(osition,)e FD(\026)2777 3676 y Fz(0)2845 3661 y FH(is)h(the)h(highest)g(w)m(eigh)m (t,)147 3782 y(and)j(t)m(w)m(o)g(distinct)f(w)m(eigh)m(ts)h(cannot)g(b) s(oth)f(b)s(e)h(highest.)446 3902 y(W)-8 b(e)29 b(ha)m(v)m(e)h(already) f(sho)m(wn)h(that)e(ev)m(ery)j(irreducible)d(represen)m(tation)h(is)f (the)i(direct)e(sum)147 4022 y(of)j(its)g(w)m(eigh)m(t)h(spaces.)45 b(Since)31 b(the)h(represen)m(tation)g(is)f(\034nite-dimensional,)e (there)j(can)g(b)s(e)g(only)147 4143 y(\034nitely)38 b(man)m(y)g(w)m(eigh)m(ts.)60 b(It)39 b(follo)m(ws)d(that)i(there)h(m)m (ust)f(exist)h(a)e(w)m(eigh)m(t)i FD(\026)3006 4158 y Fz(0)3083 4143 y FH(suc)m(h)g(that)f(there)147 4263 y(is)e(no)h(w)m (eigh)m(t)f FD(\026)e FA(6)p FE(=)h FD(\026)964 4278 y Fz(0)1039 4263 y FH(with)h FD(\026)e FA(\027)h FD(\026)1529 4278 y Fz(0)1568 4263 y FH(.)55 b(This)37 b(sa)m(ys)h(that)e(there)h (is)f(no)h(w)m(eigh)m(t)g(higher)e(than)i FD(\026)3703 4278 y Fz(0)147 4384 y FH(\(whic)m(h)d(is)f FC(not)44 b FH(the)34 b(same)f(as)h(sa)m(ying)g(the)g FD(\026)1808 4399 y Fz(0)1880 4384 y FH(is)g(highest\).)46 b(But)34 b(if)f(there)h(is)f(no)h(w)m(eigh)m(t)g(higher)147 4504 y(than)f FD(\026)434 4519 y Fz(0)473 4504 y FH(,)f(then)i(for)e(an)m(y) h(non-zero)f(w)m(eigh)m(t)h(v)m(ector)h FD(v)i FH(with)c(w)m(eigh)m(t)h FD(\026)2758 4519 y Fz(0)2797 4504 y FH(,)g(w)m(e)g(m)m(ust)g(ha)m(v)m (e)1470 4677 y FD(\031)t FE(\()p FD(X)1648 4692 y Fz(1)1687 4677 y FE(\))p FD(v)e FE(=)d FD(\031)t FE(\()p FD(X)2085 4692 y Fz(2)2124 4677 y FE(\))p FD(v)j FE(=)d(0)p FH(.)147 4851 y(\(F)-8 b(or)32 b(otherwise,)h(sa)m(y)-8 b(,)33 b FD(\031)t FE(\()p FD(X)1186 4866 y Fz(1)1226 4851 y FE(\))p FD(v)j FH(will)30 b(b)s(e)j(a)f(w)m(eigh)m(t)h(v)m(ector)g (with)g(w)m(eigh)m(t)f FD(\026)2936 4866 y Fz(0)2998 4851 y FE(+)22 b FD(\013)3159 4815 y Fz(\(1\))3280 4851 y FA(\037)29 b FD(\026)3445 4866 y Fz(0)3484 4851 y FH(.\))446 4971 y(Since)39 b FD(\031)k FH(is)c(assumed)h(irreducible,)f(the)h (smallest)d(in)m(v)-5 b(arian)m(t)38 b(subspace)j(con)m(taining)c FD(v)147 5092 y FH(m)m(ust)c(b)s(e)g(the)g(whole)f(space;)i(therefore)f (the)g(represen)m(tation)g(is)f(highest)g(w)m(eigh)m(t)h(cyclic.)43 b Fy(\003)147 5267 y FI(Prop)s(osition)36 b(129)48 b FC(Every)25 b(highest)g(weight)f(cyclic)h(r)-5 b(epr)g(esentation)24 b(of)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))31 b FC(is)25 b(irr)-5 b(e)g(ducible.)p eop %%Page: 143 143 143 142 bop 147 48 a Fx(Pro)r(of)28 b(of)f(the)h(Classi\034cation)e (Theorem)2174 b FG(143)147 298 y FI(Pro)s(of)147 496 y FH(Let)28 b FE(\()p FD(\031)t(;)17 b(V)22 b FE(\))28 b FH(b)s(e)g(a)g(highest)g(w)m(eigh)m(t)g(cyclic)g(represen)m(tation)h (with)e(highest)i(w)m(eigh)m(t)f FD(\026)3254 511 y Fz(0)3321 496 y FH(and)g(cyclic)147 616 y(v)m(ector)34 b FD(v)t FH(.)44 b(By)33 b(complete)f(reducibilit)m(y)f(\(Prop)s(osition)g (115\),)h FD(V)54 b FH(decomp)s(oses)34 b(as)f(a)g(direct)f(sum)147 736 y(of)g(irreducible)f(represen)m(tations)1699 988 y FD(V)1806 960 y FA(\030)1806 992 y FE(=)1911 893 y Fv(M)1974 1103 y Ft(i)2078 988 y FD(V)2135 1003 y Ft(i)2163 988 y FH(.)1352 b(\(6.6\))446 1315 y(By)42 b(Prop)s(osition)e(125,)k (eac)m(h)f(of)e(the)h FD(V)1943 1330 y Ft(i)1971 1315 y FH('s)h(is)e(the)h(direct)g(sum)g(of)f(its)g(w)m(eigh)m(t)h(spaces.) 147 1436 y(Th)m(us)35 b(since)e(the)h(w)m(eigh)m(t)f FD(\026)1171 1451 y Fz(0)1243 1436 y FH(o)s(ccurs)h(in)e FD(V)22 b FH(,)33 b(it)f(m)m(ust)h(o)s(ccur)g(in)f(some)h FD(V)2817 1451 y Ft(i)2845 1436 y FH(.)45 b(On)33 b(the)h(other)f (hand,)147 1556 y(Prop)s(osition)j(127)g(sa)m(ys)j(that)e(the)g(w)m (eigh)m(t)h(space)g(corresp)s(onding)f(to)g FD(\026)2853 1571 y Fz(0)2929 1556 y FH(is)g(one-dimensional,)147 1676 y(that)g(is,)h FD(v)i FH(is)d(\(up)g(to)g(a)f(constan)m(t\))i(the) g FC(only)45 b FH(v)m(ector)38 b(in)e FD(V)59 b FH(with)36 b(w)m(eigh)m(t)i FD(\026)3037 1691 y Fz(0)3076 1676 y FH(.)56 b(Th)m(us)39 b FD(V)3468 1691 y Ft(i)3533 1676 y FH(m)m(ust)147 1797 y(con)m(tain)f FD(v)t FH(.)62 b(But)39 b(then)g(that)f FD(V)1339 1812 y Ft(i)1406 1797 y FH(is)g(an)h(in)m(v) -5 b(arian)m(t)37 b(subspace)j(con)m(taing)e FD(v)t FH(,)i(so)f FD(V)3180 1812 y Ft(i)3246 1797 y FE(=)f FD(V)22 b FH(.)61 b(Th)m(us)147 1917 y(there)33 b(is)g(only)f(one)g(term)g(in)g(the)h (sum)f(\(6.6\),)h(and)f FD(V)54 b FH(is)32 b(irreducible.)42 b Fy(\003)147 2197 y FI(Prop)s(osition)36 b(130)48 b FC(Two)38 b(irr)-5 b(e)g(ducible)37 b(r)-5 b(epr)g(esentations)37 b(of)h FF(sl)16 b FE(\(3;)h Fu(C)j FE(\))44 b FC(with)38 b(the)g(same)f(highest)147 2318 y(weight)d(ar)-5 b(e)35 b(e)-5 b(quivalent.)147 2617 y FI(Pro)s(of)147 2815 y FH(W)d(e)38 b(no)m(w)f(kno)m(w)i(that)d(a)h(represen)m(tation)h(is)f (irreducible)e(if)h(and)i(only)e(if)g(it)g(is)h(highest)g(w)m(eigh)m(t) 147 2935 y(cyclic.)56 b(Supp)s(ose)38 b(that)f FE(\()p FD(\031)t(;)17 b(V)k FE(\))36 b FH(and)h FE(\()p FD(\033)n(;)17 b(W)d FE(\))37 b FH(are)g(t)m(w)m(o)g(suc)m(h)i(represen)m(tations)f (with)e(the)h(same)147 3056 y(highest)30 b(w)m(eigh)m(t)g FD(\026)846 3071 y Fz(0)885 3056 y FH(.)43 b(Let)30 b FD(v)k FH(and)c FD(w)i FH(b)s(e)e(the)h(cyclic)e(v)m(ectors)j(for)d FD(V)52 b FH(and)30 b FD(W)14 b FH(,)30 b(resp)s(ectiv)m(ely)-8 b(.)43 b(No)m(w)147 3176 y(consider)26 b(the)g(represen)m(tation)h FD(V)i FA(\010)8 b FD(W)14 b FH(,)28 b(and)e(let)f FD(U)36 b FH(b)s(e)26 b(smallest)e(in)m(v)-5 b(arian)m(t)24 b(subspace)k(of)d FD(V)30 b FA(\010)8 b FD(W)147 3297 y FH(whic)m(h)33 b(con)m(tains)g(the)g(v)m(ector)g FE(\()p FD(v)t(;)17 b(w)s FE(\))p FH(.)446 3423 y(By)29 b(de\034nition,)f FD(U)38 b FH(is)28 b(a)g(highest)g(w)m(eigh)m(t)g(cyclic)g(represen)m (tation,)i(therefore)e(irreducible)147 3544 y(b)m(y)51 b(Prop)s(osition.)94 b(129.)i(Consider)50 b(the)h(t)m(w)m(o)f(\020pro)5 b(jection\021)50 b(maps)f FD(P)2910 3559 y Fz(1)3007 3544 y FE(:)58 b FD(V)e FA(\010)34 b FD(W)71 b FA(!)57 b FD(V)22 b FH(,)147 3664 y FD(P)210 3679 y Fz(1)250 3664 y FE(\()p FD(v)t(;)17 b(w)s FE(\))45 b(=)h FD(v)i FH(and)43 b FD(P)1019 3679 y Fz(2)1105 3664 y FE(:)k FD(V)k FA(\010)30 b FD(W)60 b FA(!)47 b FD(W)14 b FH(,)46 b FD(P)1935 3679 y Fz(1)1974 3664 y FE(\()p FD(v)t(;)17 b(w)s FE(\))45 b(=)i FD(w)s FH(.)76 b(It)43 b(is)h(easy)g(to)f(c)m(hec) m(k)j(that)e FD(P)3703 3679 y Fz(1)147 3785 y FH(and)34 b FD(P)401 3800 y Fz(2)474 3785 y FH(are)g(morphisms)e(of)h(represen)m (tations.)48 b(Therefore)35 b(the)f(restrictions)f(of)h FD(P)3256 3800 y Fz(1)3329 3785 y FH(and)f FD(P)3582 3800 y Fz(2)3655 3785 y FH(to)147 3905 y FD(U)39 b FA(\032)28 b FD(V)43 b FA(\010)23 b FD(W)46 b FH(will)30 b(also)i(b)s(e)h (morphisms.)446 4032 y(Clearly)42 b(neither)53 b FD(P)1209 4047 y Fz(1)1248 4032 y FA(j)1276 4061 y Ft(U)1378 4032 y FH(nor)f FD(P)1634 4047 y Fz(2)1673 4032 y FA(j)1701 4061 y Ft(U)1803 4032 y FH(is)42 b(the)i(zero)f(map)f(\(since)h(b)s (oth)f(are)h(non-zero)g(on)147 4152 y FE(\()p FD(v)t(;)17 b(w)s FE(\))o FH(\).)58 b(Moreo)m(v)m(er,)41 b FD(U)10 b FH(,)39 b FD(V)22 b FH(,)39 b(and)e FD(W)51 b FH(are)38 b(all)d(irreducible.)58 b(Therefore,)39 b(b)m(y)g(Sc)m(h)m(ur's)g (Lemma,)157 4273 y FD(P)220 4288 y Fz(1)259 4273 y FA(j)287 4302 y Ft(U)374 4273 y FH(is)27 b(an)h(isomorphism)d(of)i FD(U)38 b FH(with)28 b FD(V)21 b FH(,)29 b(and)38 b FD(P)1983 4288 y Fz(2)2022 4273 y FA(j)2050 4302 y Ft(U)2136 4273 y FH(is)27 b(an)h(isomorphism)d(of)i FD(U)39 b FH(with)27 b FD(W)14 b FH(.)41 b(Th)m(us)147 4393 y FD(V)253 4365 y FA(\030)254 4397 y FE(=)359 4393 y FD(U)463 4365 y FA(\030)464 4397 y FE(=)568 4393 