%% intro.tex
%% Introduction to the Book of Involutions

\chapter*{Introduction}

For us an involution is an anti-automorphism of order two of an
algebra.  The most elementary example is the transpose for matrix
algebras.  A more complicated example of an algebra over~$\LQ$
admitting an involution is the multiplication algebra of a Riemann
surface (see the notes at the end of Chapter~\ref{1.chap.inv.hem} for
more details). The central problem here, to give necessary and
sufficient conditions on a division algebra over~$\LQ$ to be a
multiplication algebra, was completely solved by Albert (1934/35). To
achieve this, Albert developed a theory of central simple algebras
with involution, based on the theory of simple algebras initiated a
few years earlier by Brauer, Noether, and also Albert and Hasse, and
gave a complete classification over~$\LQ$.  This is the historical
origin of our subject, however our motivation has a different source.
The basic objects are still central simple algebras, i.e., ``forms''
of matrix algebras. As observed by Weil (1960), central simple
algebras with involution occur in relation to classical algebraic
simple adjoint groups: connected components of automorphism groups of
central simple algebras with involution are such groups (with the
exception of a quaternion algebra with an orthogonal involution, where
the connected component of the automorphism group is a torus), and, in
their turn, such groups are connected components of automorphism
groups of central simple algebras with involution.

Even if this is mainly a book on algebras, the correspondence between
algebras and groups is a constant leitmotiv.  Properties of the
algebras are reflected in properties of the groups and of related
structures, such as Dynkin diagrams, and vice versa. For example we
associate certain algebras to algebras with involution in a functorial
way, such as the Clifford algebra (for orthogonal involutions) or the
$\lambda$-powers and the discriminant algebra (for unitary
involutions). These algebras are exactly the ``Tits algebras,''
defined by Tits (1971) in terms of irreducible representations of the
groups. Another example is algebraic triality, which is historically
related with groups of type~$D_4$ (\'E.~Cartan) and whose ``algebra''
counterpart is, so far as we know, systematically approached here for
the first time.

In the first chapter we recall basic properties of central simple
algebras and involutions.  As a rule for the whole book, without
however going to the utmost limit, we try to allow base fields of
characteristic~$2$ as well as those of other characteristic.
Involutions are divided up into orthogonal, symplectic and unitary
types. A central idea of this chapter is to interpret involutions in
terms of hermitian forms over skew fields. Quadratic pairs, introduced
at the end of the chapter, give a corresponding interpretation for
quadratic forms in characteristic $2$.

In Chapter~\ref{2.chap.inv.inv} we define several invariants of
involutions; the index is defined for every type of involution.  For
quadratic pairs additional invariants are the discriminant, the (even)
Clifford algebra and the Clifford module; for unitary involutions we
introduce the discriminant algebra. The definition of the discriminant
algebra is prepared for by the construction of the $\lambda$-powers of
a central simple algebra.  The last part of this chapter is devoted to
trace forms on algebras, which represent an important tool for recent
results discussed in later parts of the book.  Our method of
definition is based on scalar extension: after specifying the
definitions ``rationally'' (i.e., over an arbitrary base field), the
main properties are proven by working over a splitting field. This is
in contrast to Galois descent, where constructions over a separable
closure are shown to be invariant under the Galois group and therefore
are defined over the base field. A main source of inspiration for
Chapters \ref{1.chap.inv.hem} and~\ref{2.chap.inv.inv} is the paper
\cite{Tits} of Tits on ``Formes quadratiques, groupes orthogonaux et
alg\`ebres de Clifford.''

In Chapter~\ref{3.chap.simil} we investigate the automorphism groups
of central simple algebras with involutions. Inner automorphisms are
induced by elements which we call similitudes.  These automorphism
groups are twisted forms of the classical projective orthogonal,
symplectic and unitary groups.  After proving results which hold for
all types of involutions, we focus on orthogonal and unitary
involutions, where additional information can be derived from the
invariants defined in Chapter~\ref{2.chap.inv.inv}.  The next two
chapters are devoted to algebras of low degree.  There exist certain
isomorphisms among classical groups, known as exceptional
isomorphisms.  From the algebra point of view, this is explained in
the first part of Chapter~\ref{4.chap.alg.four} by properties of the
Clifford algebra of orthogonal involutions on algebras of degree $3$,
$4$, $5$ and~$6$.  In the second part we focus on tensor products of
two quaternion algebras, which we call biquaternion algebras. These
algebras have many interesting properties, which could be the subject
of a monograph of its own. This idea was at the origin of our project.

