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The Book of Involutions


Max-Albert Knus
Alexander Merkurjev
Markus Rost
Jean-Pierre Tignol
Author address:
¨
Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich, Switzerland
E-mail address: knus@math.ethz.ch
URL: http://www.math.ethz.ch/˜knus/

Dept. of Mathematics, University of California at Los Angeles,
Los Angeles, California, 90095-1555, USA
E-mail address: merkurev@math.ucla.edu
URL: http://www.math.ucla.edu/˜merkurev/
¨
NWF I - Mathematik, Universitat Regensburg, D-93040 Regens-
burg, Germany
E-mail address: markus.rost@mathematik.uni-regensburg.de
URL: http://www.physik.uni-regensburg.de/˜rom03516/

D´partement de math´matique, Universit´ catholique de Louvain,
e e e
Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
E-mail address: tignol@agel.ucl.ac.be
URL: http://www.math.ucl.ac.be/tignol/
Contents

Pr´face
e .............................. vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter I. Involutions and Hermitian Forms . . . . . . . . . . . . . 1
§1. Central Simple Algebras . . . . . . . . . . . . . . . . . . . 3
1.A. Fundamental theorems . . . . . . . . . . . . . . . . . 3
1.B. One-sided ideals in central simple algebras . . . . . . . . . 5
1.C. Severi-Brauer varieties . . . . . . . . . . . . . . . . . 9
§2. Involutions . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.A. Involutions of the ¬rst kind . . . . . . . . . . . . . . . 13
2.B. Involutions of the second kind . . . . . . . . . . . . . . 20
2.C. Examples . . . . . . . . . . . . . . . . . . . . . . . 23
2.D. Lie and Jordan structures . . . . . . . . . . . . . . . . 27
§3. Existence of Involutions . . . . . . . . . . . . . . . . . . . 31
3.A. Existence of involutions of the ¬rst kind . . . . . . . . . . 32
3.B. Existence of involutions of the second kind . . . . . . . . 36
§4. Hermitian Forms . . . . . . . . . . . . . . . . . . . . . . 41
4.A. Adjoint involutions . . . . . . . . . . . . . . . . . . . 42
4.B. Extension of involutions and transfer . . . . . . . . . . . 45
§5. Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . 53
5.A. Standard identi¬cations . . . . . . . . . . . . . . . . . 53
5.B. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 56
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter II. Invariants of Involutions . . . . . . . . . . . . . . . . . 71
§6. The Index . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.A. Isotropic ideals . . . . . . . . . . . . . . . . . . . . . 72
6.B. Hyperbolic involutions . . . . . . . . . . . . . . . . . 74
6.C. Odd-degree extensions . . . . . . . . . . . . . . . . . 79
§7. The Discriminant . . . . . . . . . . . . . . . . . . . . . . 80
7.A. The discriminant of orthogonal involutions . . . . . . . . 80
7.B. The discriminant of quadratic pairs . . . . . . . . . . . . 83
§8. The Cli¬ord Algebra . . . . . . . . . . . . . . . . . . . . . 87
8.A. The split case . . . . . . . . . . . . . . . . . . . . . 87
8.B. De¬nition of the Cli¬ord algebra . . . . . . . . . . . . . 91
8.C. Lie algebra structures . . . . . . . . . . . . . . . . . . 95
iii
iv CONTENTS


8.D. The center of the Cli¬ord algebra . . . . . . . . . . . . 99
8.E. The Cli¬ord algebra of a hyperbolic quadratic pair . . . . . 106
§9. The Cli¬ord Bimodule . . . . . . . . . . . . . . . . . . . . 107
9.A. The split case . . . . . . . . . . . . . . . . . . . . . 107
9.B. De¬nition of the Cli¬ord bimodule . . . . . . . . . . . . 108
9.C. The fundamental relations . . . . . . . . . . . . . . . . 113
§10. The Discriminant Algebra . . . . . . . . . . . . . . . . . . 114
10.A. The »-powers of a central simple algebra . . . . . . . . . 115
10.B. The canonical involution . . . . . . . . . . . . . . . . 116
10.C. The canonical quadratic pair . . . . . . . . . . . . . . . 119
10.D. Induced involutions on »-powers . . . . . . . . . . . . . 123
10.E. De¬nition of the discriminant algebra . . . . . . . . . . . 126
10.F. The Brauer class of the discriminant algebra . . . . . . . . 130
§11. Trace Form Invariants . . . . . . . . . . . . . . . . . . . . 132
11.A. Involutions of the ¬rst kind . . . . . . . . . . . . . . . 133
11.B. Involutions of the second kind . . . . . . . . . . . . . . 138
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Chapter III. Similitudes . . . . . . . . . . . . . . . . . . . . . . 153
§12. General Properties . . . . . . . . . . . . . . . . . . . . . . 153
12.A. The split case . . . . . . . . . . . . . . . . . . . . . 153
12.B. Similitudes of algebras with involution . . . . . . . . . . 158
12.C. Proper similitudes . . . . . . . . . . . . . . . . . . . 163
12.D. Functorial properties . . . . . . . . . . . . . . . . . . 168
§13. Quadratic Pairs . . . . . . . . . . . . . . . . . . . . . . . 172
13.A. Relation with the Cli¬ord structures . . . . . . . . . . . 172
13.B. Cli¬ord groups . . . . . . . . . . . . . . . . . . . . . 176
13.C. Multipliers of similitudes . . . . . . . . . . . . . . . . 190
§14. Unitary Involutions . . . . . . . . . . . . . . . . . . . . . 193
14.A. Odd degree . . . . . . . . . . . . . . . . . . . . . . 193
14.B. Even degree . . . . . . . . . . . . . . . . . . . . . . 194
14.C. Relation with the discriminant algebra . . . . . . . . . . 194
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Chapter IV. Algebras of Degree Four . . . . . . . . . . . . . . . . 205
§15. Exceptional Isomorphisms . . . . . . . . . . . . . . . . . . 205
15.A. B1 ≡ C1 . . . . . . . . . . . . . . . . . . . . . . . . 207
15.B. A2 ≡ D2 . . . . . . . . . . . . . . . . . . . . . . . . 210
1
15.C. B2 ≡ C2 . . . . . . . . . . . . . . . . . . . . . . . . 216
15.D. A3 ≡ D3 . . . . . . . . . . . . . . . . . . . . . . . . 220
§16. Biquaternion Algebras . . . . . . . . . . . . . . . . . . . . 233
16.A. Albert forms . . . . . . . . . . . . . . . . . . . . . . 235
16.B. Albert forms and symplectic involutions . . . . . . . . . . 237
16.C. Albert forms and orthogonal involutions . . . . . . . . . . 245
§17. Whitehead Groups . . . . . . . . . . . . . . . . . . . . . . 253
17.A. SK1 of biquaternion algebras . . . . . . . . . . . . . . . 253
17.B. Algebras with involution . . . . . . . . . . . . . . . . 266
CONTENTS v


Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Chapter V. Algebras of Degree Three . . . . . . . . . . . . . . . . 279
´
§18. Etale and Galois Algebras . . . . . . . . . . . . . . . . . . 279
´
18.A. Etale algebras . . . . . . . . . . . . . . . . . . . . . 280
18.B. Galois algebras . . . . . . . . . . . . . . . . . . . . . 287
18.C. Cubic ´tale algebras . . . . . . . . . . .
e . . . . . . . 296
§19. Central Simple Algebras of Degree Three . . . . . . . . . . . . 302
19.A. Cyclic algebras . . . . . . . . . . . . . . . . . . . . . 302
19.B. Classi¬cation of involutions of the second kind . . . . . . . 304
´
19.C. Etale subalgebras . . . . . . . . . . . . . . . . . . . . 307
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Chapter VI. Algebraic Groups . . . . . . . . . . . . . . . . . . . 323
§20. Hopf Algebras and Group Schemes . . . . . . . . . . . . . . 324
20.A. Group schemes . . . . . . . . . . . . . . . . . . . . . 325
§21. The Lie Algebra and Smoothness . . . . . . . . . . . . . . . 334
21.A. The Lie algebra of a group scheme . . . . . . . . . . . . 334
§22. Factor Groups . . . . . . . . . . . . . . . . . . . . . . . 339
22.A. Group scheme homomorphisms . . . . . . . . . . . . . . 339
§23. Automorphism Groups of Algebras . . . . . . . . . . . . . . 344
23.A. Involutions . . . . . . . . . . . . . . . . . . . . . . 345
23.B. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 350
§24. Root Systems . . . . . . . . . . . . . . . . . . . . . . . . 352
§25. Split Semisimple Groups . . . . . . . . . . . . . . . . . . . 354
25.A. Simple split groups of type A, B, C, D, F , and G . . . . . 355
25.B. Automorphisms of split semisimple groups . . . . . . . . . 358
§26. Semisimple Groups over an Arbitrary Field . . . . . . . . . . . 359
26.A. Basic classi¬cation results . . . . . . . . . . . . . . . . 362
26.B. Algebraic groups of small dimension . . . . . . . . . . . 372
§27. Tits Algebras of Semisimple Groups . . . . . . . . . . . . . . 373
27.A. De¬nition of the Tits algebras . . . . . . . . . . . . . . 374
27.B. Simply connected classical groups . . . . . . . . . . . . 376
27.C. Quasisplit groups . . . . . . . . . . . . . . . . . . . . 377
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Chapter VII. Galois Cohomology . . . . . . . . . . . . . . . . . . 381
§28. Cohomology of Pro¬nite Groups . . . . . . . . . . . . . . . . 381
28.A. Cohomology sets . . . . . . . . . . . . . . . . . . . . 381
28.B. Cohomology sequences . . . . . . . . . . . . . . . . . 383
28.C. Twisting . . . . . . . . . . . . . . . . . . . . . . . 385
28.D. Torsors . . . . . . . . . . . . . . . . . . . . . . . . 386
§29. Galois Cohomology of Algebraic Groups . . . . . . . . . . . . 389
29.A. Hilbert™s Theorem 90 and Shapiro™s lemma . . . . . . . . 390
29.B. Classi¬cation of algebras . . . . . . . . . . . . . . . . 393
29.C. Algebras with a distinguished subalgebra . . . . . . . . . 396
vi CONTENTS


29.D. Algebras with involution . . . . . . . . . . . . . . . . 397
29.E. Quadratic spaces . . . . . . . . . . . . . . . . . . . . 404
29.F. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 406
§30. Galois Cohomology of Roots of Unity . . . . . . . . . . . . . 411
30.A. Cyclic algebras . . . . . . . . . . . . . . . . . . . . . 412
30.B. Twisted coe¬cients . . . . . . . . . . . . . . . . . . . 414
30.C. Cohomological invariants of algebras of degree three . . . . 418
§31. Cohomological Invariants . . . . . . . . . . . . . . . . . . . 421
31.A. Connecting homomorphisms . . . . . . . . . . . . . . . 421
31.B. Cohomological invariants of algebraic groups . . . . . . . . 427
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Chapter VIII. Composition and Triality . . . . . . . . . . . . . . . 447
§32. Nonassociative Algebras . . . . . . . . . . . . . . . . . . . 447
§33. Composition Algebras . . . . . . . . . . . . . . . . . . . . 451
33.A. Multiplicative quadratic forms . . . . . . . . . . . . . . 451
33.B. Unital composition algebras . . . . . . . . . . . . . . . 452
Pr´face
e