y FD(W)14 b FH(.)43 b Fy(\003)147 4673 y FI(Prop)s(osition)36 b(131)48 b FC(If)40 b FD(\031)45 b FC(is)40 b(an)g(irr)-5 b(e)g(ducible)40 b(r)-5 b(epr)g(esentation)40 b(of)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FC(,)42 b(then)e(the)h(highest)147 4793 y(weight)34 b(of)h FD(\031)k FC(is)34 b(of)h(the)g(form)1665 5033 y FD(\026)27 b FE(=)h(\()p FD(m)1978 5048 y Fz(1)2018 5033 y FD(;)17 b(m)2147 5048 y Fz(2)2186 5033 y FE(\))147 5272 y FC(with)35 b FD(m)444 5287 y Fz(1)518 5272 y FC(and)g FD(m)793 5287 y Fz(2)867 5272 y FC(non-ne)-5 b(gative)34 b(inte)-5 b(gers.)p eop %%Page: 144 144 144 143 bop 147 48 a FK(144)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)147 298 y FI(Pro)s(of)147 483 y FH(W)-8 b(e)45 b(already)f(kno)m(w)i(that)e FC(al)5 b(l)54 b FH(of)44 b(the)h(w)m(eigh)m(ts)g(of)f FD(\031)k FH(are)d(of)f(the)h(form)e FE(\()p FD(m)3063 498 y Fz(1)3102 483 y FD(;)17 b(m)3231 498 y Fz(2)3271 483 y FE(\))p FH(,)47 b(with)d FD(m)3702 498 y Fz(1)147 604 y FH(and)c FD(m)429 619 y Fz(2)509 604 y FH(in)m(tegers.)66 b(W)-8 b(e)40 b(m)m(ust)g(sho)m(w)h(that)f(if)f FD(\026)1978 619 y Fz(0)2057 604 y FE(=)h(\()p FD(m)2296 619 y Fz(1)2336 604 y FD(;)17 b(m)2465 619 y Fz(2)2505 604 y FE(\))40 b FH(is)f(the)h(highest)g(w)m(eigh)m(t,)i(then)147 724 y FD(m)232 739 y Fz(1)308 724 y FH(and)37 b FD(m)587 739 y Fz(2)663 724 y FH(are)f(b)s(oth)g(non-negativ)m(e.)55 b(F)-8 b(or)35 b(this,)i(w)m(e)g(again)e(use)i(what)g(w)m(e)g(kno)m(w)h (ab)s(out)e(the)147 844 y(represen)m(tations)29 b(of)e FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))p FH(.)48 b(The)29 b(follo)m(wing)c(result)i(can)h(b)s(e)g(obtained)g(from)e(the)i(pro)s (of)f(of)h(the)147 965 y(classi\034cation)j(of)h(the)h(irreducible)e (represen)m(tations)j(of)e FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(.)391 1196 y(Let)38 b FE(\()p FD(\031)t(;)17 b(V)k FE(\))38 b FH(b)s(e)g(an)m(y)g(\034nite-dimensional)c(represen)m (tation)39 b(of)e FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(.)65 b(Let)38 b FD(v)j FH(b)s(e)391 1316 y(an)h(eigen)m(v)m(ector)i (for)e FD(\031)t FE(\()p FD(H)8 b FE(\))42 b FH(with)g(eigen)m(v)-5 b(alue)42 b FD(\025)p FH(.)73 b(If)42 b FD(\031)t FE(\()p FD(X)8 b FE(\))p FD(v)48 b FE(=)c(0)p FH(,)h(then)e FD(\025)f FH(is)g(a)391 1436 y(non-negativ)m(e)32 b(in)m(teger.)446 1667 y(No)m(w,)g(if)d FD(\031)34 b FH(is)c(an)g(irreducible)f(represen) m(tation)i(of)f FF(sl)17 b FE(\(3;)g Fs(C)p FE(\))30 b FH(with)f(highest)i(w)m(eigh)m(t)f FD(\026)3599 1682 y Fz(0)3666 1667 y FE(=)147 1788 y(\()p FD(m)270 1803 y Fz(1)310 1788 y FD(;)17 b(m)439 1803 y Fz(2)478 1788 y FE(\))p FH(,)28 b(and)f(if)e FD(v)32 b FA(6)p FE(=)27 b(0)f FH(is)h(a)f(w)m(eigh)m(t)h(v)m(ector)g(with)f(w)m(eigh)m(t)h FD(\026)2430 1803 y Fz(0)2469 1788 y FH(,)h(then)f(w)m(e)h(m)m(ust)f (ha)m(v)m(e)h FD(\031)t FE(\()p FD(X)3511 1803 y Fz(1)3550 1788 y FE(\))p FD(v)j FE(=)147 1908 y FD(\031)t FE(\()p FD(X)325 1923 y Fz(2)364 1908 y FE(\))p FD(v)h FE(=)27 b(0)p FH(.)43 b(\(Otherwise,)32 b FD(\026)1287 1923 y Fz(0)1357 1908 y FH(w)m(ouldn't)f(b)s(e)g(highest.\))43 b(Th)m(us)33 b(applying)c(the)j(ab)s(o)m(v)m(e)f(result)g(to)147 2029 y(the)g(restrictions)f(of)h FD(\031)j FH(to)d FA(f)o FD(H)1263 2044 y Fz(1)1303 2029 y FD(;)17 b(X)1428 2044 y Fz(1)1467 2029 y FD(;)g(Y)1568 2044 y Fz(1)1607 2029 y FA(g)30 b FH(and)h(to)f FA(f)p FD(H)2123 2044 y Fz(2)2162 2029 y FD(;)17 b(X)2287 2044 y Fz(2)2326 2029 y FD(;)g(Y)2427 2044 y Fz(2)2466 2029 y FA(g)31 b FH(sho)m(ws)h(that)e FD(m)3119 2044 y Fz(1)3190 2029 y FH(and)g FD(m)3462 2044 y Fz(2)3533 2029 y FH(m)m(ust)147 2149 y(b)s(e)j(non-negativ)m(e.) 43 b Fy(\003)147 2380 y FI(Prop)s(osition)36 b(132)48 b FC(If)33 b FD(m)1155 2395 y Fz(1)1228 2380 y FC(and)g FD(m)1501 2395 y Fz(2)1573 2380 y FC(ar)-5 b(e)33 b(non-ne)-5 b(gative)32 b(inte)-5 b(gers,)33 b(then)g(ther)-5 b(e)33 b(exists)g(an)f(irr)-5 b(e-)147 2501 y(ducible)34 b(r)-5 b(epr)g(esentation)34 b(of)h FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))41 b FC(with)35 b(highest)f(weight)g FD(\026)28 b FE(=)f(\()p FD(m)2727 2516 y Fz(1)2767 2501 y FD(;)17 b(m)2896 2516 y Fz(2)2935 2501 y FE(\))p FC(.)147 2763 y FI(Pro)s(of)147 2948 y FH(Note)50 b(that)g(the)g(trivial)e(represen)m (tation)i(is)g(an)f(irreducible)g(represen)m(tation)i(with)e(highest) 147 3069 y(w)m(eigh)m(t)d FE(\(0)p FD(;)17 b FE(0\))o FH(.)81 b(So)45 b(w)m(e)h(need)h(only)d(construct)j(represen)m(tations) f(with)f(at)f(least)h(one)h(of)e FD(m)3702 3084 y Fz(1)147 3189 y FH(and)33 b FD(m)422 3204 y Fz(2)494 3189 y FH(p)s(ositiv)m(e.) 446 3310 y(First,)e(w)m(e)i(construct)g(t)m(w)m(o)g(irreducible)d (represen)m(tations)j(with)f(highest)f(w)m(eigh)m(ts)i FE(\(1)p FD(;)17 b FE(0\))147 3430 y FH(and)32 b FE(\(0)p FD(;)17 b FE(1\))o FH(.)44 b(\(These)33 b(are)g(the)f(so-called)f FI(fundamen)m(tal)k(represen)m(tations)p FH(.\))43 b(The)33 b(standard)147 3551 y(represen)m(tation)k(of)e FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))42 b FH(is)35 b(an)h(irreducible)e(represen)m (tation)j(with)e(highest)h(w)m(eigh)m(t)g FE(\(1)p FD(;)17 b FE(0\))o FH(,)147 3671 y(as)32 b(is)g(easily)f(c)m(hec)m(k)m(ed.)47 b(T)-8 b(o)32 b(construct)h(an)f(irreducible)e(represen)m(tation)j (with)f(w)m(eigh)m(t)g FE(\(0)p FD(;)17 b FE(1\))31 b FH(w)m(e)147 3791 y(mo)s(dify)g(the)i(standard)g(represen)m(tation.)44 b(Sp)s(eci\034cally)-8 b(,)31 b(w)m(e)j(de\034ne)1667 4012 y FD(\031)t FE(\()p FD(Z)7 b FE(\))28 b(=)f FA(\000)p FD(Z)2158 3971 y Ft(tr)3542 4012 y FH(\(6.7\))147 4247 y(for)32 b(all)f FD(Z)j FA(2)28 b FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))6 b FH(.)43 b(Using)32 b(the)h(fact)f(that)h FE(\()p FD(AB)5 b FE(\))2089 4199 y Ft(tr)2180 4247 y FE(=)28 b FD(B)2363 4211 y Ft(tr)2426 4247 y FD(A)2499 4211 y Ft(tr)2563 4247 y FH(,)k(it)g(is)g(easy)h(to)g(c)m(hec)m(k)h (that)1367 4468 y FA(\000)17 b FE([)p FD(Z)1555 4483 y Fz(1)1594 4468 y FD(;)g(Z)1705 4483 y Fz(2)1744 4468 y FE(])1771 4420 y Ft(tr)1862 4468 y FE(=)1966 4387 y Fv(\002)2007 4468 y FA(\000)p FD(Z)2158 4427 y Ft(tr)2151 4493 y Fz(1)2222 4468 y FD(;)g FA(\000)p FD(Z)2417 4427 y Ft(tr)2410 4493 y Fz(2)2481 4387 y Fv(\003)147 4689 y FH(so)35 b(that)g FD(\031)k FH(is)34 b(really)g(a)g(represen)m (tation.)52 b(\(This)35 b(is)f(isomorphic)f(to)h(the)i(dual)e(of)g(the) i(standard)147 4810 y(represen)m(tation,)e(as)e(de\034ned)i(in)e (Exercise)i(14)e(of)g(Chapter)i(5.\))43 b(It)32 b(is)h(easy)g(to)f(see) i(that)f FD(\031)j FH(is)c(an)147 4930 y(irreducible)f(represen)m (tation)j(with)e(highest)g(w)m(eigh)m(t)h FE(\(0)p FD(;)17 b FE(1\))o FH(.)446 5051 y(Let)44 b FE(\()p FD(\031)725 5066 y Fz(1)764 5051 y FD(;)17 b(V)865 5066 y Fz(1)904 5051 y FE(\))44 b FH(denote)g Fu(C)1377 5015 y Fz(3)1465 5051 y FH(acted)g(on)g(b)m(y)g(the)g(standard)f(represen)m(tation,)k (and)d(let)e FD(v)3702 5066 y Fz(1)147 5171 y FH(denote)29 b(a)g(w)m(eigh)m(t)f(v)m(ector)i(corresp)s(onding)e(to)g(the)h(highest) f(w)m(eigh)m(t)h FE(\(1)p FD(;)17 b FE(0\))o FH(.)42 b(\(So,)30 b FD(v)3197 5186 y Fz(1)3264 5171 y FE(=)d(\(1)p FD(;)17 b FE(0)p FD(;)g FE(0\))p FH(.\))147 5292 y(Let)48 b FE(\()p FD(\031)430 5307 y Fz(2)469 5292 y FD(;)17 b(V)570 5307 y Fz(2)609 5292 y FE(\))48 b FH(denote)g Fu(C)1089 5255 y Fz(3)1182 5292 y FH(acted)g(on)f(b)m(y)h(the)g (represen)m(tation)g(\(6.7\),)i(and)e(let)e FD(v)3277 5307 y Fz(2)3364 5292 y FH(denote)i(a)p eop %%Page: 145 145 145 144 bop 147 48 a Fx(An)28 b(Example:)37 b(Highest)28 b(W)-7 b(eigh)n(t)27 b Fe(\()q(1)p Fa(;)14 b Fe(1\))2186 b FG(145)147 298 y FH(w)m(eigh)m(t)42 b(v)m(ector)h(for)e(the)h (highest)g(w)m(eigh)m(t)g FE(\()o(0)p FD(;)17 b FE(1\))p FH(.)70 b(\(So,)44 b FD(v)2339 313 y Fz(2)2422 298 y FE(=)f(\(0)p FD(;)17 b FE(0)p FD(;)g FE(1\))p FH(.