%% boi-1.tex (short version)
%% Chapter 1 of the Book of Involutions

\chapter{Involutions and Hermitian Forms}  
\label{1.chap.inv.hem}  

Our perspective in this work is that involutions on central simple
algebras are twisted forms of symmetric or alternating bilinear forms
up to a scalar factor. To motivate this point of view, we consider the
basic, classical situation of linear algebra.

Let~$V$ be a finite dimensional vector space over a field~$F$ of
arbitrary characteristic. A bilinear form $b\col V\times V\to F$ is
called \index{A}{nonsingular bilinear form}\emph{nonsingular} if the
induced map
\[
\hat b\col V\to V^*=\Hom_F(V,F)
\]
defined by
\[
\hat b(x)(y)=b(x,y)\quad\text{for $x$, $y\in V$}
\]
is an isomorphism of vector spaces. For any $f\in\End_F(V)$ we may
then define $\sigma_b(f)\in\End_F(V)$ \index{B}{sigmab@{$\sigma_b$
    (involution adjoint to $b$)}} by
\[
\sigma_b(f)=\hat b^{-1}\circ f^t\circ\hat b
\]
where $f^t\in\End_F(V^*)$\index{B}{ft@{$f^t$ (transpose of $f$)}} is
the \index{A}{transpose of a linear map}\emph{transpose} of~$f$,
defined by mapping $\varphi \in V^*$ to $\varphi\circ f$. Alternately,
$\sigma_b(f)$ may be defined by the following property:
\begin{equation}
  \label{intro.sigama.def}  
  \tag{$*$}
  b\bigl(x,f(y)\bigr)=b\bigl(\sigma_b(f)(x),y\bigr)
  \quad\text{for $x$, $y\in V$.}
\end{equation}
The map $\sigma_b \col \End_F(V)\to\End_F(V)$ is then an
anti-automorphism of $\End_F(V)$ which is known as the
\index{A}{adjoint anti-automorphism}\emph{adjoint anti-automorphism}
with respect to the nonsingular bilinear form~$b$. The map $\sigma_b$
clearly is $F$-linear.


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