Quatre des meilleurs alg´bristes d™aujourd™hui (j™aimerais dire, comme jadis,
e
g´om`tres , au sens noble, mais h´las d´suet du terme) nous donnent ce beau
ee e e
Livre des Involutions, qu™ils me demandent de pr´facer.
e
Quel est le propos de l™ouvrage et a quels lecteurs s™adresse-t-il? Bien sˆr il y
` u
est souvent question d™involutions, mais celles-ci sont loin d™ˆtre omnipr´sentes et le
e e
titre est plus l™expression d™un ´tat d™ˆme que l™a¬rmation d™un th`me central. En
e a e
fait, les questions envisag´es sont multiples, relevant toutes de domaines importants
e
des math´matiques contemporaines ; sans vouloir ˆtre exhaustif (ceci n™est pas une
e e
introduction), on peut citer :
- les formes quadratiques et les alg`bres de Cli¬ord,
e
- les alg`bres centrales simples (ici les involutions, et notamment celles de
e
seconde esp`ce, se taillent une place de choix !) mais aussi les alg`bres
e e
alternatives et les alg`bres de Jordan,
e
- les alg`bres de Hopf,
e
- les groupes alg´briques, principalement semi-simples,
e
- la cohomologie galoisienne.
Pour ce qui est du public concern´, la lecture ou la consultation du livre sera
e
pro¬table a un large ´ventail de math´maticiens. Le non-initi´ y trouvera une
` e e e
introduction claire aux concepts fondamentaux des domaines en question ; expos´s e
le plus souvent en fonction d™applications concr`tes, ces notions de base sont pr´-
e e
sent´es de fa¸on vivante et d´pouill´e, sans g´n´ralit´s gratuites (les auteurs ne sont
e c e e ee e
pas adeptes de grandes th´ories abstraites). Le lecteur d´j` inform´, ou croyant
e ea e
l™ˆtre, pourra r´apprendre (ou d´couvrir) quelques beaux th´or`mes jadis bien
e e e ee
connus mais un peu oubli´s dans la litt´rature r´cente, ou au contraire, voir
e e e
des r´sultats qui lui sont en principe familiers expos´s sous un jour nouveau et
e e
´clairant (je pense par exemple a l™introduction des alg`bres trialitaires au dernier
e ` e
chapitre). En¬n, les sp´cialistes et les chercheurs auront a leur disposition une
e `
r´f´rence pr´cieuse, parfois unique, pour des d´veloppements r´cents, souvents dˆs
ee e e e u
aux auteurs eux-mˆmes, et dont certains sont expos´s ici pour la premi`re fois
e e e
(c™est par exemple le cas pour plusieurs r´sultats sur les invariants cohomologiques,
e
donn´s a la ¬n du chapitre 7).
e`
Malgr´ la grande vari´t´ des th`mes consid´r´s et les individualit´s tr`s mar-
e ee e ee e e
qu´es des quatre auteurs, ce Livre des Involutions a une unit´ remarquable. Le
e e
ciment un peu fragile des involutions n™est certes pas seul a l™expliquer. Il y a
`
aussi, bien sˆr, les interconnections multiples entre les sujets trait´s ; mais plus
u e
d´terminante encore est l™importance primordiale accord´e a des structures fortes,
e e`
se prˆtant par exemple a des th´or`mes de classi¬cation substantiels. Ce n™est pas
e ` ee
un hasard si les alg`bres centrales simples de petites dimensions (trois et quatre),
e
les groupes exceptionnels de type G2 et F4 (on regrette un peu que Sa Majest´ E8 e
vii
´
viii PREFACE


fasse ici ¬gure de parent pauvre), les alg`bres de composition, . . . , re¸oivent autant
e c
d™attention.
On l™a compris, ce Livre est tout a la fois un livre de lecture passionnant et
`
un ouvrage de r´f´rence d™une extrˆme richesse. Je suis reconnaissant aux auteurs
ee e
de l™honneur qu™ils m™ont fait en me demandant de le pr´facer, et plus encore de
e
m™avoir permis de le d´couvrir et d™apprendre a m™en servir.
e `