\))69 b(No)m(w)43 b(consider)f(the)147 418 y(represen)m(tation)1271 646 y FD(V)1328 661 y Fz(1)1389 646 y FA(\012)23 b FD(V)1546 661 y Fz(1)1602 646 y FA(\001)17 b(\001)g(\001)j(\012)j FD(V)1897 661 y Fz(1)1958 646 y FA(\012)g FD(V)2115 661 y Fz(2)2177 646 y FA(\012)f FD(V)2333 661 y Fz(2)2389 646 y FA(\001)17 b(\001)g(\001)e FD(V)2579 661 y Fz(2)147 874 y FH(where)35 b FD(V)487 889 y Fz(1)561 874 y FH(o)s(ccurs)g FD(m)949 889 y Fz(1)1022 874 y FH(times,)f(and)g FD(V)1560 889 y Fz(2)1634 874 y FH(o)s(ccurs)g FD(m)2021 889 y Fz(2)2095 874 y FH(times.)47 b(Note)35 b(that)f(the)g(action)f(of)h FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))147 995 y FH(on)33 b(this)f(space)h(is)763 1223 y FD(Z)h FA(!)28 b FE(\()o FD(\031)1084 1238 y Fz(1)1124 1223 y FE(\()p FD(Z)7 b FE(\))22 b FA(\012)h FD(I)h FA(\001)17 b(\001)g(\001)k(\012)h FD(I)8 b FE(\))859 1368 y(+)22 b(\()p FD(I)30 b FA(\012)22 b FD(\031)1222 1383 y Fz(1)1262 1368 y FE(\()p FD(Z)7 b FE(\))22 b FA(\012)h FD(I)h FA(\001)17 b(\001)g(\001)j(\012)j FD(I)8 b FE(\))22 b(+)g FA(\001)17 b(\001)g(\001)j FE(+)i(\()p FD(I)30 b FA(\012)23 b(\001)17 b(\001)g(\001)e FD(I)30 b FA(\012)22 b FD(\031)2855 1383 y Fz(2)2895 1368 y FE(\()p FD(Z)7 b FE(\)\))17 b FH(.)415 b(\(6.8\))147 1596 y(Let)33 b FD(\031)377 1611 y Ft(m)439 1620 y Fr(1)474 1611 y Ft(;m)556 1620 y Fr(2)628 1596 y FH(denote)g(this)f(represen)m(tation.) 446 1719 y(Consider)h(the)g(v)m(ector)1036 1947 y FD(v)1083 1962 y Ft(m)1145 1971 y Fr(1)1180 1962 y Ft(;m)1262 1971 y Fr(2)1329 1947 y FE(=)27 b FD(v)1479 1962 y Fz(1)1541 1947 y FA(\012)22 b FD(v)1687 1962 y Fz(1)1744 1947 y FA(\001)17 b(\001)g(\001)j(\012)j FD(v)2029 1962 y Fz(1)2090 1947 y FA(\012)g FD(v)2237 1962 y Fz(2)2299 1947 y FA(\012)f FD(v)2445 1962 y Fz(2)2502 1947 y FA(\001)17 b(\001)g(\001)j(\012)j FD(v)2787 1962 y Fz(2)2826 1947 y FH(.)147 2176 y(Then)34 b(applying)d(\(6.8\))h(sho)m(ws)i(that)1317 2404 y FD(\031)1372 2419 y Ft(m)1434 2428 y Fr(1)1469 2419 y Ft(;m)1551 2428 y Fr(2)1590 2404 y FE(\()p FD(H)1709 2419 y Fz(1)1748 2404 y FE(\))p FD(v)1833 2419 y Ft(m)1895 2428 y Fr(1)1931 2419 y Ft(;m)2013 2428 y Fr(2)2079 2404 y FE(=)28 b FD(m)2268 2419 y Fz(1)2307 2404 y FD(v)2354 2419 y Ft(m)2416 2428 y Fr(1)2451 2419 y Ft(;m)2533 2428 y Fr(2)1317 2549 y FD(\031)1372 2564 y Ft(m)1434 2573 y Fr(1)1469 2564 y Ft(;m)1551 2573 y Fr(2)1590 2549 y FE(\()p FD(H)1709 2564 y Fz(2)1748 2549 y FE(\))p FD(v)1833 2564 y Ft(m)1895 2573 y Fr(1)1931 2564 y Ft(;m)2013 2573 y Fr(2)2079 2549 y FE(=)g FD(m)2268 2564 y Fz(2)2307 2549 y FD(v)2354 2564 y Ft(m)2416 2573 y Fr(1)2451 2564 y Ft(;m)2533 2573 y Fr(2)1317 2694 y FD(\031)1372 2709 y Ft(m)1434 2718 y Fr(1)1469 2709 y Ft(;m)1551 2718 y Fr(2)1590 2694 y FE(\()p FD(X)1709 2709 y Fz(1)1748 2694 y FE(\))p FD(v)1833 2709 y Ft(m)1895 2718 y Fr(1)1931 2709 y Ft(;m)2013 2718 y Fr(2)2079 2694 y FE(=)g(0)1317 2840 y FD(\031)1372 2855 y Ft(m)1434 2864 y Fr(1)1469 2855 y Ft(;m)1551 2864 y Fr(2)1590 2840 y FE(\()p FD(X)1709 2855 y Fz(2)1748 2840 y FE(\))p FD(v)1833 2855 y Ft(m)1895 2864 y Fr(1)1931 2855 y Ft(;m)2013 2864 y Fr(2)2079 2840 y FE(=)g(0)p FH(.)1283 b(\(6.9\))446 3070 y(No)m(w,)35 b(the)f(represen)m(tation)g FD(\031)1559 3085 y Ft(m)1621 3094 y Fr(1)1656 3085 y Ft(;m)1738 3094 y Fr(2)1811 3070 y FH(is)f FC(not)43 b FH(irreducible)32 b(\(unless)j FE(\()p FD(m)3024 3085 y Fz(1)3063 3070 y FD(;)17 b(m)3192 3085 y Fz(2)3232 3070 y FE(\))29 b(=)h(\(1)p FD(;)17 b FE(0\))32 b FH(or)147 3191 y FE(\(0)p FD(;)17 b FE(1\))o FH(\).)55 b(Ho)m(w)m(ev)m(er,)40 b(if)35 b(w)m(e)i(let)f FD(W)50 b FH(denote)37 b(the)g(smallest)e(in)m (v)-5 b(arian)m(t)34 b(subspace)39 b(con)m(taining)c(the)147 3311 y(v)m(ector)44 b FD(v)497 3326 y Ft(m)559 3335 y Fr(1)594 3326 y Ft(;m)676 3335 y Fr(2)715 3311 y FH(,)i(then)d(in)g (ligh)m(t)e(of)h(\(6.9\),)j FD(W)57 b FH(will)41 b(b)s(e)i(highest)g(w) m(eigh)m(t)g(cyclic)g(with)f(highest)147 3432 y(w)m(eigh)m(t)32 b FE(\()p FD(m)578 3447 y Fz(1)617 3432 y FD(;)17 b(m)746 3447 y Fz(2)786 3432 y FE(\))p FH(.)43 b(Therefore)32 b(b)m(y)g(Prop)s(osition)d(129,)i FD(W)45 b FH(is)30 b(irreducible)g(with)h(highest)g(w)m(eigh)m(t)147 3552 y FE(\()p FD(m)270 3567 y Fz(1)310 3552 y FD(;)17 b(m)439 3567 y Fz(2)478 3552 y FE(\))p FH(.)446 3675 y(Th)m(us)34 b FD(W)46 b FH(is)32 b(the)h(represen)m(tation)h(w)m(e)f(w)m(an)m(t.)44 b Fy(\003)446 3921 y FH(W)-8 b(e)33 b(ha)m(v)m(e)h(no)m(w)f(completed)f (the)h(pro)s(of)f(of)g(Theorem)g(1.)147 4193 y FI(6.5)112 b(An)38 b(Example:)47 b(Highest)36 b(W)-9 b(eigh)m(t)36 b FE(\(1)p FD(;)17 b FE(1\))147 4363 y FH(T)-8 b(o)37 b(obtain)f(the)h(irreducible)f(represen)m(tation)i(with)e(highest)h(w)m (eigh)m(t)g FE(\(1)p FD(;)17 b FE(1\))37 b FH(w)m(e)h(are)f(supp)s (osed)147 4484 y(to)44 b(tak)m(e)h(the)f(tensor)h(pro)s(duct)f(of)g (the)h(irreducible)d(represen)m(tations)k(with)d(highest)h(w)m(eigh)m (ts)147 4604 y FE(\(1)p FD(;)17 b FE(0\))28 b FH(and)i FE(\(0)p FD(;)17 b FE(1\))o FH(,)30 b(and)f(then)h(extract)g(a)f (certain)f(in)m(v)-5 b(arian)m(t)28 b(subspace.)44 b(Let)30 b(us)g(establish)e(some)147 4724 y(notation)i(for)h(the)h(represen)m (tations)h FE(\()o(1)p FD(;)17 b FE(0\))31 b FH(and)h FE(\(0)p FD(;)17 b FE(1\))o FH(.)43 b(In)32 b(the)f(standard)h (represen)m(tation,)h(the)147 4845 y(w)m(eigh)m(t)g(v)m(ectors)h(for) 961 5186 y FD(H)1042 5201 y Fz(1)1109 5186 y FE(=)1212 4985 y Fv(0)1212 5165 y(@)1341 5064 y FE(1)121 b(0)h(0)1341 5185 y(0)83 b FA(\000)p FE(1)g(0)1341 5305 y(0)121 b(0)h(0)1772 4985 y Fv(1)1772 5165 y(A)1876 5186 y FE(;)83 b FD(H)2067 5201 y Fz(2)2134 5186 y FE(=)2238 4985 y Fv(0)2238 5165 y(@)2366 5064 y FE(0)g(0)122 b(0)2366 5185 y(0)83 b(1)122 b(0)2366 5305 y(0)83 b(0)g FA(\000)p FE(1)2798 4985 y Fv(1)2798 5165 y(A)2901 5186 y FE(;)p eop %%Page: 146 146 146 145 bop 147 48 a FK(146)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)147 298 y FH(are)40 b(the)g(standard)g(basis)g(elemen)m(ts)g(for)f Fu(C)1781 262 y Fz(3)1826 298 y FH(,)j(namely)-8 b(,)40 b FD(e)2306 313 y Fz(1)2346 298 y FH(,)h FD(e)2459 313 y Fz(2)2499 298 y FH(,)g(and)f FD(e)2809 313 y Fz(3)2849 298 y FH(.)65 b(The)40 b(corresp)s(onding)147 418 y(w)m(eigh)m(ts)34 b(are)e FE(\(1)p FD(;)17 b FE(0\))o FH(,)33 b FE(\()p FA(\000)p FE(1)p FD(;)17 b FE(1\))o FH(,)33 b(and)g FE(\(0)p FD(;)17 b FA(\000)p FE(1\))o FH(.)44 b(The)33 b(highest)g(w)m(eigh)m(t) f(is)h FE(\()o(1)p FD(;)17 b FE(0\))p FH(.)446 539 y(Recall)31 b(that)1062 855 y FD(Y)1119 870 y Fz(1)1186 855 y FE(=)1290 654 y Fv(0)1290 834 y(@)1418 733 y FE(0)83 b(0)g(0)1418 854 y(1)g(0)g(0)1418 974 y(0)g(0)g(0)1772 654 y Fv(1)1772 834 y(A)1876 855 y FE(;)g FD(Y)2043 870 y Fz(2)2110 855 y FE(=)2213 654 y Fv(0)2213 834 y(@)2342 733 y FE(0)g(0)g(0)2342 854 y(0)g(0)g(0)2342 974 y(0)g(1)g(0)2696 654 y Fv(1)2696 834 y(A)2800 855 y FH(.)147 1189 y(Th)m(us)1278 1383 y FD(Y)1335 1398 y Fz(1)1375 1383 y FE(\()p FD(e)1458 1398 y Fz(1)1497 1383 y FE(\))g(=)g FD(e)1822 1398 y Fz(2)2028 1383 y FD(Y)2085 1398 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Ft(w)1229 2145 y FE(\()p FD(X)8 b FE(\))28 b(=)f FD(\031)1601 2064 y Fv(\000)1646 2145 y FE(Ad)q FD(w)1847 2104 y Fw(\000)p Fz(1)1940 2145 y FE(\()p FD(X)8 b FE(\))2105 2064 y Fv(\001)2178 2145 y FE(=)28 b