% This is a plain tex file: % % oideals.tex % % Ideals in the Integral Octaves % FONTS % % title font (scaled, not a larger font, for portability) \font\titlefont=cmb10 scaled \magstep3 % calligraphic font \newfam\calfont \font\tencal=eusm10 \font\sevencal=eusm7 \font\fivecal=eusm5 \textfont\calfont=\tencal \scriptfont\calfont=\sevencal \scriptscriptfont\calfont=\fivecal \def\cal{\fam=\calfont} % blackboard bold (includes many math symbols) \newfam\bbbfont \font\tenbbb=msbm10 \font\sevenbbb=msbm7 \font\fivebbb=msbm5 \textfont\bbbfont=\tenbbb \scriptfont\bbbfont=\sevenbbb \scriptscriptfont\bbbfont=\fivebbb \def\bbb{\fam=\bbbfont} % PROGRAMMING STUFF % % `if not defined' : \long\def\ifndef#1{\expandafter\ifx\csname#1\endcsname\relax} % sometimes more convenient, but you can't say % ``\ifdef...\else...\fi'' \long\def\ifdef#1{\ifndef{#1}\else} % FRONT MATTER % % note: \author, \title and \note get redefined when % \bibliography is called. \long\def\author#1{\def\Zauthor{#1}} \long\def\title#1{\def\Ztitle{#1}} \long\def\address#1{\def\Zaddress{#1}} \long\def\email#1{\def\Zemail{#1}} \long\def\homepage#1{\def\Zhomepage{#1}} \long\def\date#1{\def\Zdate{#1}} \long\def\subject#1{\def\Zsubject{#1}} \long\def\note#1{\def\Znote{#1}} \def\plaintitlepage{{\parindent=0pt \ifdef{Ztitle}\par{\titlefont\Ztitle}\medskip\fi \ifdef{Zauthor}\par{\Zauthor}\fi \ifdef{Zdate}\par{\Zdate}\fi \smallskip \ifdef{Zemail}\par{\it \Zemail}\fi \ifdef{Zhomepage}\par{web page: \it\Zhomepage}\fi \ifdef{Zaddress}\par{\Zaddress}\fi \ifdef{Zsubject}\smallskip\par{1991 mathematics subject classification: \Zsubject}\fi \ifndef{Znote}\else\smallskip\par{\Znote}\fi \vskip 0pt plus 10pt}} \def\abstract{\bigbreak\noindent{\bf Abstract.}\par\noindent} % SECTIONS, THEOREMS, ETC. % %\let\section=\beginsection \outer\def\section#1\par{\bigbreak\noindent{\bf #1}\nobreak\medskip\noindent} % Proclaimed Things (theorems, lemmas, etc.) % (modeled on plain tex's ``\proclaim'', but theorems can be more % than one paragraph long.) \outer\def\beginproclaim#1. {\medbreak \noindent{\bf #1.\enspace}\begingroup\sl} \def\endproclaim{\endgroup\par\ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} % Proofs (n.b. \drawbox is defined in `OTHER') \def\beginproof#1{\smallskip{\it#1\/}} \def\endproof{\leavevmode\vrule height0pt width0pt depth0pt\nobreak\hfill\proofbox\smallskip} \def\QED{\endproof} \def\proofbox{\drawbox{1.2ex}{1.2ex}{.1ex}} % Remarks (n.b. remarks are only one par long) \def\remark#1#2\par{\ifdim\lastskip<\smallskipamount \removelastskip\penalty55\smallskip\fi {\it #1\/}#2\smallbreak} % CROSS-REFERENCES AND BIBLIOGRAPHY % % If the tag has not been defined (say by \deftag), then % \tag warns and inserts question % marks. 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We prove that every one-sided ideal is in fact two-sided, principal, and generated by a rational integer. \section{\Tag{sec-intro} Introduction} The nonassociative ring $\k$ of integral octaves is a discrete subring of the nonassociative field $\O$ of octaves---it is the natural analogue of the the rational integers in $\Z$. Dickson introduced $\k$ in \cite{dickson:new-simple-theory}; one can use $\k$ to construct the finite simple groups now called $G_2(p)$ for $p$ a prime number. Much later, Coxeter \cite{cox:integer-octaves} rediscovered the ring and obtained a number of new results concerning it. The purpose of this note is to obtain a complete description of the ideals in $\k$: every one-sided ideal is actually two-sided and generated by a rational integer. This substantially improves a result of Mahler \cite{mahler:ideals-cayley-dickson-algebra}, who proved that any one-sided ideal is generated by a rational integer multiple of an element of one of three possible norms. Our arguments rely on geometric properties of the $E_8$ lattice and its automorphism group, rather than the sort of explicit computation that Mahler used. Our definitions of $\O$ and $\k$ are those of \ecite{ATLAS}{p.