Jacques Tits
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 Q admitting an involution is the multiplication algebra
of a Riemann surface (see the notes at the end of Chapter ?? for more details).
The central problem here, to give necessary and su¬cient conditions on a division
algebra over Q 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 classi¬cation
over Q. This is the historical origin of our subject, however our motivation has a
di¬erent source. The basic objects are still central simple algebras, i.e., “forms”
of matrix algebras. As observed by Weil (1960), central simple algebras with in-
volution 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 involu-
tion, 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 alge-
bras and groups is a constant leitmotiv. Properties of the algebras are re¬‚ected in
properties of the groups and of related structures, such as Dynkin diagrams, and
conversely. For example we associate certain algebras to algebras with involution
in a functorial way, such as the Cli¬ord algebra (for orthogonal involutions) or the
»-powers and the discriminant algebra (for unitary involutions). These algebras are
exactly the “Tits algebras”, de¬ned by Tits (1971) in terms of irreducible represen-
tations of the groups. Another example is algebraic triality, which is historically
related with groups of type D4 (E. Cartan) and whose “algebra” counterpart is, so
far as we know, systematically approached here for the ¬rst time.
In the ¬rst chapter we recall basic properties of central simple algebras and in-
volutions. As a rule for the whole book, without however going to the utmost limit,
we try to allow base ¬elds 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
¬elds. Quadratic pairs, introduced at the end of the chapter, give a corresponding
interpretation for quadratic forms in characteristic 2.
In Chapter ?? we de¬ne several invariants of involutions; the index is de¬ned for
every type of involution. For quadratic pairs additional invariants are the discrim-
inant, the (even) Cli¬ord algebra and the Cli¬ord module; for unitary involutions
we introduce the discriminant algebra. The de¬nition of the discriminant algebra
ix
x INTRODUCTION


is prepared for by the construction of the »-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
de¬nition is based on scalar extension: after specifying the de¬nitions “rationally”
(i.e., over an arbitrary base ¬eld), the main properties are proven by working over
a splitting ¬eld. 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 de¬ned over the base ¬eld. A main source of inspiration for Chapters ?? and ??
is the paper [?] of Tits on “Formes quadratiques, groupes orthogonaux et alg`bres e
de Cli¬ord.”
In Chapter ?? we investigate the automorphism groups of central simple alge-
bras with involutions. Inner automorphisms are induced by elements which we call
similitudes. These automorphism groups are twisted forms of the classical projec-
tive 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 de¬ned in Chapter ??.
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 ¬rst part of Chapter ?? by prop-
erties of the Cli¬ord 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 proper-
ties, 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 A2 . Chapter ?? gives a new presentation of results of Albert
and a complete classi¬cation of these algebras. In preparation for this, we recall
general results on ´tale and Galois algebras.
e
The aim of Chapter ?? is to give the classi¬cation of semisimple algebraic groups
over arbitrary ¬elds. We use the functorial approach to algebraic groups, although
we quote without proof some basic results on algebraic groups over algebraically
closed ¬elds. In the central section we describe in detail Weil™s correspondence [?]
between central simple algebras with involution and classical groups. Exceptional
isomorphisms are reviewed again in terms of this correspondence. In the last section
we de¬ne Tits algebras of semisimple groups and give explicit constructions of them
in classical cases.
The theme of Chapter ?? 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 G2 , F4
and to D4 , which, in view of triality, is also exceptional. In the Weil correspon-
dence, octonion algebras play the algebra role for G2 and exceptional simple Jordan
algebras the algebra role for F4 .
Octonion algebras are an important class of composition algebras and Chap-
ter ?? 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 A2 or type G2 . Triality
INTRODUCTION xi


is de¬ned through the Cli¬ord algebra of symmetric 8-dimensional compositions.
As a step towards exceptional simple Jordan algebras, we introduce twisted compo-
sitions, which are de¬ned over cubic ´tale algebras. This generalizes a construction
e
of Springer. The corresponding group of automorphisms in the split case is the
semidirect product Spin8 S3 .
In Chapter ?? we describe di¬erent constructions of exceptional simple Jordan
algebras, due to Freudenthal, Springer and Tits (the algebra side) and give in-
terpretations from the algebraic group side. The Springer construction arises from
twisted compositions, de¬ned in Chapter ??, and basic ingredients of Tits construc-
tions are algebras of degree 3 with unitary involutions, studied in Chapter ??. We
conclude this chapter by de¬ning cohomological invariants for exceptional simple
Jordan algebras.
The last chapter deals with trialitarian actions on simple adjoint groups of
type D4 . To complete Weil™s program for outer forms of D4 (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 ´tale algebra. The trialitarian condition relates the algebra
e
to its Cli¬ord algebra. Trialitarian algebras also occur in the construction of Lie
algebras of type D4 . 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 com-
plements 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) e¬ort of four very
di¬erent people. We are aware that the result is still quite heterogeneous; however,
we ¬‚atter ourselves that the di¬erences in style may be viewed as a positive feature.
Our work started out as an attempt to understand Tits™ de¬nition of the Cli¬ord
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, . . . Not only was Jacques Tits a constant source of
inspiration through his work, but he also had a direct personal in¬‚uence, 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´face. For all that we thank him
e
wholeheartedly.
This book could not have been written without the help and the encourage-
ment of many friends. They are too numerous to be listed here individually, but
we hope they will recognize themselves and ¬nd 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 Dejai¬e, Alberto Elduque, Darrell
Haile, Luc Haine, Pat Morandi, Holger Petersson, Ahmed Serhir, Tony Springer,
xii INTRODUCTION


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 bro-
ken 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
¬rst chapters of this book.
Our project immensely bene¬ted 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´veloppement Scienti¬que and the
e
Institut de Math´matique Pure et Appliqu´e of the Universit´ catholique de Lou-
e e e
vain, and Tignol™s stay in Z¨rich for the winter semester of 1995“96, with support
u
from the Eidgen¨ssische Technische Hochschule. Moreover, Merkurjev gratefully
o
acknowledges support from the Alexander von Humboldt foundation and the hos-
pitality of the Bielefeld university for the year 1995“96, and Jean-Pierre Tignol is
grateful to the National Fund for Scienti¬c Research of Belgium for partial support.
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 D4 . 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 our wishes for the publication.
Conventions and Notations

Maps. The image of an element x under a map f is generally denoted f (x);
the notation xf is also used however, notably for homomorphisms of left modules.
In that case, we also use the right-hand rule for mapping composition; for the image
f g
of x ∈ X under the composite map X ’ Y ’ Z we set either g —¦ f (x) or xf g and
’’
the composite is thus either g —¦ f or f g.
As a general rule, module homomorphisms are written on the opposite side of
the scalars. (Right modules are usually preferred.) Thus, if M is a module over a
ring R, it is also a module (on the opposite side) over EndR (M ), and the R-module
structure de¬nes a natural homomorphism:

R ’ EndEndR (M ) (M ).