FD(\031)t FE(\()p FD(w)2452 2104 y Fw(\000)p Fz(1)2546 2145 y FD(X)8 b(w)s FE(\))p FH(.)147 2319 y(Since)39 b FE(Ad)q FD(w)609 2283 y Fw(\000)p Fz(1)741 2319 y FH(is)g(a)g(Lie)f (algebra)g(automorphism,)g FD(\031)2187 2334 y Ft(w)2283 2319 y FH(will)f(in)h(fact)h(b)s(e)g(a)g(represen)m(tation)g(of)147 2440 y FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))5 b FH(.)446 2560 y(Recall)30 b(that)h(since)g FF(SU)q FE(\(3\))g FH(is)f(simply)g(connected,)j(then)f(for)f(eac)m(h)h(represen)m(tation) f FD(\031)k FH(of)147 2681 y FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))40 b FH(there)35 b(is)e(an)h(asso)s(ciated)h(represen)m(tation)f FE(\005)h FH(of)e FF(SU)q FE(\(3\))h FH(\(acting)f(on)h(the)h(same)f (space\))147 2801 y(suc)m(h)g(that)1627 2975 y FE(\005)1717 2895 y Fv(\000)1762 2975 y FD(e)1807 2934 y Ft(X)1875 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b(,)45 b(if)d FD(\026)i FH(is)f(a)g(w)m(eigh)m(t)h(for)f(some)g (represen)m(tation)h FD(\031)t FH(,)i(and)e FD(w)i FH(is)d(an)147 5171 y(elemen)m(t)j(of)g FD(W)14 b FH(,)49 b(then)e(there)g(will)d(b)s (e)j(some)f(new)h(w)m(eigh)m(t)g(whic)m(h)f(m)m(ust)h(also)e(b)s(e)i(a) f(w)m(eigh)m(t)147 5292 y(of)d FD(\031)t FH(.)75 b(W)-8 b(e)43 b(will)e(denote)i(this)g(new)h(w)m(eigh)m(t)f FD(w)32 b FA(\001)d FD(\026)p FH(.)75 b(F)-8 b(or)42 b(example,)j(if)d FD(\026)j FE(=)h(\()p FD(m)3234 5307 y Fz(1)3273 5292 y FD(;)17 b(m)3402 5307 y Fz(2)3442 5292 y FE(\))p FH(,)45 b(then)p eop %%Page: 150 150 150 149 bop 147 48 a FK(150)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)147 298 y FD(w)217 313 y Fz(1)279 298 y FA(\001)c FD(\026)29 b FE(=)g(\()p FA(\000)p FD(m)723 313 y Fz(1)786 298 y FA(\000)23 b FD(m)971 313 y Fz(2)1011 298 y FD(;)17 b(m)1140 313 y Fz(1)1179 298 y FE(\))p FH(.)47 b(\(W)-8 b(e)33 b(de\034ne)i FD(w)25 b FA(\001)e FD(\026)33 b 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FD(\026)g FC(is)g(a)f(weight)h(for)g FD(\031)k FC(if)36 b(and)h(only)g(if)g FD(w)26 b FA(\001)d FD(\026)37 b FC(is)g(a)g(weight)f(for)h FD(\031)t FC(.)51 b(The)36 b(multiplicity)i(of)e FD(\026)147 2155 y FC(is)f(the)g(same)f(as)g(the) h(multiplicity)g(of)f FD(w)25 b FA(\001)d FD(\026)p FC(.)446 2366 y FH(If)44 b(w)m(e)i(think)e(of)g(the)g(w)m(eigh)m(ts)i FD(\026)h FE(=)h(\()p FD(m)1995 2381 y Fz(1)2034 2366 y FD(;)17 b(m)2163 2381 y Fz(2)2203 2366 y FE(\))44 b FH(as)g(sitting)f(inside)h Fu(R)3087 2330 y Fz(2)3132 2366 y FH(,)j(then)e(w)m(e)h(can)147 2486 y(think)35 b(of)f(\(6.15\))h(as)g(a)f(\034nite)h(group)g(of)f(linear)g (transformations)f(of)h Fu(R)2812 2450 y Fz(2)2858 2486 y FH(.)50 b(\(The)36 b(fact)f(that)f(this)147 2607 y(is)44 b(a)g FC(gr)-5 b(oup)50 b FH(of)44 b(transformations)e(follo)m(ws)h (form)g(\(6.16\).\))77 b(Since)45 b(this)f(is)f(a)h FC(\034nite)52 b FH(group)43 b(of)147 2727 y(transformations,)30 b(it)h(is)g(p)s 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b(section)h(giv)m(es)f(a)g(brief)f(synopsis)j(of)d (the)i(structure)g(theory)g(and)f(represen)m(tation)h(theory)f(of)147 3877 y(complex)k(semisimple)e(Lie)i(algebras.)41 b(The)31 b(moral)c(of)i(the)h(story)g(is)f(that)g(all)e(suc)m(h)k(Lie)e (algebras)147 3997 y(lo)s(ok)j(and)g(feel)g(a)h(lot)e(lik)m(e)h FF(sl)17 b FE(\()o(3;)g Fu(C)j FE(\))6 b FH(.)43 b(This)33 b(section)g(will)d(not)i(con)m(tain)g(an)m(y)h(pro)s(ofs.)446 4118 y(If)h Fk(g)g FH(is)f(a)g(Lie)h(algebra,)f(a)g(subspace)j FD(I)h FA(\032)31 b Fk(g)i FH(is)h(said)f(to)g(b)s(e)h(an)g FI(ideal)f FH(if)g FE([)p FD(X)r(;)17 b(Y)k FE(])30 b FA(2)g FD(I)42 b FH(for)147 4238 y(all)33 b FD(X)40 b FA(2)32 b Fk(g)j FH(and)h(all)c FD(Y)54 b FA(2)32 b FD(I)8 b FH(.)51 b(A)35 b(Lie)f(algebra)g Fk(g)h FH(is)g(a)g(said)f(to)h(b)s (e)g FI(simple)c FH(if)j FE(dim)15 b Fk(g)33 b FA(\025)f FE(2)j FH(and)g Fk(g)147 4358 y FH(has)j(no)f(ideals)f(other)h(than)g FA(f)p FE(0)p FA(g)f FH(and)i Fk(g)p FH(.)57 b(A)37 b(Lie)f(algebra)g Fk(g)i FH(is)e(said)h(to)g(b)s(e)g FI(semisim)o(ple)31 b FH(if)36 b Fk(g)147 4479 y FH(can)d(b)s(e)g(written)f(as)h(the)g (direct)f(sum)h(of)f(simple)f(Lie)g(algebras.)446 4599 y(In)d(this)g(section)g(w)m(e)h(consider)g(semisimple)c(Lie)j(algebras) f(o)m(v)m(er)i(the)g(complex)e(n)m(um)m(b)s(ers.)147 4719 y(Examples)32 b(of)g(complex)g(semisimple)e(Lie)h(algebras)h (include)g FF(sl)17 b FE(\()o FD(n)p FE(;)g Fu(C)j FE(\))6 b FH(,)33 b FF(so)p FE(\()p FD(n)p FE(;)17 b Fu(C)j FE(\))38 b FH(\()p FD(n)28 b FA(\025)g FE(3)p FH(\),)k(and)147 4840 y FF(sp)q FE(\()p FD(n)p FE(;)17 b Fu(C)i FE(\))p FH(.)56 b(All)33 b(of)h(these)h(are)g(actually)e(simple,)h(except)i (for)e FF(so)p FE(\(4;)17 b Fu(C)i FE(\))41 b FH(whic)m(h)35 b(is)f(isomorphic)e(to)147 4960 y FF(sl)q FE(\(2;)17 b Fu(C)i FE(\))28 b FA(\010)23 b FF(sl)p FE(\(2;)17 b Fu(C)i FE(\))p FH(.)147 5171 y FI(De\034nition)36 b(136)49 b FC(L)-5 b(et)30 b Fk(g)g FC(b)-5 b(e)29 b(a)h(c)-5 b(omplex)29 b(semisimple)e(Lie)j(algebr)-5 b(a.)42 b(A)30 b(subsp)-5 b(ac)g(e)29 b Fk(h)h FC(of)f Fk(g)h FC(is)g(said)147 5292 y(to)35 b(b)-5 b(e)35 b(a)f FB(Cart\341n)43 b(sub)-6 b(algebr)g(a)35 b FC(if)p eop %%Page: 151 151 151 150 bop 147 48 a Fx(Complex)27 b(Semisimple)h(Lie)g(Algebras)2231 b FG(151)263 298 y FC(1.)48 b Fk(h)35 b FC(is)f(ab)-5 b(elian.)44 b(That)35 b(is,)f FE([)p FD(H)1430 313 y Fz(1)1469 298 y FD(;)17 b(H)1594 313 y Fz(2)1633 298 y FE(])28 b(=)g(0)34 b FC(for)h(al)5 b(l)34 b FD(H)2251 313 y Fz(1)2291 298 y FD(;)17 b(H)2416 313 y Fz(2)2482 298 y FA(2)28 b Fk(h)p FC(.)263 503 y(2.)48 b Fk(h)33 b FC(is)g(maximal)f(ab)-5 b(elian.)43 b(That)33 b(is,)h(if)f FD(X)i FA(2)28 b Fk(g)34 b FC(satis\034es)e FE([)p FD(H)r(;)17 b(X)8 b FE(])28 b(=)f(0)33 b FC(for)h(al)5 b(l)33 b FD(H)i FA(2)28 b Fk(h)p FC(,)33 b(then)391 623 y FD(X)j FA(2)28 b Fk(h)p FC(.)263 828 y(3.)48 b(F)-7 b(or)34 b(al)5 b(l)34 b FD(H)i FA(2)28 b Fk(h)p FC(,)34 b FE(ad)p FD(H)i FE(:)27 b Fk(g)h FA(!)f Fk(g)35 b FC(is)g(diagonalizable.)446 1058 y FH(Since)22 b(all)e(the)i FD(H)8 b FH('s)21 b(comm)m(ute,)j(so)e (do)f(the)h FE(ad)p FD(H)8 b FH('s.)40 b(\(I.e.,)25 b FE([ad)p FD(H)2733 1073 y Fz(1)2772 1058 y FD(;)17 b FE(ad)p FD(H)3000 1073 y Fz(2)3039 1058 y FE(])28 b(=)f(ad)17 b([)p FD(H)3425 1073 y Fz(1)3464 1058 y FD(;)g(H)3589 1073 y Fz(2)3628 1058 y FE(])28 b(=)147 1179 y(0)p FH(.\))74 b(By)43 b(assumption,)h(eac)m(h)g FE(ad)p FD(H)50 b FH(is)42 b(diagonalizable,)g(and)g(they)i(comm)m(ute,)g(therefore)g(the)147 1299 y FE(ad)p FD(H)8 b FH('s)31 b(are)g(sim)m(ultaneously)f (diagonalizable.)39 b(\(Using)30 b(a)h(standard)g(linear)e(algebra)h (fact.\))43 b(Let)147 1419 y Fk(h)199 1383 y Fw(\003)271 1419 y FH(denote)33 b(the)g(dual)f(of)g Fk(h)p FH(,)h(namely)-8 b(,)31 b(the)i(space)h(of)e(linear)f(functionals)g(on)i Fk(h)p FH(.)147 1650 y FI(De\034nition)j(137)49 b FC(If)31 b Fk(g)h FC(is)f(a)h(c)-5 b(omplex)30 b(semisimple)g(Lie)h(algebr)-5 b(a)31 b(and)g Fk(h)g FC(a)h(Cart\341n)f(sub)-5 b(algebr)g(a,)147 1771 y(then)33 b(an)g(element)g FD(\013)h FC(of)f Fk(h)1121 1734 y Fw(\003)1194 1771 y FC(is)g(said)g(to)h(b)-5 b(e)33 b(a)g FB(r)-6 b(o)g(ot)36 b FC(\(for)d Fk(g)h FC(with)f(r)-5 b(esp)g(e)g(ct)33 b(to)g Fk(h)p FC(\))h(if)f FD(\013)h FC(is)f(non-zer)-5 b(o)147 1891 y(and)34 b(ther)-5 b(e)35 b(exists)g FD(Z)f FA(6)p FE(=)28 b(0)34 b FC(in)h Fk(g)g FC(such)g(that)1601 2112 y FE([)p FD(H)r(;)17 b(Z)7 b FE(])28 b(=)f FD(\013)q FE(\()p FD(H)8 b FE(\))p FD(Z)1211 b FH(\(6.17\))147 2333 y FC(for)43 b(al)5 b(l)43 b FD(H)51 b FA(2)43 b Fk(h)p FC(.)70 b(\(Thus)43 b(a)g(r)-5 b(o)g(ot)43 b(is)g(a)g(non-zer)-5 b(o)42 b(set)i(of)f(simultane)-5 b(ous)42 b(eigenvalues)g(for)h(the)147 2453 y FE(ad)p FD(H)8 b FC('s.