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% 
% Identifying Models of the Octave Projective Plane
%
% Daniel Allcock

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\title{Identifying Models of the Octave Projective Plane}

\author{Daniel Allcock}

\email{allcock@math.berkeley.edu}
\address{Department of Mathematics, \par
University of California, \par
Berkeley, CA 94720.}

\date{15 August 1995}
\note{Published in {\it Geometriae Dedicata} {\bf65}(1997) 215--217.}
\subject{51A35 (17C40)}
\plaintitlepage

%
% Definitions for cross-references
%
% Do not tamper with the following line.
% ---begin deftags---
% ---end deftags---
% Do not tamper with the previous line or the following line.
% ---begin defcites---
\defcite{1}{asl:op2}
\defcite{2}{freud:oktav-geometry}
\defcite{3}{jordan:op2}
% ---end defcites---
% Do not tamper with the preceeding line.

\abstract
We provide a convenient identification between two models of the
projective plane over the alternative field of octaves: Aslaksen's
coordinate approach and the classic approach via Jordan algebras. We
do this by modifying a 1949 lemma of P. Jordan.

\section{The Octave Plane}

The projective plane $ \calo P^2 $ over the alternative field $\calo$
of octaves (also called Cayley numbers) may be viewed from several
perspectives.  Two particularly attractive models are the elegant
coordinatization due to H.~Aslaksen using `restricted homogeneous
coordinates' \cite{asl:op2}, and the model developed extensively by
H.~Freudenthal, in which the points of $ \calo P^2 $ are identified
with a set of idempotents in $\calj$, a certain Jordan algebra
\cite{freud:oktav-geometry}.  What is missing is a convenient means to
pass between these two languages. This paper makes the observation
that a lemma due to P.~Jordan \cite{jordan:op2}, when suitably
modified, yields a beautiful identification. Jordan's paper seems to
have received little attention, despite being the first paper linking
$\calj$ and $\calo P^2$.

Briefly, here are the models. The points of Aslaksen's plane are the
nonzero triples $(x_1,x_2,x_3)$ of octaves with at least one real
element, modulo the relation that two such triples are equivalent if
they differ by left multiplication by an element of $\calo$.  Lines
may be defined as follows.  Declare two points to be {\it orthogonal}
if we have $x_1\bar y_1 + x_2\bar y_2 + x_3\bar y_3 = 0$ when we
choose representative triples $(x_i)$, $(y_i)$ for the points, with at
least 2 of the sets $\{x_i,y_i\}\> (i=1,2,3) $ containing a real
number. (This choice may always be made.)  The lines of the geometry
are the sets orthogonal to the points. Clever computations in
\cite{asl:op2} show that these conditions do actually yield a
projective plane.  ({\it Note:} Aslaksen required one coordinate to
be unity, but this is inessential; he also defined the same set of
lines without reference to the ``innner product'' above. His lines and
ours coincide.)

The exceptional Jordan algebra $\calj$ is the (real) algebra of
$3\times3$ Hermitian matrices with elements in $\calo$, under the
multiplication defined by $A*B=(AB+BA)/2$. The points of $\calo P^2$
are the trace 1 idempotents, and two such idempotents are called {\it
orthogonal} if their Jordan product vanishes. Again, the lines of the
geometry are the point-sets orthogonal to points. It is convenient to
identify an idempotent of $\calj$ with the vector subspace of $\calj$
consisting of its real scalar multiples.

Generalizing a construction of P. Jordan, we define a map from
Aslaksen's plane to $\calj$ by $(x_1,x_2,x_3)\mapsto e$ where $e$ is
the matrix defined by $(e_{ij})= \bar x_i x_j $. This is well-defined
up to real scalar multiplication, and it is easy to check that $e$ is
a trace 1 idempotent exactly when $|x_1|^2 +|x_2|^2 +|x_3|^2 =1$.

\beginproclaim Theorem.
The map defined above is an isomorphism from Aslaksen's
model of $\calo P^2$ to Freudenthal's.
\endproclaim

\beginproof{Proof: }
It is observed above that the map is well-defined, and it is trivial
to check that it is injective. To show that it is surjective, one need
only find suitable $(x_1,x_2,x_3)$, given a trace 1 idempotent in
$\calj$, which is easy.  The heart of the theorem is proving that the
notions of orthogonality between points of $\calo P^2$ coincide. We
accomplish this in the following lemma, which is a modification of
Jordan's Hilfsatz 2. We indicate a proof (Jordan didn't) because the
calculation is very tedious if approached incorrectly.
\endproof

\beginproclaim Lemma.
Let $(x_1,x_2,x_3)$, $(y_1,y_2,y_3)$ be two triples of elements of
$\calo$, each with at least one real element, and such that at least
two of the sets $\{x_i,y_i\}\> (i=1,2,3)$ contain a real number. Then,
defining elements $e,f$ of $\calj$ by $e_{ij}=\bar x_ix_j$,
$f_{ij}=\bar y_i y_j$, we have $e*f=0$ if and only if $x_1\bar y_1
+x_2\bar y_2 +x_3\bar y_3 = 0$.
\endproclaim

\beginproof{Proof: }
We know that $e$ and $f$ are scalar multiples of trace 1 idempotents,
and therefore (see \cite{freud:oktav-geometry}) $e*f=0$ if and only if
${\rm Tr}(e*f)=0$.  We have
$$
2{\rm Tr}(e*f) =  \sum_{i,j}(e_{ij}f_{ji} + f_{ij}e_{ji}) 
	=  2\sum_{i,j} {\rm Re}(e_{ij}f_{ji}) 
	=  2\sum_{i,j} {\rm Re}((\bar x_i x_j)(\bar y_j y_i)).
$$
Without loss of generality we may assume that $x_1$ and $y_2$ are
real, and so every term (except the $i=j=3$ term) contains a real
number.  By using the octave identities ${\rm Re}(a(bc))={\rm
Re}((ab)c)={\rm Re}((bc)a)$, we may replace each such term of the sum
by ${\rm Re}((x_j\bar y_j)(y_i\bar x_i))$.  We may also do this in the
case $i=j=3$, for the reason that any two elements of $\calo$ lie in
an associative subalgebra of $\calo$. So we have
$$\eqalign{
{\rm Tr}(e*f) & = \sum_{i,j} {\rm Re}((x_j\bar y_j)(y_i\bar x_i))
	=  {\rm Re} \Bigl( \Bigl( \sum_{j}x_j\bar y_j \Bigr) \Bigl( 
		\sum_{i}y_i\bar x_i \Bigr) \Bigr) \cr
     &	=  |x_1\bar y_1 + x_2\bar y_2 + x_3\bar y_3|^2,
\cr}$$
which completes the proof.
\endproof

\section{References}

% Do not tamper with the following line
% ---begin bibentries---

\bibitem{asl:op2}
H.~Aslaksen.
 Restricted homogeneous coordinates for the {C}ayley projective plane.
 {\it Geometriae Dedicata}, 40:245--50, 1991.

\bibitem{freud:oktav-geometry}
H.~Freudenthal.
 {O}ktaven, {A}usnahmegruppen, und {O}ktavengeometrie.
 {\it Geometriae Dedicata}, 19:1--73, 1985.
 (Informally published, Utrecht 1951).

\bibitem{jordan:op2}
P.~Jordan.
 {\"{U}}ber eine nicht-desaurguessche ebene projective {G}eometrie.
 {\it Abhandlungen Mathematischen Seminar der Universit{\"{a}}t
  Hamburg}, 16:74--6, 1949.

% ---end bibentries---
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