Note therefore that if S ‚ EndR (M ) is a subring, and if we endow M with its
natural S-module structure, then EndS (M ) is the opposite of the centralizer of S
in EndR (M ):
op
EndS (M ) = CEndR (M ) S .

Of course, if R is commutative, every right R-module MR may also be regarded as a
left R-module R M , and every endomorphism of MR also is an endomorphism of R M .
Note however that with the convention above, the canonical map EndR (MR ) ’
EndR (R M ) is an anti-isomorphism.
The characteristic polynomial and its coe¬cients. Let F denote an ar-
bitrary ¬eld. The characteristic polynomial of a matrix m ∈ Mn (F ) (or an endo-
morphism m of an n-dimensional F -vector space) is denoted

Pm (X) = X n ’ s1 (m)X n’1 + s2 (m)X n’2 ’ · · · + (’1)n sn (m).
(0.1)

The trace and determinant of m are denoted tr(m) and det(m) :

tr(m) = s1 (m), det(m) = sn (m).
We recall the following relations between coe¬cients of the characteristic polyno-
mial:

(0.2) Proposition. For m, m ∈ Mn (F ), we have s1 (m)2 ’ s1 (m2 ) = 2s2 (m) and

s1 (m)s1 (m ) ’ s1 (mm ) = s2 (m + m ) ’ s2 (m) ’ s2 (m ).

Proof : It su¬ces to prove these relations for generic matrices m = (xij )1¤i,j¤n ,
m = (xij )1¤i,j¤n whose entries are indeterminates over Z; the general case follows
by specialization. If »1 , . . . , »n are the eigenvalues of the generic matrix m (in
xiii
xiv CONVENTIONS AND NOTATIONS


an algebraic closure of Q(xij | 1 ¤ i, j ¤ n)), we have s1 (m) = »i and
1¤i¤n
s2 (m) = 1¤i<j¤n »i »j , hence

s1 (m)2 ’ 2s2 (m) = »2 = s1 (m2 ),
i
1¤i¤n

proving the ¬rst relation. The second relation follows by linearization, since 2 is
not a zero-divisor in Z[xij , xij | 1 ¤ i, j ¤ n].
If L is an associative and commutative F -algebra of dimension n and ∈ L,
the characteristic polynomial of multiplication by , viewed as an F -endomorphism
of L, is called the generic polynomial of and is denoted
PL, (X) = X n ’ s1 ( )X n’1 + s2 ( )X n’2 ’ · · · + (’1)n sn ( ).
The trace and norm of are denoted TL/F ( ) and NL/F ( ) (or simply T ( ), N ( )):
TL/F ( ) = s1 ( ), NL/F ( ) = sn ( ).
We also denote
(0.3) SL/F ( ) = S( ) = s2 ( ).
The characteristic polynomial is also used to de¬ne a generic polynomial for central
simple algebras, called the reduced characteristic polynomial : see (??). Generaliza-
tions to certain nonassociative algebras are given in § ??.
Bilinear forms. A bilinear form b : V — V ’ F on a ¬nite dimensional vector
space V over an arbitrary ¬eld F is called symmetric if b(x, y) = b(y, x) for all
x, y ∈ V , skew-symmetric if b(x, y) = ’b(y, x) for all x, y ∈ V and alternating
if b(x, x) = 0 for all x ∈ V . Thus, the notions of skew-symmetric and alternating
(resp. symmetric) form coincide if char F = 2 (resp. char F = 2). Alternating forms
are skew-symmetric in every characteristic.
If b is a symmetric or alternating bilinear form on a (¬nite dimensional) vector
space V , the induced map
ˆ V ’ V — = HomF (V, F )
b:
is de¬ned by ˆ b(x)(y) = b(x, y) for x, y ∈ V . The bilinear form b is nonsingular (or
regular , or nondegenerate) if ˆ is bijective. (It su¬ces to require that ˆ be injective,
b b
i.e., that the only vector x ∈ V such that b(x, y) = 0 for all y ∈ V is x = 0, since
we are dealing with ¬nite dimensional vector spaces over ¬elds.) Alternately, b is
nonsingular if and only if the determinant of its Gram matrix with respect to an
arbitrary basis of V is nonzero:
det b(ei , ej ) = 0.
1¤i,j¤n
In that case, the square class of this determinant is called the determinant of b :
· F —2 ∈ F — /F —2 .
det b = det b(ei , ej ) 1¤i,j¤n
The discriminant of b is the signed determinant:
disc b = (’1)n(n’1)/2 det b ∈ F — /F —2 where n = dim V .
For ±1 , . . . , ±n ∈ F , the bilinear form ±1 , . . . , ±n on F n is de¬ned by
±1 , . . . , ±n (x1 , . . . , xn ), (y1 , . . . , yn ) = ±1 x1 y1 + · · · + ±n xn yn .
We also de¬ne the n-fold P¬ster bilinear form ±1 , . . . , ±n by
±1 , . . . , ±n = 1, ’±1 — · · · — 1, ’±n .
CONVENTIONS AND NOTATIONS xv