\))446 2574 y(The)45 b(ve)-5 b(ctor)45 b FD(Z)52 b FC(is)45 b(c)-5 b(al)5 b(le)-5 b(d)44 b(a)h FB(r)-6 b(o)g(ot)55 b(ve)-6 b(ctor)46 b FC(c)-5 b(orr)g(esp)g(onding)43 b(to)i(the)h(r)-5 b(o)g(ot)45 b FD(\013)q FC(,)i(and)e(the)147 2694 y(sp)-5 b(ac)g(e)33 b(of)g(al)5 b(l)33 b FD(Z)i FA(2)28 b Fk(g)33 b FC(satisfying)g(\(6.17\))g(is)g(the)h FB(r)-6 b(o)g(ot)41 b(sp)-6 b(ac)g(e)34 b FC(c)-5 b(orr)g(esp)g(onding) 32 b(to)h FD(\013)q FC(.)44 b(This)33 b(sp)-5 b(ac)g(e)147 2815 y(is)35 b(denote)-5 b(d)34 b Fk(g)658 2778 y Ft(\013)708 2815 y FC(.)446 2935 y(The)g(set)h(of)g(al)5 b(l)34 b(r)-5 b(o)g(ots)35 b(wil)5 b(l)34 b(b)-5 b(e)35 b(denote)-5 b(d)34 b FE(\001)p FC(.)446 3166 y FH(Note)c(that)g(if)f Fk(g)f FE(=)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(,)31 b(then)g(one)f(Cart\341n)g(subalgebra)g(is)g(the)g(space)i(spanned)f(b) m(y)147 3286 y FD(H)228 3301 y Fz(1)298 3286 y FH(and)f FD(H)566 3301 y Fz(2)605 3286 y FH(.)43 b(The)31 b(ro)s(ots)f(\(with)f (resp)s(ect)j(to)e(this)g(Cart\341n)g(subalgebra\))g(ha)m(v)m(e)h(b)s (een)g(calculated)147 3406 y(in)h(\(6.3\).)147 3637 y FI(Theorem)j(138)49 b FC(If)40 b Fk(g)i FC(is)e(a)h(c)-5 b(omplex)40 b(semisimple)f(Lie)i(algebr)-5 b(a,)41 b(then)g(a)f (Cart\341n)h(sub)-5 b(algebr)g(a)147 3757 y Fk(h)44 b FC(exists.)70 b(If)42 b Fk(h)737 3772 y Fz(1)820 3757 y FC(and)h Fk(h)1070 3772 y Fz(2)1153 3757 y FC(ar)-5 b(e)43 b(two)g(Cart\341n)g(sub)-5 b(algebr)g(as,)45 b(then)e(ther)-5 b(e)43 b(is)g(an)g(automorphism)147 3878 y(of)38 b Fk(g)h FC(which)e(takes)h Fk(h)933 3893 y Fz(1)1011 3878 y FC(to)g Fk(h)1183 3893 y Fz(2)1223 3878 y FC(.)55 b(In)37 b(p)-5 b(articular,)39 b(any)g(two)f(Cart\341n)g(sub)-5 b(algebr)g(as)37 b(have)h(the)g(same)147 3998 y(dimension.)446 4229 y FH(F)-8 b(rom)34 b(no)m(w)i(on,)h Fk(g)f FH(will)d(denote)j(a)g (complex)f(semisimple)e(Lie)i(algebra,)g(and)h Fk(h)g FH(a)f(\034xed)147 4349 y(Cart\341n)e(subalgebra)f(in)g Fk(g)p FH(.)147 4580 y FI(De\034nition)k(139)49 b FC(The)37 b FB(r)-6 b(ank)38 b FC(of)e(a)h(c)-5 b(omplex)36 b(semisimple)g(Lie)h (algebr)-5 b(a)36 b(is)h(the)g(dimension)e(of)147 4700 y(a)g(Cart\341n)f(sub)-5 b(algebr)g(a.)446 4930 y FH(F)d(or)31 b(example,)g(the)h(rank)g(of)g FF(sl)17 b FE(\()o FD(n)p FE(;)g Fu(C)j FE(\))38 b FH(is)31 b FD(n)21 b FA(\000)g FE(1)p FH(.)42 b(One)32 b(Cart\341n)g(subalgebra)f(in)g FF(sl)17 b FE(\()p FD(n)p FE(;)g Fu(C)j FE(\))147 5051 y FH(is)37 b(the)g(space)h(of)e(diagonal)e(matrices)i(with)h(trace)g (zero.)57 b(\(Note)36 b(that)h(in)f(the)h(case)h FD(n)d FE(=)g(3)i FH(the)147 5171 y(space)c(of)e(diagonal)e(matrices)i(with)g (trace)h(zero)f(is)h(precisely)f(the)h(span)g(of)f FD(H)3044 5186 y Fz(1)3115 5171 y FH(and)h FD(H)3385 5186 y Fz(2)3424 5171 y FH(.\))43 b(Both)147 5292 y FF(so)p FE(\(2)p FD(n)p FE(;)17 b Fu(C)j FE(\))38 b FH(and)33 b FF(so)p FE(\(2)p FD(n)22 b FE(+)g(1;)17 b Fu(C)j FE(\))38 b FH(ha)m(v)m(e)c(rank)f FD(n)p FH(.)p eop %%Page: 152 152 152 151 bop 147 48 a FK(152)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)147 298 y FI(De\034nition)36 b(140)49 b FC(L)-5 b(et)36 b FE(\()p FD(\031)t(;)17 b(V)k FE(\))36 b FC(b)-5 b(e)36 b(a)g (\034nite-dimensional,)d(c)-5 b(omplex-line)g(ar)34 b(r)-5 b(epr)g(esentation)35 b(of)147 418 y Fk(g)p FC(.)45 b(Then)34 b FD(\026)27 b FA(2)i Fk(h)759 382 y Fw(\003)833 418 y FC(is)34 b(c)-5 b(al)5 b(le)-5 b(d)34 b(a)h FB(weight)f FC(for)h FD(\031)k FC(if)34 b(ther)-5 b(e)35 b(exists)g FD(v)c FA(6)p FE(=)d(0)34 b FC(in)h FD(V)56 b FC(such)35 b(that)1605 637 y FD(\031)t FE(\()p FD(H)8 b FE(\))p FD(v)31 b FE(=)c FD(\026)p FE(\()p FD(H)8 b FE(\))p FD(v)147 856 y FC(for)35 b(al)5 b(l)34 b FD(H)h FA(2)29 b Fk(h)p FC(.)44 b(The)34 b(ve)-5 b(ctor)35 b FD(v)k FC(is)34 b(c)-5 b(al)5 b(le)-5 b(d)34 b(a)h FB(weight)40 b(ve)-6 b(ctor)35 b FC(for)g(the)g(weight)f FD(\026)p FC(.)446 1083 y FH(Note)k(that)g(the)g(ro)s(ots)g(are)g(precisely)g(the)g (non-zero)g(w)m(eigh)m(ts)h(for)e(the)i(adjoin)m(t)e(repre-)147 1203 y(sen)m(tation.)147 1430 y FI(Lemma)c(141)49 b FC(L)-5 b(et)39 b FD(\013)g FC(b)-5 b(e)37 b(a)i(r)-5 b(o)g(ot)38 b(and)g FD(Z)45 b FC(a)38 b(c)-5 b(orr)g(esp)g(onding)36 b(r)-5 b(o)g(ot)39 b(ve)-5 b(ctor.)54 b(L)-5 b(et)39 b FD(\026)f FC(b)-5 b(e)38 b(a)g(weight)147 1551 y(for)g(a)f(r)-5 b(epr)g(esentation)37 b FD(\031)42 b FC(and)37 b FD(v)42 b FC(a)37 b(c)-5 b(orr)g(esp)g(onding)36 b(weight)h(ve)-5 b(ctor.)53 b(Then)37 b(either)h FD(\031)t FE(\()p FD(Z)7 b FE(\))p FD(v)36 b FE(=)d(0)147 1671 y FC(or)i(else)f FD(\031)t FE(\()p FD(Z)7 b FE(\))p FD(v)38 b FC(is)d(a)g(weight)f(ve)-5 b(ctor)35 b(with)f(weight)h FD(\026)21 b FE(+)h FD(\013)q FC(.)147 1930 y FI(Pro)s(of)147 2115 y FH(Same)32 b(as)h(for)f FF(sl)17 b FE(\(3;)g Fs(C)p FE(\))o FH(.)44 b Fy(\003)147 2342 y FI(De\034nition)36 b(142)49 b FC(A)35 b(set)g(of)g(r)-5 b(o)g(ots)35 b FA(f)o FD(\013)1610 2357 y Fz(1)1650 2342 y FD(;)17 b FA(\001)g(\001)g(\001)d FD(\013)1888 2357 y Ft(l)1914 2342 y FA(g)35 b FC(is)g(c)-5 b(al)5 b(le)-5 b(d)34 b(a)g FB(simple)40 b(system)34 b FC(\(or)g FB(b)-6 b(asis)p FC(\))35 b(if)263 2569 y(1.)48 b FA(f)p FD(\013)503 2584 y Fz(1)542 2569 y FD(;)17 b FA(\001)g(\001)g(\001)e FD(\013)781 2584 y Ft(l)807 2569 y FA(g)51 b FC(is)35 b(a)f(ve)-5 b(ctor)35 b(sp)-5 b(ac)g(e)34 b(b)-5 b(asis)34 b(for)h Fk(h)2081 2533 y Fw(\003)2120 2569 y FC(.)263 2772 y(2.)48 b(Every)35 b(r)-5 b(o)g(ot)35 b FD(\013)28 b FA(2)g FE(\001)36 b FC(c)-5 b(an)34 b(b)-5 b(e)34 b(written)h(in)g (the)g(form)1446 2991 y FD(\013)28 b FE(=)g FD(n)1698 3006 y Fz(1)1738 2991 y FD(\013)1800 3006 y Fz(1)1861 2991 y FE(+)22 b FD(n)2017 3006 y Fz(2)2057 2991 y FD(\013)2119 3006 y Fz(2)2180 2991 y FE(+)g FA(\001)17 b(\001)g(\001)k FE(+)h FD(n)2573 3006 y Ft(l)2599 2991 y FD(\013)2661 3006 y Ft(l)391 3210 y FC(with)32 b(e)-5 b(ach)32 b FD(n)875 3225 y Ft(i)935 3210 y FC(an)g(inte)-5 b(ger,)32 b(and)g(such)g(that)g (the)h FD(n)2242 3225 y Ft(i)2270 3210 y FC('s)f(ar)-5 b(e)32 b(either)g(al)5 b(l)32 b(non-ne)-5 b(gative)30 b(or)i(al)5 b(l)391 3330 y(non-p)-5 b(ositive.)446 3557 y(A)33 b(r)-5 b(o)g(ot)32 b FD(\013)i FC(is)e(said)f(to)i(b)-5 b(e)32 b FB(p)-6 b(ositive)33 b FC(\(with)f(r)-5 b(esp)g(e)g(ct)32 b(to)g(the)h(given)e(simple)h(system\))g(if)g(the)147 3678 y FD(n)205 3693 y Ft(i)234 3678 y FC('s)i(ar)-5 b(e)35 b(non-ne)-5 b(gative;)33 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4832 y FE(=)k FD(a)1707 4847 y Fz(1)1747 4832 y FD(\013)1809 4847 y Fz(1)1870 4832 y FE(+)22 b FD(a)2019 4847 y Fz(2)2059 4832 y FD(\013)2121 4847 y Fz(2)2183 4832 y FE(+)g FA(\001)17 b(\001)g(\001)j FE(+)i FD(a)2568 4847 y Ft(l)2594 4832 y FD(\013)2656 4847 y Ft(l)147 5051 y FC(with)35 b FD(a)410 5066 y Ft(i)466 5051 y FA(\025)28 b FE(0)p FD(:)35 b FC(This)f(r)-5 b(elation)34 b(is)h(denote)-5 b(d)34 b FD(\026)1783 5066 y Fz(1)1850 5051 y FA(\027)28 b FD(\026)2014 5066 y Fz(2)2088 5051 y FC(or)35 b FD(\026)2273 5066 y Fz(2)2339 5051 y FA(\026)29 b FD(\026)2504 5066 y Fz(1)2543 5051 y FC(.)446 5171 y(A)35 b(weight)g FD(\026)914 5186 y Fz(0)988 5171 y FC(for)f(a)h(r)-5 b(epr)g(esentation)34 b FD(\031)39 b FC(is)c FB(highest)g FC(if)g(al)5 b(l)34 b(the)h(weights)f FD(\026)h FC(of)f FD(\031)39 b FC(satisfy)147 5292 y FD(\026)28 b FA(\026)g FD(\026)398 5307 y Fz(0)437 5292 y FC(.)p eop %%Page: 153 153 153 152 bop 147 48 a Fx(Complex)27 b(Semisimple)h(Lie)g(Algebras)2231 b FG(153)446 298 