~85}. The nonassociative field $\O$ of octaves is an 8-dimensional algebra over the real numbers $\R$, with basis $e_\infty=1$, $e_0,\ldots,e_6$. Multiplication is defined by the relations that for each $n=0,\ldots,6$, $e_n^2=-1$ and the span of $e_\infty$, $e_n$, $e_{n+1}$ and $e_{n+3}$ is a copy of the the (associative) field of quaternions, with $e_ne_{n+1}e_{n+3}=-1$. Here the subscripts should be read modulo 7. If $x=x_\infty e_\infty+x_0e_0+\cdots+ x_6e_6$ with each $x_n\in\R$ then the conjugate of $x$ is $\bar x=x_\infty e_\infty-x_0e_0-\cdots -x_6e_6$. The absolute value $|x|$ of $x$ is $\sqrt{x\bar x}$, which is a nonnegative real number; $|x|=0$ just if $x=0$. The identity $|xy|=|x|\,|y|$ (for any $x,y\in\O$) is important for the proof of theorem~\tag{thm-Euclidean-domain}. The real part of $x$ is $x_\infty$; if this vanishes then we say that $x$ is imaginary. The integral octaves $\k$ are the discrete subring of $\O$ consisting of those $x=\sum x_ne_n$ satisfying $x_n\in{1\over2}\Z$ for all $n=\infty,0,\ldots,6$ and that the set of $n$ for which $x_n\notin\Z$ is one of $$ \displaylines{ \{0124\},\quad \{0235\},\quad \{0346\},\quad \{0156\}, \cr \{\infty013\},\quad \{\infty026\},\quad \{\infty045\},\quad \{\infty0123456\}, \cr} $$ or the complement of one of these. One may check \cite{cox:integer-octaves} that $\k$ is closed under addition and multiplication. Furthermore, if $x\in\k$ then $|x|^2\in\Z$, and under the metric on $\k$ induced by the absolute value function, $\k$ is a scaled copy of the $E_8$ lattice \ecite{splag}{ch.~4,~\S8.1}. In particular, the minimal distance between elements of $\k$ is 1 and the covering radius of $\k$ is $1/\sqrt2$. (This means that every element of $\O$ lies within $1/\sqrt2$ of $\k$ and that $1/\sqrt2$ is the smallest number for which this holds.) There are 240 elements of $\k$ of absolute value 1, which are the units of $\k$. We denote the set of these by $\units$. The first step in applying the geometry of the $E_8$ lattice to the study of ideals in $\k$ is the following theorem and its corollary. \beginproclaim Theorem {\Tag{thm-Euclidean-domain}} \cite{cox:integer-octaves} \cite{mahler:ideals-cayley-dickson-algebra}. The function $x\mapsto|x|^2$ is a Euclidean norm on $\k$. \endproclaim \remark{Remark:} By this we mean that for all $x,m\in\k$ there are $q,r\in\k$ such that $x=qm+r$ with $|r|^2<|m|^2$, and that there are also $q',r'\in\k$ such that $x=mq'+r'$ with $|r'|^2<|m|^2$. \beginproof{Proof:} In the notation of the remark, let $q\in\k$ by such that $qm$ is an element of $\k m$ nearest $x$, and let $r=x-qm$. The right-multiplication map of $m$ increases distances by a factor of $|m|$, so $\k m$ has covering radius $|m|/\sqrt2$. Therefore $|r|\leq |m|/\sqrt2$. A similar argument applied to $m\k$ completes the proof. \endproof \beginproclaim Corrollary {\Tag{thm-pid}}. Any left (resp. right) ideal in $\k$ has the form $\k m$ (resp. $m\k$) for some $m\in\k$. \endproclaim \beginproof{Proof:} This follows from the usual argument that a Euclidean domain is a principal ideal domain. \endproof Of course, since $\k$ is not associative, if $m$ is a random integral octave then $\k m$ and $m\k$ might fail to be ideals. Our main theorem is essentially the assertion that $\k m$ or $m\k$ is an ideal if and only if $m$ is a product of a rational integer and a unit of $\k$: \def\statement{Any one-sided ideal $\ideal$ in $\k$ is two-sided, principal, and has the form $\k n=n\k$ for some rational integer $n$.