If b : V — V ’ F is a symmetric bilinear form, we denote by qb : V ’ F the
associated quadratic map, de¬ned by
qb (x) = b(x, x) for x ∈ V .
Quadratic forms. If q : V ’ F is a quadratic map on a ¬nite dimensional
vector space over an arbitrary ¬eld F , the associated symmetric bilinear form b q is
called the polar of q; it is de¬ned by
bq (x, y) = q(x + y) ’ q(x) ’ q(y) for x, y ∈ V ,
hence bq (x, x) = 2q(x) for all x ∈ V . Thus, the quadratic map qbq associated to bq
is qbq = 2q. Similarly, for every symmetric bilinear form b on V , we have bqb = 2b.
Let V ⊥ = { x ∈ V | bq (x, y) = 0 for y ∈ V }. The quadratic map q is called
nonsingular (or regular , or nondegenerate) if either V ⊥ = {0} or dim V ⊥ = 1 and
q(V ⊥ ) = {0}. The latter case occurs only if char F = 2 and V is odd-dimensional.
Equivalently, a quadratic form of dimension n is nonsingular if and only if it is
n/2 2
equivalent over an algebraic closure to i=1 x2i’1 x2i (if n is even) or to x0 +
(n’1)/2
x2i’1 x2i (if n is odd).
i=1
The determinant and the discriminant of a nonsingular quadratic form q of
dimension n over a ¬eld F are de¬ned as follows: let M be a matrix representing q
in the sense that
q(X) = X · M · X t
where X = (x1 , . . . , xn ) and t denotes the transpose of matrices; the condition that
q is nonsingular implies that M + M t is invertible if n is even or char F = 2, and
has rank n ’ 1 if n is odd and char F = 2. The matrix M is uniquely determined by
q up to the addition of a matrix of the form N ’ N t ; therefore, M + M t is uniquely
determined by q.
If char F = 2 we set
+ M t ) · F —2 ∈ F — /F —2
1
det q = det 2 (M

and

disc q = (’1)n(n’1)/2 det q ∈ F — /F —2 .
Thus, the determinant (resp. the discriminant) of a quadratic form is the determi-
nant (resp. the discriminant) of its polar form divided by 2n .
If char F = 2 and n is odd we set
det q = disc q = q(y) · F —2 ∈ F — /F —2
(0.4)
where y ∈ F n is a nonzero vector such that (M + M t ) · y = 0. Such a vector y is
uniquely determined up to a scalar factor, since M + M t has rank n ’ 1, hence the
de¬nition above does not depend on the choice of y.
If char F = 2 and n is even we set
det q = s2 (M + M t )’1 M + „˜(F ) ∈ F/„˜(F )

and
m(m’1)
disc q = + det q ∈ F/„˜(F )
2
xvi CONVENTIONS AND NOTATIONS


where m = n/2 and „˜(F ) = { x + x2 | x ∈ F }. (More generally, for ¬elds of
characteristic p = 0, „˜ is de¬ned as „˜(x) = x + xp , x ∈ F .) The following lemma
shows that the de¬nition of det q does not depend on the choice of M :
(0.5) Lemma. Suppose char F = 2. Let M, N ∈ Mn (F ) and W = M + M t . If W
is invertible, then
2
s2 W ’1 (M + N + N t ) = s2 (W ’1 M ) + s1 (W ’1 N ) + s1 (W ’1 N ) .
Proof : The second relation in (??) yields

s2 W ’1 M + W ’1 (N + N t ) =
s2 (W ’1 M ) + s2 W ’1 (N + N t ) + s1 (W ’1 M )s1 W ’1 (N + N t )
+ s1 W ’1 M W ’1 (N + N t ) .
In order to prove the lemma, we show below:
2
s2 W ’1 (N + N t ) = s1 (W ’1 N )
(0.6)
s1 (W ’1 M )s1 W ’1 (N + N t ) = 0
(0.7)
s1 W ’1 M W ’1 (N + N t ) = s1 (W ’1 N ).
(0.8)
Since a matrix and its transpose have the same characteristic polynomial, the traces
of W ’1 N and (W ’1 N )t = N t W ’1 are the same, hence
s1 (W ’1 N t ) = s1 (N t W ’1 ) = s1 (W ’1 N ).
Therefore, s1 W ’1 (N + N t ) = 0, and (??) follows.
Similarly, we have
s1 (W ’1 M W ’1 N t ) = s1 (N W ’1 M t W ’1 ) = s1 (W ’1 M t W ’1 N ),
hence the left side of (??) is
s1 (W ’1 M W ’1 N ) + s1 (W ’1 M t W ’1 N ) = s1 W ’1 (M + M t )W ’1 N .
Since M + M t = W , (??) follows.
The second relation in (??) shows that the left side of (??) is
s2 (W ’1 N ) + s2 (W ’1 N t ) + s1 (W ’1 N )s1 (W ’1 N t ) + s1 (W ’1 N W ’1 N t ).
Since W ’1 N and W ’1 (W ’1 N )t W (= W ’1 N t ) have the same characteristic poly-
nomial, we have si (W ’1 N ) = si (W ’1 N t ) for i = 1, 2, hence the ¬rst two terms
cancel and the third is equal to s1 (W ’1 N )2 . In order to prove (??), it therefore
su¬ces to show
s1 (W ’1 N W ’1 N t ) = 0.
Since W = M + M t , we have W ’1 = W ’1 M W ’1 + W ’1 M t W ’1 , hence
s1 (W ’1 N W ’1 N t ) = s1 (W ’1 M W ’1 N W ’1 N t ) + s1 (W ’1 M t W ’1 N W ’1 N t ),
and (??) follows if we show that the two terms on the right side are equal. Since
W t = W we have (W ’1 M W ’1 N W ’1 N t )t = N W ’1 N t W ’1 M t W ’1 , hence
s1 (W ’1 M W ’1 N W ’1 N t ) = s1 (N W ’1 N t )(W ’1 M t W ’1 )
= s1 (W ’1 M t W ’1 N W ’1 N t ).
CONVENTIONS AND NOTATIONS xvii