y FH(The)28 b(follo)m(wing)d(deep)j(theorem)f (captures)i(m)m(uc)m(h)f(of)e(the)i(structure)h(theory)f(of)f(semisim-) 147 418 y(ple)32 b(Lie)g(algebras.)147 652 y FI(Theorem)j(144)49 b FC(L)-5 b(et)43 b Fk(g)g FC(b)-5 b(e)42 b(a)h(c)-5 b(omplex)41 b(semisimple)g(Lie)h(algebr)-5 b(a,)44 b Fk(h)e FC(a)h(Cart\341n)f(sub)-5 b(algebr)g(a,)147 772 y(and)34 b FE(\001)i FC(the)e(set)h(of)g(r)-5 b(o)g(ots.)45 b(Then)263 1005 y(1.)j(F)-7 b(or)34 b(e)-5 b(ach)34 b(r)-5 b(o)g(ot)35 b FD(\013)28 b FA(2)g FE(\001)p FC(,)36 b(the)e(c)-5 b(orr)g(esp)g(onding)33 b(r)-5 b(o)g(ot)35 b(sp)-5 b(ac)g(e)34 b Fk(g)2599 969 y Ft(\013)2684 1005 y FC(is)g(one-dimensional.)263 1212 y(2.)48 b(If)34 b FD(\013)i FC(is)e(a)h(r)-5 b(o)g(ot,)35 b(then)g(so)f(is)h FA(\000)p FD(\013)q FC(.)263 1418 y(3.)48 b(A)35 b(simple)f(system)h(of)f(r)-5 b(o)g(ots)35 b FA(f)p FD(\013)1585 1433 y Fz(1)1624 1418 y FD(;)17 b FA(\001)g(\001)g(\001)e FD(\013)1863 1433 y Ft(l)1889 1418 y FA(g)35 b FC(exists.)446 1651 y FH(W)-8 b(e)41 b(no)m(w)f(need)h(to)f(iden)m(tify)f(the)i(correct)g(set)f(of)g(w)m (eigh)m(ts)h(to)e(b)s(e)i(highest)f(w)m(eigh)m(ts)g(of)147 1771 y(irreducible)31 b(represen)m(tations.)147 2005 y FI(Theorem)k(145)49 b FC(L)-5 b(et)31 b FA(f)p FD(\013)1115 2020 y Fz(1)1154 2005 y FD(;)17 b FA(\001)g(\001)g(\001)e FD(\013)1393 2020 y Ft(l)1419 2005 y FA(g)30 b FC(denote)g(a)g(simple)f (system)i(of)f(r)-5 b(o)g(ots,)31 b FD(X)2955 2020 y Ft(i)3013 2005 y FC(an)f(element)g(of)g(the)147 2125 y(r)-5 b(o)g(ot)35 b(sp)-5 b(ac)g(e)34 b Fk(g)649 2089 y Ft(\013)694 2099 y Fn(i)760 2125 y FC(and)g FD(Y)1006 2140 y Ft(i)1069 2125 y FC(an)g(element)g(of)h(the)g(r)-5 b(o)g(ot)35 b(sp)-5 b(ac)g(e)34 b Fk(g)2348 2089 y Fw(\000)p Ft(\013)2448 2099 y Fn(i)2479 2125 y FC(.)44 b(De\034ne)1655 2347 y FD(H)1736 2362 y Ft(i)1792 2347 y FE(=)28 b([)p FD(X)2004 2362 y Ft(i)2032 2347 y FD(;)17 b(Y)2133 2362 y Ft(i)2160 2347 y FE(])g FC(.)147 2569 y(Then)34 b(it)h(is)g(p)-5 b(ossible)33 b(to)i(cho)-5 b(ose)34 b FD(X)1463 2584 y Ft(i)1526 2569 y FC(and)h FD(Y)1773 2584 y Ft(i)1835 2569 y FC(such)g(that)263 2802 y(1.)48 b(Each)34 b FD(H)717 2817 y Ft(i)780 2802 y FC(is)h(non-zer)-5 b(o)34 b(and)g(c)-5 b(ontaine)g(d)34 b(in)g Fk(h)p FC(.)263 3009 y(2.)48 b(The)31 b(sp)-5 b(an)31 b(of)g FA(f)p FD(H)1050 3024 y Ft(i)1078 3009 y FD(;)17 b(X)1203 3024 y Ft(i)1230 3009 y FD(;)g(Y)1331 3024 y Ft(i)1359 3009 y FA(g)31 b FC(is)g(a)h(sub)-5 b(algebr)g(a)30 b(of)h Fk(g)h FC(isomorphic)e (\(in)h(the)g(obvious)g(way\))g(to)391 3129 y FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FC(.)263 3336 y(3.)48 b(The)34 b(set)h FA(f)p FD(H)873 3351 y Fz(1)912 3336 y FD(;)17 b FA(\001)g(\001)g(\001)e FD(H)1170 3351 y Ft(l)1196 3336 y FA(g)34 b FC(is)h(a)g(b)-5 b(asis)34 b(for)g Fk(h)p FC(.)446 3569 y FH(Note)h(that)g(\(in)f(most)g(cases\))j(the)e(set)h (of)e(all)f FD(H)2236 3584 y Ft(i)2264 3569 y FH('s,)j FD(X)2473 3584 y Ft(i)2501 3569 y FH('s,)g(and)f FD(Y)2878 3584 y Ft(i)2906 3569 y FH('s)h(\()p FD(i)c FE(=)f(1)p FD(;)17 b FE(2)p FD(;)g FA(\001)g(\001)g(\001)d FD(l)r FH(\))35 b(do)147 3689 y FC(not)49 b FH(span)41 b Fk(g)p FH(.)65 b(In)40 b(the)g(case)h Fk(g)f FE(=)g FF(sl)16 b FE(\(3;)h Fu(C)j FE(\))6 b FH(,)41 b FD(l)h FE(=)e(2)p FH(,)i(and)d(the)i(span)f(of)f FD(H)2931 3704 y Fz(1)2970 3689 y FD(;)17 b(X)3095 3704 y Fz(1)3135 3689 y FD(;)g(Y)3236 3704 y Fz(1)3274 3689 y FD(;)g(H)3399 3704 y Fz(2)3438 3689 y FD(;)g(X)3563 3704 y Fz(2)3602 3689 y FD(;)g(Y)3703 3704 y Fz(2)147 3809 y FH(represen)m(ts)37 b(only)e(six)g(of)f(the)h (eigh)m(t)g(dimensions)f(of)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(.)51 b(Nev)m(ertheless)37 b(the)e(subalgebras)147 3930 y FA(f)p FD(H)278 3945 y Ft(i)306 3930 y FD(;)17 b(X)431 3945 y Ft(i)459 3930 y FD(;)g(Y)560 3945 y Ft(i)587 3930 y FA(g)33 b FH(pla)m(y)f(an)g(imp)s(ortan)m(t)f(role.)446 4051 y(W)-8 b(e)33 b(are)g(no)m(w)g(ready)g(to)f(state)h(the)g(main)e (theorem.)147 4284 y FI(Theorem)k(146)49 b FC(L)-5 b(et)43 b Fk(g)g FC(b)-5 b(e)42 b(a)h(c)-5 b(omplex)41 b(semisimple)g(Lie)h (algebr)-5 b(a,)44 b Fk(h)e FC(a)h(Cart\341n)f(sub)-5 b(algebr)g(a,)147 4405 y(and)45 b FA(f)o FD(\013)458 4420 y Fz(1)498 4405 y FD(;)17 b FA(\001)g(\001)g(\001)d FD(\013)736 4420 y Ft(l)762 4405 y FA(g)45 b FC(a)g(simple)f(system)h (of)f(r)-5 b(o)g(ots.)76 b(L)-5 b(et)45 b FA(f)p FD(H)2337 4420 y Fz(1)2376 4405 y FD(;)17 b FA(\001)g(\001)g(\001)d FD(H)2633 4420 y Ft(l)2659 4405 y FA(g)45 b FC(b)-5 b(e)45 b(as)f(in)h(The)-5 b(or)g(em)44 b(145.)147 4525 y(Then)263 4758 y(1.)k(In)35 b(e)-5 b(ach)35 b(irr)-5 b(e)g(ducible)34 b(r)-5 b(epr)g(esentation)35 b FD(\031)k FC(of)c Fk(g)p FC(,)h(the)f FD(\031)t FE(\()p FD(H)8 b FE(\))p FC('s)35 b(ar)-5 b(e)35 b(simultane)-5 b(ously)35 b(diago-)391 4878 y(nalizable.)263 5085 y(2.)48 b(Each)34 b(irr)-5 b(e)g(ducible)35 b(r)-5 b(epr)g(esentation)34 b(of)g Fk(g)h FC(has)f(a)h(unique)g(highest)f(weight.)263 5292 y(3.)48 b(Two)27 b(irr)-5 b(e)g(ducible)27 b(r)-5 b(epr)g(esentations) 26 b(of)i Fk(g)g FC(with)f(the)h(same)e(highest)h(weight)g(ar)-5 b(e)28 b(e)-5 b(quivalent.)p eop %%Page: 154 154 154 153 bop 147 48 a FK(154)1899 b FG(The)27 b(Represen)n(tations)f(of) i Ff(SU)p Fe(\(3\))p FG(,)g(and)f(Bey)n(ond)263 298 y FC(4.)48 b(If)42 b FD(\026)560 313 y Fz(0)641 298 y FC(is)g(the)h (highest)f(weight)f(of)h(an)g(irr)-5 b(e)g(ducible)42 b(r)-5 b(epr)g(esentation)42 b(of)g Fk(g)p FC(,)i(then)e(for)g FD(i)g FE(=)391 418 y(1)p FD(;)17 b FE(2)p FD(;)g FA(\001)g(\001)g (\001)d FD(l)r FC(,)35 b FD(\026)864 433 y Fz(0)903 418 y FE(\()p FD(H)1022 433 y Ft(i)1050 418 y FE(\))g FC(is)f(a)h(non-ne)-5 b(gative)33 b(inte)-5 b(ger.)263 615 y(5.)48 b(Conversely,)h(if)e FD(\026)1099 630 y Fz(0)1189 615 y FA(2)k Fk(h)1358 579 y Fw(\003)1445 615 y FC(is)c(such)g(that)h FD(\026)2067 630 y Fz(0)2106 615 y FE(\()p FD(H)2225 630 y Ft(i)2253 615 y FE(\))f FC(is)g(a)g(non-ne)-5 b(gative)46 b(inte)-5 b(ger)46 b(for)h(al)5 b(l)391 735 y FD(i)28 b FE(=)g(1)p FD(;)17 b FE(2)p FD(;)g FA(\001)g(\001)g(\001)c FD(l)r FC(,)32 b(then)f(ther)-5 b(e)31 b(is)f(an)h(irr)-5 b(e)g(ducible)30 b(r)-5 b(epr)g(esentation)30 b(of)h Fk(g)g FC(with)f(highest)h(weight) 391 856 y FD(\026)450 871 y Fz(0)489 856 y FC(.)446 1057 y FH(The)j(w)m(eigh)m(ts)f FD(\026)1053 1072 y Fz(0)1124 1057 y FH(as)g(in)f(4\))g(and)h(5\))f(are)h(called)e FI(dominan)m(t)j(in)m(tegral)i(w)m(eigh)m(ts)p FH(.)147 1316 y FI(6.8)112 b(Exercises)266 1483 y FH(1.)49 b(Sho)m(w)33 b(that)g(for)f(an)m(y)h(pair)e(of)i FD(n)22 b FA(\002)h FD(n)32 b FH(matrices)g FD(X)40 b FH(and)33 b FD(Y)21 b FH(,)1560 1599 y Fv(\002)1602 1680 y FD(X)1691 1639 y Ft(tr)1754 1680 y FD(;)c(Y)1876 1639 y Ft(tr)1939 1599 y Fv(\003)2009 1680 y FE(=)27 b FA(\000)17 b FE([)p FD(X)r(;)g(Y)22 b FE(])2466 1632 y Ft(tr)2546 1680 y FH(.)391 1877 y(Using)43 b(this)h(fact)g(and)g(the)g(fact)f(that)h FD(X)1977 1840 y Ft(tr)1969 1901 y(i)2087 1877 y FE(=)j FD(Y)2267 1892 y Ft(i)2338 1877 y FH(for)d FD(i)j FE(=)g(1)p FD(;)17 b FE(2)p FD(;)g FE(3)p FH(,)45 b(explain)e(the)h(sym-)391 1997 y(metry)j(b)s(et)m(w)m(een)i FD(X)8 b FH('s)47 b(and)g FD(Y)21 b FH('s)48 b(in)e(the)h(comm)m(utation)e(relations)h(for)g FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(.)87 b(F)-8 b(or)391 2117 y(example,)35 b(sho)m(w)h(that)f(the)g(relation)e FE([)q FD(Y)1876 2132 y Fz(1)1915 2117 y FD(;)17 b(Y)2016 2132 y Fz(2)2054 2117 y FE(])32 b(=)g FA(\000)p FD(Y)2355 2132 y Fz(3)2429 2117 y FH(can)k(b)s(e)f(obtained)f(from)g(the)h(rela-) 391 2238 