Algebras with unitary involutions are also of interest for odd
degrees, the lowest case being degree~$3$. From the group point of
view algebras with unitary involutions of degree~$3$ are of
type~$A_2$.  Chapter~\ref{5.chap.alg.three} gives a new presentation
of results of Albert and a complete classification of these algebras.
In preparation for this, we recall general results on \'etale and
Galois algebras.

The aim of Chapter~\ref{6.chap.alg.grp} is to give the classification
of semisimple algebraic groups over arbitrary fields. We use the
functorial approach to algebraic groups, although we quote without
proof some basic results on algebraic groups over algebraically closed
fields.  In the central section we describe in detail Weil's
correspondence \cite{We} between central simple algebras with
involution and classical groups.  Exceptional isomorphisms are
reviewed again in terms of this correspondence.  In the last section
we define Tits algebras of semisimple groups and give explicit
constructions of them in classical cases.

The theme of Chapter~\ref{7.chap.nonab.coh} is Galois cohomology. We
introduce the formalism and describe many examples. Previous results
are reinterpreted in this setting and cohomological invariants are
discussed.  Most of the techniques developed here are also needed for
the following chapters.

The last three chapters are dedicated to the exceptional groups of
type $G_2$, $F_4$ and to~$D_4$, which, in view of triality, is also
exceptional. In the Weil correspondence, octonion algebras play the
algebra role for~$G_2$ and exceptional simple Jordan algebras the
algebra role for~$F_4$.

Octonion algebras are an important class of composition algebras and
Chapter~\ref{8.chap.comp.tria} gives an extensive discussion of
composition algebras.  Of special interest from the group point of
view are ``symmetric'' compositions. In dimension~$8$ these are of two
types, corresponding to algebraic groups of type~$A_2$ or type~$G_2$.
Triality is defined through the Clifford algebra of symmetric
$8$-dimensional compositions. As a step towards exceptional simple
Jordan algebras, we introduce twisted compositions, which are defined
over cubic \'etale algebras.  This generalizes a construction of
Springer. The corresponding group of automorphisms in the split case
is the semidirect product $\Spin_8\rtimes S_3$.

In Chapter~\ref{9.chap.cub.jord} we describe different constructions
of exceptional simple Jordan algebras, due to Freudenthal, Springer
and Tits (the algebra side) and give interpretations from the
algebraic group side. The Springer construction arises from twisted
compositions, defined in Chapter~\ref{8.chap.comp.tria}, and basic
ingredients of Tits constructions are algebras of degree~$3$ with
unitary involutions, studied in Chapter~\ref{3.chap.simil}.  We
conclude this chapter by defining cohomological invariants for
exceptional simple Jordan algebras.

The last chapter deals with trialitarian actions on simple adjoint
groups of type~$D_4$.  To complete Weil's program for outer forms
of~$D_4$ (a case not treated by Weil), we introduce a new notion,
which we call a trialitarian algebra. The underlying structure is a
central simple algebra with an orthogonal involution, of degree~$8$
over a cubic \'etale algebra. The trialitarian condition relates the
algebra to its Clifford algebra.  Trialitarian algebras also occur in
the construction of Lie algebras of type~$D_4$.  Some indications in
this direction are given in the last section.

Exercises and notes can be found at the end of each chapter. Omitted
proofs sometimes occur as exercises. Moreover we included as exercises
some results we like, but which we did not wish to develop fully.  In
the notes we wanted to give complements and to look at some results
from a historical perspective.  We have tried our best to be useful;
we cannot, however, give strong guarantees of completeness or even
fairness.

This book is the achievement of a joint (and very exciting) effort of
four very different people. We are aware that the result is still
quite heterogeneous; however, we flatter ourselves that the
differences in style may be viewed as a positive feature.

Our work started out as an attempt to understand Tits' definition of
the Clifford algebra of a generalized quadratic form, and ended up
including many other topics to which Tits made fundamental
contributions, such as linear algebraic groups, exceptional algebras,
triality, \dots \ Not only was Jacques Tits a constant source of
inspiration through his work, but he also had a direct personal
influence, notably through his threat --- early in the inception of
our project --- to speak evil of our work if it did not include the
characteristic~$2$ case.  Finally he also agreed to bestow his
blessings on our book sous forme de pr\'eface. For all that we thank
him wholeheartedly.