Quadratic forms are called equivalent if they can be transformed into each other
by invertible linear changes of variables. The various quadratic forms representing a
quadratic map with respect to various bases are thus equivalent. It is easily veri¬ed
that the determinant det q (hence also the discriminant disc q) is an invariant of the
equivalence class of the quadratic form q; the determinant and the discriminant are
therefore also de¬ned for quadratic maps. The discriminant of a quadratic form (or
map) of even dimension in characteristic 2 is also known as the pseudodiscriminant
or the Arf invariant of the form. See §?? for the relation between the discriminant
and the even Cli¬ord algebra.
Let ±1 , . . . , ±n ∈ F . If char F = 2 we denote by ±1 , . . . , ±n the diagonal
quadratic form
± 1 , . . . , ± n = ± 1 x2 + · · · + ± n x2
1 n
which is the quadratic form associated to the bilinear form ±1 , . . . , ±n . We also
de¬ne the n-fold P¬ster quadratic form ±1 , . . . , ±n by
±1 , . . . , ±n = 1, ’±1 — · · · — 1, ’±n
where — = —F is the tensor product over F . If char F = 2, the quadratic forms
[±1 , ±2 ] and [±1 ] are de¬ned by
2 2
[±1 ] = ±1 X 2 ,
[±1 , ±2 ] = ±1 X1 + X1 X2 + ±2 X2 and
and the n-fold P¬ster quadratic form ±1 , . . . , ±n ]] by
±1 , . . . , ±n ]] = ±1 , . . . , ±n’1 — [1, ±n ].
(See Baeza [?, p. 5] or Knus [?, p. 50] for the de¬nition of the tensor product of a
bilinear form and a quadratic form.) For instance,
±1 , ±2 ]] = (x2 + x1 x2 + ±2 x2 ) + ±1 (x2 + x3 x4 + ±2 x2 ).
1 2 3 4
xviii CONVENTIONS AND NOTATIONS
CHAPTER I


Involutions and Hermitian Forms

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 ¬nite dimensional vector space over a ¬eld F of arbitrary char-
acteristic. A bilinear form b : V — V ’ F is called nonsingular if the induced
map
ˆ V ’ V — = HomF (V, F )
b:
de¬ned by
ˆ
b(x)(y) = b(x, y) for x, y ∈ V
is an isomorphism of vector spaces. For any f ∈ EndF (V ) we may then de¬ne
σb (f ) ∈ EndF (V ) by
σb (f ) = ˆ’1 —¦ f t —¦ ˆ
b b
where f t ∈ EndF (V — ) is the transpose of f , de¬ned by mapping • ∈ V — to • —¦ f .
Alternately, σb (f ) may be de¬ned by the following property:
(—) b x, f (y) = b σb (f )(x), y for x, y ∈ V .
The map σb : EndF (V ) ’ EndF (V ) is then an anti-automorphism of EndF (V )
which is known as the adjoint anti-automorphism with respect to the nonsingular
bilinear form b. The map σb clearly is F -linear.
The basic result which motivates our approach and which will be generalized
in (??) is the following:
Theorem. The map which associates to each nonsingular bilinear form b on V its
adjoint anti-automorphism σb induces a one-to-one correspondence between equiv-
alence classes of nonsingular bilinear forms on V modulo multiplication by a factor
in F — and linear anti-automorphisms of EndF (V ). Under this correspondence, F -
linear involutions on EndF (V ) (i.e., anti-automorphisms of period 2) correspond
to nonsingular bilinear forms which are either symmetric or skew-symmetric.
Proof : From relation (—) it follows that for ± ∈ F — the adjoint anti-automorphism
σ±b with respect to the multiple ±b of b is the same as the adjoint anti-automor-
phism σb . Therefore, the map b ’ σb induces a well-de¬ned map from the set
of nonsingular bilinear forms on V up to a scalar factor to the set of F -linear
anti-automorphisms of End(V ).
To show that this map is one-to-one, note that if b, b are nonsingular bilinear
forms on V , then the map v = ˆ’1 —¦ b ∈ GL(V ) satis¬es
ˆ
b
b (x, y) = b v(x), y for x, y ∈ V .
1
2 I. INVOLUTIONS AND HERMITIAN FORMS