y(tion)f FE([)p FD(X)702 2253 y Fz(1)741 2238 y FD(;)17 b(X)866 2253 y Fz(2)906 2238 y FE(])31 b(=)h FD(X)1153 2253 y Fz(3)1227 2238 y FH(b)m(y)k(taking)e(transp)s(oses.)52 b(Sho)m(w)36 b(that)e(the)i(relation)d FE([)p FD(H)3301 2253 y Fz(1)3340 2238 y FD(;)17 b(Y)3441 2253 y Fz(2)3480 2238 y FE(])31 b(=)h FD(Y)3703 2253 y Fz(2)391 2358 y FH(follo)m(ws)f(from)h(the)h(relation)d FE([)p FD(H)1575 2373 y Fz(1)1614 2358 y FD(;)17 b(X)1739 2373 y Fz(2)1779 2358 y FE(])27 b(=)h FA(\000)p FD(X)2095 2373 y Fz(2)2135 2358 y FH(.)266 2555 y(2.)49 b(Recall)37 b(the)i(de\034nition)f(of)g (the)h(dual)f FD(\031)1878 2519 y Fw(\003)1956 2555 y FH(of)g(a)g(represen)m(tation)i FD(\031)i FH(from)37 b(Exercise)j(14)e(of)391 2675 y(Chapter)33 b(5.)44 b(Consider)32 b(this)h(for)f(the)h(case)g(of)f(represen)m(tations)i(of)e FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(.)391 2834 y(a\))32 b(Sho)m(w)i(that)e(the)h(w)m(eigh)m(ts)g(of)f FD(\031)1664 2798 y Fw(\003)1736 2834 y FH(are)h(the)g(negativ)m(es)g(of)f(the)h(w)m (eigh)m(ts)g(of)f FD(\031)t FH(.)391 2992 y(b\))25 b(Sho)m(w)g(that)f (if)g FD(\031)k FH(is)c(the)h(irreducible)e(represen)m(tation)i(of)f FF(sl)17 b FE(\(3;)g Fu(C)j FE(\))30 b FH(with)24 b(highest)h(w)m(eigh) m(t)391 3113 y FE(\()p FD(m)514 3128 y Fz(1)554 3113 y FD(;)17 b(m)683 3128 y Fz(2)722 3113 y FE(\))27 b FH(then)h FD(\031)1063 3076 y Fw(\003)1130 3113 y FH(is)f(the)g(irreducible)f (represen)m(tation)i(with)f(highest)g(w)m(eigh)m(t)h FE(\()p FD(m)3469 3128 y Fz(2)3509 3113 y FD(;)17 b(m)3638 3128 y Fz(1)3677 3113 y FE(\))p FH(.)391 3271 y FC(Hint)9 b FH(:)48 b(If)35 b(y)m(ou)g(iden)m(tify)f FD(V)56 b FH(and)35 b FD(V)1690 3235 y Fw(\003)1764 3271 y FH(b)m(y)h(c)m(ho)s (osing)e(a)g(basis)h(for)f FD(V)21 b FH(,)35 b(then)h FD(A)3214 3235 y Ft(tr)3312 3271 y FH(is)e(just)h(the)391 3392 y(usual)d(matrix)f(transp)s(ose.)266 3588 y(3.)49 b(Let)41 b Fk(h)f FH(denote)h(the)g(subspace)i(of)d FF(sl)17 b FE(\()o(3;)g Fu(C)j FE(\))46 b FH(spanned)c(b)m(y)f FD(H)2668 3603 y Fz(1)2748 3588 y FH(and)g FD(H)3027 3603 y Fz(2)3066 3588 y FH(.)67 b(Let)41 b FD(G)f FH(denote)391 3709 y(the)f(group)g(of)f(all)e(matrices)i FD(A)g FA(2)h FF(SU)p FE(\(3\))g FH(suc)m(h)h(that)e FE(Ad)q FD(A)g FH(preserv)m(es)k Fk(h)p FH(.)61 b(No)m(w)40 b(let)e FD(G)3703 3724 y Fz(0)391 3829 y FH(denote)32 b(the)f(group)g(of)f(all) f(matrices)h FD(A)e FA(2)g FF(SU)q FE(\(3\))i FH(suc)m(h)j(that)d FE(Ad)q FD(A)h FH(is)f(the)h(iden)m(tit)m(y)g(on)g Fk(h)p FH(,)391 3949 y(i.e.,)36 b(suc)m(h)g(that)f FE(Ad)p FD(A)p FE(\()p FD(H)1334 3964 y Fz(1)1374 3949 y FE(\))d(=)g FD(H)1633 3964 y Fz(1)1707 3949 y FH(and)j FE(Ad)p FD(A)p FE(\()p FD(H)2218 3964 y Fz(2)2258 3949 y FE(\))d(=)g FD(H)2517 3964 y Fz(2)2556 3949 y FH(.)51 b(Sho)m(w)36 b(that)f FD(G)3185 3964 y Fz(0)3260 3949 y FH(is)f(a)h(normal)391 4070 y(subgroup)27 b(of)f FD(G)p FH(.)42 b(Compute)26 b FD(G)h FH(and)g FD(G)1840 4085 y Fz(0)1879 4070 y FH(.)42 b(Sho)m(w)27 b(that)f FD(G=G)2607 4085 y Fz(0)2673 4070 y FH(is)g(isomorphic)f(to)h(the)h(W)-8 b(eyl)391 4190 y(group)32 b FD(W)14 b FH(.)266 4387 y(4.)49 b(a\))32 b(V)-8 b(erify)32 b(Theorems)i(144)d(and)i(145)f(explicitly)f(for)h (the)g(case)i Fk(g)28 b FE(=)f FF(sl)17 b FE(\()p FD(n)p FE(;)g Fu(C)j FE(\))6 b FH(.)391 4545 y(b\))38 b(Consider)h(the)g(task) g(of)f(trying)f(to)h(pro)m(v)m(e)i(Theorem)e(146)g(for)g(the)g(case)h (of)f FF(sl)17 b FE(\()p FD(n)p FE(;)g Fu(C)j FE(\))6 b FH(.)391 4666 y(No)m(w)29 b(that)g(y)m(ou)g(ha)m(v)m(e)h(done)f (\(a\),)g(what)g(part)f(of)g(the)h(pro)s(of)f(go)s(es)h(through)f(the)h (same)g(w)m(a)m(y)391 4786 y(as)37 b(for)f FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(?)56 b(A)m(t)36 b(what)h(p)s(oin)m(ts)f (in)g(the)h(pro)s(of)f(of)g(the)h(corresp)s(onding)f(theorem)h(for)391 4907 y FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))39 b FH(did)32 b(w)m(e)h(use)h(sp)s(ecial)d(prop)s(erties)i(of)f FF(sl)17 b FE(\(3;)g Fu(C)i FE(\))6 b FH(?)391 5065 y FC(Hint)j FH(:)45 b(Most)33 b(of)f(it)f(is)i(the)g(same,)f(but)h(there)g(is)g (one)f(critical)f(p)s(oin)m(t)g(whic)m(h)i(w)m(e)h(do)f(some-)391 5185 y(thing)f(whic)m(h)h(do)s(es)g(not)f(generalize)g(to)g FF(sl)17 b FE(\()p FD(n)p FE(;)g Fu(C)j FE(\))6 b FH(.)p eop %%Page: 155 155 155 154 bop 1688 1047 a FL(Chapter)37 b(7)1304 1222 y(FINAL)i(EXAMINA) -10 b(TION)266 1491 y FH(1.)49 b(Let)30 b FD(G)g FH(b)s(e)g(a)g (connected)h(matrix)e(Lie)g(group,)h(and)g(let)f FE(Ad)f(:)g FD(G)g FA(!)f FF(GL)p FE(\()p Fk(g)p FE(\))j FH(b)s(e)g(the)g(adjoin)m (t)391 1612 y(represen)m(tation)j(of)f FD(G)p FH(.)44 b(Sho)m(w)33 b(that)1721 1791 y FE(k)m(er)q(\(Ad\))28 b(=)f FD(Z)7 b FE(\()p FD(G)p FE(\))391 1971 y FH(where)34 b FD(Z)7 b FE(\()p FD(G)p FE(\))32 b FH(denotes)i(the)f(cen)m(ter)h(of) e FD(G)p FH(.)44 b(If)32 b FD(G)c FE(=)g FF(O)p FE(\(2\))p FH(,)k(compute)h FE(k)m(er)q(\(Ad\))g FH(and)f FD(Z)7 b FE(\()p FD(G)p FE(\))391 2092 y FH(and)33 b(sho)m(w)g(that)g(they)g (are)g(not)f(equal.)391 2248 y FC(Hint)9 b FH(:)44 b(Y)-8 b(ou)32 b(should)f(use)i(the)f(fact)g(that)g(if)e FD(G)i FH(is)f(connected,)j(then)e(ev)m(ery)i FD(A)28 b FA(2)g FD(G)k FH(can)g(b)s(e)391 2368 y(written)g(in)g(the)h(form)e FD(A)d FE(=)g FD(e)1494 2332 y Ft(X)1552 2341 y Fr(1)1591 2368 y FD(e)1636 2332 y Ft(X)1694 2341 y Fr(2)1749 2368 y FA(\001)17 b(\001)g(\001)e FD(e)1927 2332 y Ft(X)1985 2340 y Fn(n)2032 2368 y FH(,)33 b(with)f FD(X)2395 2383 y Ft(i)2451 2368 y FA(2)c Fk(g)p FH(.)266 2560 y(2.)49 b(Let)39 b FD(G)h FH(b)s(e)f(a)g(\034nite,)i(comm)m(utativ)m(e)d (group.)63 b(Sho)m(w)40 b(that)f(the)h(n)m(um)m(b)s(er)f(of)g(equiv)-5 b(alence)391 2680 y(classes)41 b(of)f(irreducible)f(complex)h(represen) m(tations)h(of)f FD(G)g FH(is)g(equal)g(to)g(the)h(n)m(um)m(b)s(er)f (of)391 2801 y(elemen)m(ts)33 b(in)f FD(G)p FH(.)391 2957 y FC(Hint)9 b FH(:)50 b(Use)36 b(the)g(fact)f(that)g(ev)m(ery)i (\034nite,)e(comm)m(utativ)m(e)f(group)h(is)g(a)g(pro)s(duct)h(of)e (cyclic)391 3077 y(groups.)266 3269 y(3.)49 b(a\))39 b(Sho)m(w)h(that)e(if)g FD(R)i FA(2)f FF(O)p FE(\(2\))p FH(,)i(and)e FE(det)17 b FD(R)40 b FE(=)e FA(\000)p FE(1)p FH(,)j(then)f FD(R)g FH(has)f(t)m(w)m(o)h(real,)g(orthogonal)391 3390 y(eigen)m(v)m(ectors)34 b(with)e(eigen)m(v)-5 b(alues)33 b FE(1)f FH(and)h FA(\000)p FE(1)p FH(.)391 3546 y(b\))44 b(Let)g FD(R)g FH(b)s(e)g(in)f FF(O)p FE(\()p FD(n)p FE(\))p FH(.)77 b(Sho)m(w)44 b(that)g(there)g(exists)g(a)g(subspace)h FD(W)58 b FH(of)43 b Fu(R)3290 3509 y Ft(n)3386 3546 y FH(whic)m(h)h(is)391 3666 y(in)m(v)-5 b(arian)m(t)37 b(under)j(b)s(oth)e FD(R)i FH(and)f FD(R)1710 3630 y Fw(\000)p Fz(1)1805 3666 y FH(,)h(and)f(suc)m(h)h(that)f FE(dim)15 b FD(W)52 b FE(=)38 b(1)g FH(or)h FE(2)p FH(.)62 b(Sho)m(w)39 b(that)391 3786 y FD(W)497 3750 y Fw(?)579 3786 y FH(\(the)24 b(orthogonal)d(complemen)m(t)i(of)g FD(W)14 b FH(\))23 b(is)g(also)f(in)m(v)-5 b(arian)m(t)22 b(under)i FD(R)h FH(and)e FD(R)3355 3750 y Fw(\000)p Fz(1)3450 3786 y FH(.)40 b(Sho)m(w)391 3907 y(that)34 b(the)h(restrictions)f(of)g FD(R)h FH(and)f FD(R)1769 3871 y Fw(\000)p Fz(1)1898 3907 y FH(to)g FD(W)48 b FH(and)35 b(to)f FD(W)2578 3871 y Fw(?)2671 3907 y FH(are)g(orthogonal.)47 b(\(That)34 b(is,)391 4027 y(sho)m(w)g(that)e(these)i(restrictions)e (preserv)m(e)j(inner)d(pro)s(ducts.