} \beginproclaim Theorem {\tag{thm-main}}. \statement \endproclaim \noindent In order to prove the theorem we need to identify the group generated by the left (or right) multiplication maps of units of $\k$. We do this in the next section, and prove the theorem in section~\tag{sec-main-theorem}. \section{\Tag{sec-triality}. Triality } The best way to understand the group generated by the left (or right) multiplication maps of units of $\k$ is by considering the ``isotopy'' group of $\k$ and its ``triality'' automorphism. An isotopy is a triple $(A,B,C)$ of $\Z$-linear maps of $\k$ such that for all $x,y,z\in\k$, the equation $xyz=1$ holds just if $A(x)B(y)C(z)=1$ holds. These equations make sense because if either $xy\cdot z=1$ or $x\cdot yz=1$ then $x$, $y$ and $z$ all lie in an associative algebra, so that $xyz$ is unambiguously defined and equal to 1. The product of two isotopies is defined by $$ (A,B,C)(A',B',C')=(A\circ A',B\circ B', C\circ C'). $$ By \ecite{ATLAS}{p.~85}, the group of isotopies is isomorphic to an extension $\spin$ of the simple group $\pso=O_8^+(2)$. Furthermore, $\spin$ is generated by the triples $(L_u,R_u,B_u)$ for $u\in\units$, where $L_u$ (resp. $R_u$) is the map given by left (resp. right) multiplication by $u$, and $B_u$ is the ``bimultiplication'' map $x\mapsto u^{-1}xu^{-1}$. The fact that $(L_u,R_u,B_u)$ is an isotopy follows from the Moufang identity: $u(xy)u=(ux)(yu)$ for all $u,x,y\in\O$. For each $u\in\O\setminus\{0\}$ the maps $L_u$, $R_u$ and $B_u$ are orientation-preserving maps of $\O$, since this is obviously true for $u=1$ and $\O\setminus\{0\}$ is connected. This implies that there are three different maps $\pi_0$, $\pi_1$ and $\pi_2$ from $\spin$ to the rotation group of $\k$; these carry $(A,B,C)$ to $A$, $B$ and $C$, respectively. By the rotation group of $\k$ we mean the full group of orientation-preserving isometries of $\k$; this is the commutator subgroup of the $E_8$ Weyl group. It is a central extension $\so$ of $O_8^+(2)$. Finally, the description of $\spin$ in terms of isotopies makes visible the ``triality'' automorphism $\tau:(A,B,C)\mapsto(B,C,A)$. It is obvious that $\pi_m\circ\tau^n=\pi_{m+n}$, where the subscripts should be read modulo 3. We define $H$ to be the subgroup of $\spin$ generated by those triples $(L_u,R_u,B_u)$ with $u\in\im\units$, and we set $H_\ell=\pi_0(H)$, $H_r=\pi_1(H)$ and $H_b=\pi_2(H)$. That is, $H_\ell$, $H_r$ and $H_b$ are respectively the subgroups of $\so$ generated by the left, right and bimultiplication maps of elements of $\im\units$. \beginproclaim Lemma {\Tag{thm-has-index-120}}. $H_b$ is a maximal subgroup of $\so$ and preserves $\{\pm1\}$. \endproclaim \beginproof{Proof:} For $u\in\im\units$ it is easy to check that $B_u$ negates $1$ and $u$ and fixes $u^\perp\cap\im\O$ pointwise. (This uses the fact that orthogonal imaginary octaves anticommute.) Therefore $H_b$ acts on $\im\k$ as the $E_7$ Weyl group, since $\im\k$ is a copy of the $E_7$ lattice, being the orthogonal complement of a minimal vector in a copy of the $E_8$ lattice. Furthermore, an element of $H_b$ exchanges $1$ and $-1$ just if it reverses orientation on $\im\O$. Therefore $H_b$ is isomorphic to the $E_7$ Weyl group, and a computation reveals that $|\so|/|H_b|=120$. The maximality follows because by \ecite{ATLAS}{p.