This book could not have been written without the help and the
encouragement of many friends. They are too numerous to be listed here
individually, but we hope they will recognize themselves and find here
our warmest thanks. Richard Elman deserves a special mention for his
comment that the most useful book is not the one to which nothing can
be added, but the one which is published. This no-nonsense statement
helped us set limits to our endeavor. We were fortunate to get useful
advice on various points of the exposition from Ottmar Loos, Antonio
Paques, Parimala, Michel Racine, David Saltman, Jean-Pierre Serre and
Sridharan. We thank all of them for lending helping hands at the right
time.  A number of people were nice enough to read and comment on
drafts of parts of this book: Eva Bayer-Fluckiger, Vladimir Chernousov,
Ingrid Dejaiffe, Alberto Elduque, Darrell Haile, Luc Haine, Pat
Morandi, Holger Petersson, Ahmed Serhir, Tony Springer, Paul Swets and
Oliver Villa.  We know all of them had better things to do, and we are
grateful.  Skip Garibaldi and Adrian Wadsworth actually summoned
enough grim self-discipline to read a draft of the whole book,
detecting many shortcomings, making shrewd comments on the
organization of the book and polishing our broken English. Each
deserves a medal.  However, our capacity for making mistakes certainly
exceeds our friends' sagacity. We shall gratefully welcome any comment
or correction.

%%Jean-Pierre 
Tignol had the privilege to give a series of lectures on
``Central simple algebras, involutions and quadratic forms'' in
April~1993 at the National Taiwan University. He wants to thank
Ming-chang Kang and the National Research Council of China for this
opportunity to test high doses of involutions on a very patient
audience, and Eng-Tjioe Tan for making his stay in Taiwan a most
pleasant experience. The lecture notes from this crash course served
as a blueprint for the first chapters of this book.
 
%Our project immensely benefited by reciprocal visits among the
%authors.  We should like to mention with particular gratitude
%Merkurjev's stay in Louvain-la-Neuve in~1993, with support from the
%Fonds de D\'eveloppement Scientifique and the Institut de
%Math\'ematique Pure et Appliqu\'ee of the Universit\'e catholique de
%Louvain, and Tignol's stay in Z\"urich for the winter semester of
%1995--96, with support from the Eidgen\"ossische Technische
%Hochschule. Moreover, Merkurjev gratefully acknowledges support from
%the Alexander von Humboldt foundation and the hospitality of the
%Bielefeld university for the year 1995--96,
%and Tignol is grateful to the National Fund for Scientific Research
%of Belgium and to the European Commission for partial support under 
%the ``cr\'edit aux chercheurs'' and the TMR programmes respectively
%(contract ERB-FMRX-CT97-0107).
 
Our project immensely benefited by reciprocal visits among the
authors.  We should like to mention with particular gratitude
Merkurjev's stay in Louvain-la-Neuve in~1993, with support from the
Fonds de D\'eveloppement Scientifique and the Institut de
Math\'ematique Pure et Appliqu\'ee of the Universit\'e catholique de
Louvain, Rost's stays in Z\"urich at the Mathematics Research
Institute (FIM) in winter 1995--96 and at the Mathematics Department
of the ETH Z\"urich for the winter semester of 1996--97, and Tignol's
stay at the Mathematics Department of the ETH Z\"urich for the winter
semester of 1995--96, both with support from the Eidgen\"ossische
Technische Hochschule. Moreover, Merkurjev gratefully acknowledges
support from the Alexander von Humboldt foundation and the hospitality
of the Bielefeld university for the year 1995--96, and Tignol is
grateful to the National Fund for Scientific Research of Belgium and
to the European Commission for partial support under the ``cr\'edit
aux chercheurs'' and the TMR programmes respectively (contract
ERB-FMRX-CT97-0107).
 
The four authors enthusiastically thank Herbert Rost (Markus' father)
for the design of the cover page, in particular for his wonderful and
colorful rendition of the Dynkin diagram $D_{4}$.  They also give
special praise to Sergei Gelfand, Director of Acquisitions of the
American Mathematical Society, for his helpfulness and patience in
taking care of all their wishes for the publication.
 

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