\))391 4183 y(c\))d(Let)f FD(R)h FH(b)s(e)f(in)g FF(O)p FE(\()p FD(n)p FE(\))p FH(.)42 b(Sho)m(w)29 b(that)f Fu(R)1813 4147 y Ft(n)1894 4183 y FH(can)g(b)s(e)h(written)f(as)g(the)h(orthogonal)d(direct)i(sum) 391 4304 y(of)k(subspaces)j FD(W)1040 4319 y Ft(i)1101 4304 y FH(suc)m(h)f(that)432 4509 y(\(a\))49 b(1\))32 b(Eac)m(h)h FD(W)1059 4524 y Ft(i)1120 4509 y FH(is)f(in)m(v)-5 b(arian)m(t)31 b(under)j FD(R)f FH(and)g FD(R)2276 4473 y Fw(\000)p Fz(1)2370 4509 y FH(,)427 4659 y(\(b\))49 b(2\))32 b(Eac)m(h)h FD(W)1059 4674 y Ft(i)1120 4659 y FH(has)g(dimension)e FE(1)h FH(or)h FE(2)p FH(,)f(and)438 4810 y(\(c\))49 b(3\))32 b(If)g FE(dim)16 b FD(W)1094 4825 y Ft(i)1150 4810 y FE(=)27 b(2)p FH(,)32 b(then)i(the)f (restriction)e(of)h FD(R)i FH(to)e FD(W)2648 4825 y Ft(i)2709 4810 y FH(has)h(determinan)m(t)f(one.)391 5015 y(d\))38 b(Sho)m(w)h(that)e(the)h(exp)s(onen)m(tial)g(mapping)e(for)h FF(SO)p FE(\()p FD(n)p FE(\))h FH(is)g(on)m(to.)59 b(Mak)m(e)39 b(sure)g(y)m(ou)f(use)391 5135 y(the)33 b(fact)f(that)h(the)g(elemen)m (ts)f(of)h FF(SO)p FE(\()p FD(n)p FE(\))f FH(ha)m(v)m(e)i(determinan)m (t)e(one.)391 5292 y FC(Note)7 b FH(:)44 b(This)33 b(pro)m(vides)g(an)g (alternativ)m(e)e(pro)s(of)h(that)g(the)h(group)g FF(SO)p FE(\()p FD(n)p FE(\))f FH(is)g(connected.)p eop %%Page: 156 156 156 155 bop 147 48 a FK(156)2780 b FG(Final)27 b(Examination)266 298 y FH(4.)49 b(Determine,)25 b(up)e(to)f(equiv)-5 b(alence,)25 b(all)20 b(of)i(the)i(\034nite-dimensional,)d(irreducible)g(\(complex-) 391 418 y(linear\))37 b(represen)m(tations)i(of)f(the)g(Lie)g(algebra)f FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))32 b FA(\010)26 b FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(.)66 b(Can)39 b(y)m(our)g(answ)m(er)391 539 y(b)s(e)33 b(expressed)i(in)d(terms)g(of)g(a)h(sort)f(of)g (\020highest)h(w)m(eigh)m(t\021?)391 700 y FC(Hint)9 b FH(:)56 b(Imitate)36 b(the)j(pro)s(of)e(of)h(the)g(classi\034cation)f (of)h(the)g(irreducible)f(represen)m(tations)391 821 y(of)32 b FF(sl)p FE(\(2;)17 b Fu(C)j FE(\))p FH(.)266 1024 y(5.)49 b(Consider)41 b(the)g(irreducible)e(represen)m(tation)i FE(\()p FD(\031)t(;)17 b(V)k FE(\))41 b FH(of)f FF(sl)16 b FE(\(3;)h Fu(C)j FE(\))46 b FH(with)40 b(highest)h(w)m(eigh)m(t)391 1145 y FE(\(0)p FD(;)17 b FE(2\))o FH(.)44 b(F)-8 b(ollo)m(wing)29 b(the)k(pro)s(cedure)h(in)d(Chapter)j(6,)e(Section)g(5,)h(determine)391 1306 y(1\))f(The)i(dimension)d(of)h FD(V)21 b FH(.)391 1468 y(2\))32 b(All)f(of)h(the)h(w)m(eigh)m(ts)g(of)f FD(\031)t FH(.)391 1630 y(3\))47 b(The)i(m)m(ultiplicit)m(y)44 b(of)j(eac)m(h)i(of)e(the)h(w)m(eigh)m(ts.)90 b(\(That)48 b(is,)j(the)d(dimension)e(of)i(the)391 1751 y(corresp)s(onding)32 b(w)m(eigh)m(t)h(spaces.\))p eop %%Page: 157 157 157 156 bop 1688 1047 a FL(Chapter)37 b(8)1477 1222 y(BIBLIOGRAPHY)266 1491 y FH(1.)49 b FI(Theo)s(dor)30 b(Br\366c)m(k)m(er)f(and)h(T)-9 b(ammo)24 b(tom)j(Diec)m(k)p FH(,)f FC(R)-5 b(epr)g(esentations)27 b(of)i(Comp)-5 b(act)27 b(Lie)391 1612 y(Gr)-5 b(oups)p FH(.)44 b(Springer-V)-8 b(erlag,)30 b(1985.)391 1772 y(A)36 b(go)s(o)s(d)f(reference)j(for)e(basic)g(facts)g(on)h(compact)e (groups)i(and)f(their)g(represen)m(tations,)391 1892 y(including)23 b(c)m(haracters)k(and)e(orthogonalit)m(y)e(relations.)40 b(Analyzes)26 b(represen)m(tations)g(from)391 2012 y(a)32 b(more)g(analytic)f(and)i(less)g(algebraic)e(viewp)s(oin)m(t)h(than)g (other)h(authors.)266 2212 y(2.)49 b FI(William)42 b(F)-9 b(ulton)48 b(and)g(Jo)s(e)h(Harris)p FH(,)43 b FC(R)-5 b(epr)g(esentation)42 b(the)-5 b(ory.)71 b(A)43 b(First)h(Course.)391 2332 y FH(Graduate)49 b(T)-8 b(exts)51 b(in)e(Mathematics,)k(129.)93 b(Readings)49 b(in)g(Mathematics,)k(Springer-)391 2453 y(V)-8 b(erlag,)32 b(1991.)42 b(Has)33 b(lots)f(of)g(examples.)43 b(W)-8 b(ritten)32 b(from)f(an)i(algebraic)e(p)s(oin)m(t)g(of)h(view.) 266 2652 y(3.)49 b FI(Sigurdur)25 b(Helgason)p FH(,)e FC(Di\033er)-5 b(ential)24 b(Ge)-5 b(ometry,)27 b(Lie)e(Gr)-5 b(oups,)27 b(and)d(Symmetric)g(Sp)-5 b(ac)g(es)p FH(.)391 2772 y(Academic)32 b(Press,)i(1978.)391 2932 y(A)40 b(go)s(o)s(d)e (reference)k(for)d(a)g(lot)g(of)g(things.)64 b(Includes)41 b(structure)g(theory)f(of)f(semisimple)391 3053 y(groups.)266 3252 y(4.)49 b FI(James)30 b(E.)h(Humphreys)p FH(,)c FC(Intr)-5 b(o)g(duction)30 b(to)g(Lie)g(A)n(lgebr)-5 b(as)30 b(and)f(R)-5 b(epr)g(esentation)30 b(The-)391 3373 y(ory)p FH(.)44 b(Springer-V)-8 b(erlag,)30 b(1972.)391 3532 y(A)j(standard)g(reference)h(for)e(the)h(Lie)f(algebra)f(side)h (of)h(things)f(\(no)g(Lie)g(groups\).)266 3732 y(5.)49 b FI(N.)35 b(Jacobson)p FH(,)d FC(Lie)g(A)n(lgebr)-5 b(as)p FH(.)42 b(In)m(terscience)32 b(T)-8 b(racts)32 b(No.)42 b(10,)31 b(John)f(Wiley)g(and)g(Sons,)391 3852 y(1962.)391 4012 y(Another)j(go)s(o)s(d)e(reference)j(for)e(Lie)g (algebras.)266 4212 y(6.)49 b FI(An)m(thon)m(y)38 b(W.)g(Knapp)p FH(,)d FC(Lie)g(gr)-5 b(oups:)46 b(b)-5 b(eyond)35 b(an)g(intr)-5 b(o)g(duction.)45 b FH(Birkhauser,)34 b(1996.)391 4332 y(Go)s(o)s(d)50 b(complemen)m(t)g(to)h(Helgason)g(on)g(suc)m(h)i (matters)e(as)g(structure)i(theory)f(of)f(Lie)391 4452 y(groups.)44 b(As)33 b(title)e(suggests,)j(not)e(the)h(place)f(to)h (start,)f(but)h(a)f(go)s(o)s(d)g(reference.)266 4652 y(7.)49 b FI(W.)37 b(Miller)p FH(,)30 b FC(Symmetry)35 b(Gr)-5 b(oups)35 b(and)g(Their)f(Applic)-5 b(ations)p FH(.)42 b(Academic)32 b(Press.)391 4812 y(Orien)m(ted)h(to)m(w)m(ard)g (applications)d(to)i(ph)m(ysics.)45 b(Includes)34 b(theory)f(of)f (\034nite)g(groups.)266 5011 y(8.)49 b FI(Jean-Pierre)34 b(Serre)p FH(,)c FC(Complex)h(Semisimple)f(Lie)i(A)n(lgebr)-5 b(as)p FH(.)42 b(Springer-V)-8 b(erlag,)28 b(1987.)391 5171 y(A)22 b(v)m(ery)h(concise)f(summary)f(of)h(structure)g(theory)h (and)f(represen)m(tation)g(theory)g(of)f(semisim-)391 5292 y(ple)32 b(Lie)g(algebras.)p eop %%Page: 158 158 158 157 bop 147 48 a FK(158)2995 b FG(Bibliograph)n(y)266 298 y FH(9.)49 b FI(Jean-Pierre)27 b(Serre)p FH(,)g FC(Line)-5 b(ar)26 b(R)-5 b(epr)g(esentations)27 b(of)f(Finite)h(Gr)-5 b(oups)p FH(.)41 b(Springer-V)-8 b(erlag.)391 460 y(An)31 b(in)m(tro)s(duction)e(to)h(b)s(oth)g(complex)g(and)h(mo)s(dular)d (represen)m(tations)k(of)e(\034nite)g(groups.)218 663 y(10.)48 b FI(Barry)k(Simon)p FH(,)45 b FC(R)-5 b(epr)g(esentations)46 b(of)g(\034nite)g(and)g(c)-5 b(omp)g(act)46 b(Lie)g(gr)-5 b(oups)p FH(,)48 b(American)391 783 y(Mathematical)29 b(So)s(ciet)m(y)-8 b(,)32 b(1996.)42 b(Co)m(v)m(ers)33 b(m)m(uc)m(h)e(of)g(the)h(same)f(material)d(as)j(Br\366c)m(k)m(er)i (and)391 904 y(tom)f(Diec)m(k,)h(but)g(from)e(a)i(more)e(analytical)f (p)s(ersp)s(ectiv)m(e.)218 1107 y(11.)48 b FI(F)-9 b(rank)39 b(W.)f(W)-9 b(arner)p FH(,)33 b FC(F)-7 b(oundations)34 b(of)h(Di\033er)-5 b(entiable)34 b(Manifolds)h(and)f(Lie)i(Gr)-5 b(oups)p FH(.)391 1228 y(Springer-V)d(erlag,)31 b(1983.)391 1389 y(Key)d(w)m(ord)f(in)g(the)g(title)f(is)g FC(foundations)p FH(.)41 b(Giv)m(es)27 b(a)g(mo)s(dern)g(treatmen)m(t)g(of)f(di\033eren) m(tiable)391 1510 y(manifolds,)47 b(and)f(then)h(pro)m(v)m(es)h(some)e (imp)s(ortan)m(t,)i(non-trivial)43 b(theorems)j(ab)s(out)g(Lie)391 1630 y(groups,)31 b(including)d(the)j(relationship)d(b)s(et)m(w)m(een) 33 b(subgroups)e(and)f(subalgebras,)h(and)g(the)391 1751 y(relationship)g(b)s(et)m(w)m(een)j(represen)m(tations)g(of)e(the)h (Lie)f(algebra)f(and)i(of)f(the)h(Lie)f(group.)218 1954 y(12.)48 b FI(V.S.)59 b(V)-9 b(aradara)6 b(jan)p FH(,)58 b FC(Lie)51 b(Gr)-5 b(oups,)57 b(Lie)51 b(A)n(lgebr)-5 b(as,)56 b(and)51 b(Their)g(R)-5 b(epr)g(esentations)p FH(.)391 2074 y(Springer-V)d(erlag,)31 b(1974.)391 2236 y(A)i(comprehensiv)m(e)g(treatmen)m(t)g(of)f(b)s(oth)g(Lie)g(groups)h (and)f(Lie)g(algebras.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0005301412214--