%% boi-2.tex (short version)
%% Chapter 2 of the Book of Involutions

\chapter{Invariants of Involutions} 
\label{2.chap.inv.inv}

In this chapter, we define various kinds of invariants of central
simple algebras with involution (or with quadratic pair) and we
investigate their basic properties. The invariants considered here are
analogues of the classical invariants of quadratic forms: the Witt
index, the discriminant, the Clifford algebra and the signature. How
far the analogy can be pushed depends of course on the nature of the
involution: an index is defined for arbitrary central simple algebras
with involution or quadratic pair, but the discriminant is defined
only for orthogonal involutions and quadratic pairs, and the Clifford
algebra just for quadratic pairs. The Clifford algebra construction
actually splits into two parts: while it is impossible to define a
full Clifford algebra for quadratic pairs, the even and the odd parts
of the Clifford algebra can be recovered in the form of an algebra and
a bimodule. For unitary involutions, the notion of discriminant turns
out to lead to a rich structure: we associate
in~\S\ref{discalgebra.sec} a discriminant algebra (with involution) to
every unitary involution on a central simple algebra of even degree.
Finally, signatures can be defined for arbitrary involutions through
the associated trace forms.  These trace forms also have relations
with the discriminant or discriminant algebra. They yield higher
invariants for algebras with unitary involution of degree~$3$ in
Chapter~\ref{5.chap.alg.three} and for Jordan algebras in
Chapter~\ref{9.chap.cub.jord}.

The invariants considered in this chapter are produced by various
techniques. The index is derived from a representation of the algebra
with involution as the endomorphism algebra of some hermitian or
skew-hermitian space over a division algebra, while the definitions of
discriminant and Clifford algebra are based on the fact that scalar
extension reduces the algebra with quadratic pair to the endomorphism
algebra of a quadratic space. We show that the discriminant and even
Clifford algebra of the corresponding quadratic form can be defined in
terms of the adjoint quadratic pair, and that the definitions
generalize to yield invariants of arbitrary quadratic pairs. A similar
procedure is used to define the discriminant algebra of a central
simple algebra of even degree with unitary involution. Throughout most
of this chapter, our method of investigation is thus based on scalar
extension: after specifying the definitions ``rationally'' (i.e., over
an arbitrary base field), the main properties are proven by extending
scalars to a splitting field. This method contrasts with Galois
descent, where constructions over a separable closure are shown to be
invariant under the action of the absolute Galois group and are
therefore defined over the base field.

\section{The Index}
\label{index.sec}

According to~\thmref{invherm}, every central simple $F$-algebra with
involution $(A,\sigma)$ can be represented as
$\bigl(\End_D(V),\sigma_h\bigr)$ for some division algebra~$D$, some
$D$-vector space~$V$ and some nonsingular hermitian form~$h$ on~$V$.
Since this representation is essentially unique, it is not difficult
to check that the Witt index $w(V,h)$ of the hermitian space $(V,h)$,
defined as the maximum of the dimensions of totally isotropic
subspaces of~$V$, is an invariant of $(A,\sigma)$. In this section, we
give an alternate definition of this invariant which does not depend
on a representation of $(A,\sigma)$ as
$\bigl(\End_D(V),\sigma_h\bigr)$, and we characterize the involutions
which can be represented as adjoint involutions with respect to a
hyperbolic form. We define a slightly more general notion of index
which takes into account the Schur index of the algebra. A (weak)
analogue of Springer's theorem on odd degree extensions is discussed
in the final subsection.

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