~85}, $\pso$ has no proper subgroups of index $<120$. \endproof \beginproclaim Lemma {\Tag{thm-scalar-mults-generates-all}}. The maps $L_u$ (or alternately, the $R_u$ or the $B_u$), for $u\in\units$, generate all of $\so$. \endproclaim \beginproof{Proof:} By lemma~\tag{thm-has-index-120}, $H_b$ is a maximal subgroup of $\so$. Since there are $u\in\units$ such that $B_u(1)\notin\{\pm1\}$, the group generated by all the $B_u$ is all of $\so$. This is just the assertion that $\pi_2(\spin)=\so$. By triality, we see that each of $\pi_0(\spin)$ and $\pi_1(\spin)$ is also all of $\so$. The lemma follows. \endproof We remark the the groups $\spin$ and $H$ are respectively ${\rm Spin}_8^+(\F_2)$, the universal central extension of $O_8^+(2)$, and ${\rm Pin}_7(\F_2)$, the direct product of $\Z/2$ and the universal central extension of $O_7(2)$. Furthermore, the groups $H_b$, $H_\ell$ and $H_r$ are isomorphic to $2\times O_7(2)$, ${\rm Spin}_7(\F_2)$ and ${\rm Spin}_7(\F_2)$. One may also consider the isotopy group of $\O$ and the obvious analogues of $H$, $H_b$, $H_\ell$ and $H_r$. These five groups (after discarding Euclidean factors) are respectively isomorphic to ${\rm Spin}_8(\R)$, ${\rm Pin}_7(\R)$, $O_7(\R)$, ${\rm Spin}_7(\R)$ and ${\rm Spin}_7(\R)$. It is remarkable that the exceptional phenomena (triality, etc.) arising in the $\F_2$-versions of these groups can be ``embedded'' in the real versions. In particular, the triality automorphism $\tau$ acts in both cases as the order-three automorphism of the Dynkin diagram $D_4$, the Dynkin diagram of an orthogonal group in 8 dimensions. \section{\Tag{sec-main-theorem}. The main theorem} The proof of our main theorem is now quite short. We continue using the notation of section~\tag{sec-triality}. \beginproclaim Theorem {\Tag{thm-main}}. \statement \endproclaim \beginproof{Proof:} The result for right ideals follows formally from that for left ideals, since the identity $\overline{xy}=\bar y\bar x$ (for any $x,y\in\O$) implies that the conjugate of a right ideal is a left ideal. So suppose $\ideal$ is a nonzero left ideal. By corollary~\tag{thm-pid} we have $\ideal=\k m$ where $m$ is some minimal (nonzero) element of $\ideal$. Since $\ideal$ is a left ideal it is preserved by the group generated by the maps $L_u$ for $u\in\units$. If $\k$ were associative then this group would have order 240. But $\k$ is not, and by lemma~\tag{thm-scalar-mults-generates-all}, the group is the full rotation group $\so$ of $\k$. Observe that $\ideal$ has 240 minimal vectors, namely the vectors $um$ for $u\in\units$. It is obvious that group generated by the $L_u$ acts transitively on these, so the subgroup preserving $m$ has index 240 and order 1,451,520. The only elements of the $E_8$ lattice with a stabilizer this large are the rational integral multiples of minimal lattice vectors. One sees this by considering the full isometry group of $\k$, the $E_8$ Weyl group. The stabilizer of any vector is (conjugate to) a Weyl group corresponding to a subdiagram of the $E_8$ Dynkin diagram. The only subdiagram whose associated Weyl group is large enough is the (unique) $E_7$ subdiagram. The conjugates of this subgroup are just the stabilizers of the various minimal vectors $u$ of the lattice. For each such $u$, the only vectors stabilized by the stabilizer of $u$ are the real multiples of $u$. Therefore $m=un$ for some $u\in\units$, $n\in\Z$. Then $\ideal=\k m=\k\cdot un=\k n$ and the proof is complete. \endproof \section{References} % Do not tamper with the following line % ---begin bibentries--- \bibitem{ATLAS} J.~H. Conway, R.~T. Curtis, S.~P. Norton, R.~A. Parker, and R.~A. Wilson. {\it {A}{T}{L}{A}{S} of Finite Groups}. Oxford, 1985. \bibitem{splag} J.~H. Conway and N.~J.~A. Sloane. {\it Sphere Packings, Lattices, and Groups}. Springer-Verlag, 1988. \bibitem{cox:integer-octaves} H.~S.~M. Coxeter. Integral {C}ayley numbers. {\it Duke Math. J.}, 13:561--78, 1946. \bibitem{dickson:new-simple-theory} L.~E. Dickson. A new simple theory of hypercomplex integers. {\it Journal de Math\'ematiques Pures et Appliqu\'ees}, 2:281--326, 1923. \bibitem{mahler:ideals-cayley-dickson-algebra} K.~Mahler. On ideals in the {C}ayley-{D}ickson algebra. {\it Proc. Roy. Irish Acad.}, 48:123--133, 1943. % ---end bibentries--- % Do not tamper with the previous line. \bye