%% boi-9t.tex
%% Tits' resume in Oberwolfach, 17-26.8.1967
%% Input file for Chapter 9 of the Book of Involutions

\begin{Tits} 
  \noindent %
%%%  TITS, J.: \underline{Exceptional simple Jordan Algebras} 
  Exceptional simple Jordan Algebras
  \par \smallskip \noindent %
  (I) Denote by~$k$ a field of characteristic not~$2$, by $A$ a
  central simple algebra of degree~$3$ over~$k$, by $n\col A \to A$,
  $tr \col A\to A$ the reduced norm and reduced trace, and by
  $\times\col A\times A\to A$ the symmetric bilinear product defined
  by $(x\times x)x=n(x)$. For $x \in A$, set $\overline{x}=
  \frac{1}{2} (tr(x)-x)$.  Let $c \in k^*$. In the sum $A_0 + A_1
  +A_2$ of three copies of~$A$, introduce the following product:
  \[ \setlength{\extrarowheight}{1ex}
  \begin{array}{l|ccc}
    & x_0 & y_1 & z_2 
    \\[.5ex] \hline 
    x'_0 & \frac{1}{2}(xx' +x'x)_0 & (\overline{x'}y)_1&
    (z\overline{x'})_2 
    \\[1ex] \hline 
    y'_1& (\overline{x}y')_1 & c(y\times y')_2
    & (\overline{y'}z)_0 
    \\[1ex] \hline 
    z'_2& (z'\overline{x})_2 & (\overline{yz'})_0 & 
    \frac{1}{c}(z \times z')_1 
    \\[1ex]
  \end{array}
  \] 
  \noindent %
  (II) Denote by $\ell$\/ a quadratic extension of~$k$, by $B$ a central
  simple algebra of degree~$3$ over~$\ell$, and by $\sigma\col B \to
  B$ an involution of the second kind kind such that $k= \bmidb{x \in
    \ell}{ x^\sigma=x}$. Set $B^{\Sym }= \bmidb{ x\in B}{x^\sigma=x}$.
  Let $b \in B^{\Sym }$ and $c \in l^*$ be such that $n(b)= c^\sigma
  c$. In the sum $B^{\Sym } + B_*$ of~$B^{\Sym }$ and a copy~$B_*$
  of~$B$, define a product by
  \label{tits2}
  \[ \setlength{\extrarowheight}{1ex}
  \begin{array}{l|cc}
    &x &y 
    \\[.5ex] \hline 
    x' & \frac{1}{2}(xx' +x'x) & (\overline{x'}y)_* 
    \\[1ex] \hline
    y' & (\overline{x}y')_* & (\overline{yb{y'}^\sigma +
      y'by^\sigma}) + \bigr(c^\sigma(y^\sigma 
    \times {y'}^\sigma)b\inv\bigl)_* 
    \\[1ex]
  \end{array}
  \] 
  \textbf{Theorem 1.} % 
  \textit{The\/ $27$-dimensional algebras described under \textup{(I)}
    and \textup{(II)} are exceptional simple Jordan algebras
    over\/~$k$. Every such algebra is thus obtained.}
  \\[2ex]
  \textbf{Theorem 2.} % 
  \textit{The algebra \textup{(I)} is split if\/ $c \in n(A)$ and
    division otherwise.}  \textit{The algebra \textup{(II)} is reduced
    if\/ $c \in n(B)$ and division otherwise.}
  \\[2ex]
  \textbf{Theorem 3.} % 
  \textit{There exists an algebra of type \textup{(II)} which does not
    split over any cyclic extension of degree\/ $2$ or\/~$3$ of\/~$k$.
    \textup{(Notice that such an algebra is necessarily division and
      is not of type~(I)).}}
  \par \medskip \noindent %
  (For more details, cf.\ N. Jacobson.\ Jordan algebras, a forthcoming
  book).\\[1ex]
  \hspace*{\fill} J. Tits \hspace*{12ex}
\end{Tits}

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