%% boi-9.tex (short version)
%% Chapter 9 of the Book of Involutions

\chapter{Cubic Jordan Algebras}  
\label{9.chap.cub.jord}  

The set of symmetric elements in an associative algebra with
involution admits the structure of a Jordan algebra. One aim of this
chapter is to give some insight into the relationship between
involutions on central simple algebras and Jordan algebras.  After a
short survey on central simple Jordan algebras in
\S \ref{Jordan}, we specialize to Jordan algebras of degree~$3$
in \S \ref{Cubic}; in particular, we discuss extensively
``Freudenthal algebras,'' a class of Jordan algebras connected with
Hurwitz algebras and we describe the Springer
construction, which ties twisted compositions with cubic Jordan
algebras. On the other hand, cubic Jordan algebras are also related to
cubic associative algebras through the Tits constructions
(\S \ref{Tits}).  Of special interest, and the main object of
study of this chapter, are the exceptional simple Jordan algebras of
dimension~$27$, whose automorphism groups are of type~$F_4$. The
different constructions mentioned above are related to interesting
subgroups of~$F_4$.  For example, the automorphism group of a split
twisted composition is a subgroup of~$F_4$ and outer actions
on~$\Spin_8$ (triality)  become inner over~$F_4$.  Tits constructions
are related to the action of the cyclic group $\LZ/3\LZ$ on~$\Spin_8$
which yields invariant subgroups of classical type~$A_2$, and
Freudenthal algebras are related to the action of the group $S_3$
on~$\Spin_8$ which yields invariant subgroups of exceptional
type~$G_2$.

Cohomological invariants of exceptional simple Jordan algebras are
discussed in the last section.

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\begin{Notes}

  \itemN{\ref{Tits}} Tits constructions for fields of characteristic
  not~$2$ first appeared in print in Jacobson's book \cite{JacJord},
  as did the fact, also due to Tits, that any Albert algebra is a
  first or second Tits construction. These results were announced by
  Tits in a talk at the Oberwolfach meeting ``Jordan-Algebren und
  nicht-assoziativen Algebren, 17--26.8.1967.''  With the kind
  permission of J.~Tits and the Research Institute in Oberwolfach, we
  reproduce Tits' R\'esum\'e:
  
  \input{boi-9t.tex}
  
  Observe that the $\times$-product used by Tits is our
  $\times$-product divided by~$2$.  The extension of Tits
  constructions to cubic structures was carried out by McCrimmon
  \cite{McCFSTrev}. Tits constructions were systematically used by
  Petersson and Racine, see for example \cite{PeRaTits1} and
  \cite{PeRaTits2}. Petersson and Racine showed in particular that
  (with a few exceptions) simple cubic Jordan structures can be
  constructed by iteration of the Tits process (\cite{PeRaTits2},
  Theorem~3.1).  The result can be viewed as a cubic analog to the
  theorem of Hurwitz, proved by iterating the Cayley-Dickson process.
  
  Tits constructions can be used to give simple examples of
  exceptional division Jordan algebras of dimension~$27$.  The first
  examples of such division algebras were constructed by Albert
  \cite{Albertdivjord}.  They were significantly more complicated than
  those through Tits constructions. Assertions~\eqref{9.cohfere.2}
  and~\eqref{9.cohfere.3} of Theorem~\eqref{9.cohfere}
  and~\eqref{9.cohfere2} are due to Tits.  The nice cohomological
  proof given here is due to Ferrar and Petersson \cite{FePe} (for
  first Tits constructions).
  
  \itemN{\ref{Cohomojord}} The existence of the invariants $f_3$
  and~$f_5$ was first noticed by Serre (see 
  \cite{SerreBourb} and \cite[Fith Edition, Annexe, \S3]{Serrecoho}).
  The direct computation of the trace form given
  here, as well as Propositions~\eqref{9.cohtits1}
  and~\eqref{9.cohtits2} are due to Rost.  Serre suggested the
  existence of the invariant~$g_3$. Its definition is due to Rost
  \cite{RostJord}. An elementary approach to that invariant can be
  found in Petersson-Racine \cite{PeRaRost} and a description in
  characteristic~$3$ can be found in Petersson-Racine
  \cite{PeRaRost3}.
  
\end{Notes}

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