From this relation, it follows that the adjoint anti-automorphisms σb , σb are related
by
σb (f ) = v —¦ σb (f ) —¦ v ’1 for f ∈ EndF (V ),
or equivalently
σb = Int(v) —¦ σb ,
where Int(v) denotes the inner automorphism of EndF (V ) induced by v:
Int(v)(f ) = v —¦ f —¦ v ’1 for f ∈ EndF (V ).
Therefore, if σb = σb , then v ∈ F — and b, b are scalar multiples of each other.
Moreover, if b is a ¬xed nonsingular bilinear form on V with adjoint anti-
automorphism σb , then for any linear anti-automorphism σ of EndF (V ), the com-
’1
posite σb —¦ σ is an F -linear automorphism of EndF (V ). Since these automor-
phisms are inner, by the Skolem-Noether theorem (see (??) below), there exists
’1
u ∈ GL(V ) such that σb —¦ σ = Int(u). Then σ is the adjoint anti-automorphism
with respect to the bilinear form b de¬ned by
b (x, y) = b u(x), y .
Thus, the ¬rst part of the theorem is proved.
Observe also that if b is a nonsingular bilinear form on V with adjoint anti-
automorphism σb , then the bilinear form b de¬ned by
b (x, y) = b(y, x) for x, y ∈ V
’1 2
has adjoint anti-automorphism σb = σb . Therefore, σb = Id if and only if b and b
are scalar multiples of each other; since the scalar factor µ such that b = µb clearly
satis¬es µ2 = 1, this condition holds if and only if b is symmetric or skew-symmetric.
This shows that F -linear involutions correspond to symmetric or skew-sym-
metric bilinear forms under the bijection above.

The involution σb associated to a nonsingular symmetric or skew-symmetric
bilinear form b under the correspondence of the theorem is called the adjoint in-
volution with respect to b. Our aim in this ¬rst chapter is to give an analogous
interpretation of involutions on arbitrary central simple algebras in terms of hermit-
ian forms on vector spaces over skew ¬elds. We ¬rst review basic notions concerning
central simple algebras. The ¬rst section also discusses Severi-Brauer varieties, for
use in §??. In §?? we present the basic de¬nitions concerning involutions on cen-
tral simple algebras. We distinguish three types of involutions, according to the
type of pairing they are adjoint to over an algebraic closure: involutions which are
adjoint to symmetric (resp. alternating) bilinear forms are called orthogonal (resp.
symplectic); those which are adjoint to hermitian forms are called unitary. Invo-
lutions of the ¬rst two types leave the center invariant; they are called involutions
of the ¬rst kind. Unitary involutions are also called involutions of the second kind ;
they restrict to a nontrivial automorphism of the center. Necessary and su¬cient
conditions for the existence of an involution on a central simple algebra are given
in §??.
The theorem above, relating bilinear forms on a vector space to involutions
on the endomorphism algebra, is generalized in §??, where hermitian forms over
simple algebras are investigated. Relations between an analogue of the Scharlau
§1. CENTRAL SIMPLE ALGEBRAS 3


transfer for hermitian forms and extensions of involutions are also discussed in this
section.
When F has characteristic 2, it is important to distinguish between bilinear
and quadratic forms. Every quadratic form de¬nes (by polarization) an alternating
form, but not conversely since a given alternating form is the polar of various quad-
ratic forms. The quadratic pairs introduced in the ¬nal section may be regarded
as twisted analogues of quadratic forms up to a scalar factor in the same way that
involutions may be thought of as twisted analogues of nonsingular symmetric or
skew-symmetric bilinear forms. If the characteristic is di¬erent from 2, every or-
thogonal involution determines a unique quadratic pair since a quadratic form is
uniquely determined by its polar bilinear form. By contrast, in characteristic 2 the
involution associated to a quadratic pair is symplectic since the polar of a quadratic
form is alternating, and the quadratic pair is not uniquely determined by its asso-
ciated involution. Quadratic pairs play a central rˆle in the de¬nition of twisted
o
forms of orthogonal groups in Chapter ??.

§1. Central Simple Algebras
Unless otherwise mentioned, all the algebras we consider in this work are ¬nite-
dimensional with 1. For any algebra A over a ¬eld F and any ¬eld extension K/F ,
we write AK for the K-algebra obtained from A by extending scalars to K:
AK = A —F K.
We also de¬ne the opposite algebra Aop by
Aop = { aop | a ∈ A },
with the operations de¬ned as follows:
aop + bop = (a + b)op , aop bop = (ba)op , ± · aop = (± · a)op
for a, b ∈ A and ± ∈ F .
A central simple algebra over a ¬eld F is a (¬nite dimensional) algebra A = {0}
with center F (= F · 1) which has no two-sided ideals except {0} and A. An algebra
A = {0} is a division algebra (or a skew ¬eld ) if every non-zero element in A is
invertible.

1.A. Fundamental theorems. For the convenience of further reference, we
summarize without proofs some basic results from the theory of central simple
algebras. The structure of these algebras is determined by the following well-known
theorem of Wedderburn:
(1.1) Theorem (Wedderburn). For an algebra A over a ¬eld F , the following
conditions are equivalent:
(1) A is central simple.
(2) The canonical map A —F Aop ’ EndF (A) which associates to a — bop the linear
map x ’ axb is an isomorphism.
(3) There is a ¬eld K containing F such that AK is isomorphic to a matrix algebra
over K, i.e., AK Mn (K) for some n.
(4) If „¦ is an algebraically closed ¬eld containing F ,
A„¦ Mn („¦) for some n.
4 I. INVOLUTIONS AND HERMITIAN FORMS


(5) There is a ¬nite dimensional central division algebra D over F and an integer r
such that A Mr (D).
Moreover, if these conditions hold, all the simple left (or right) A-modules are
isomorphic, and the division algebra D is uniquely determined up to an algebra
isomorphism as D = EndA (M ) for any simple left A-module M .
References: See for instance Scharlau [?, Chapter 8] or Draxl [?, §3].
The ¬elds K for which condition (??) holds are called splitting ¬elds of A.
Accordingly, the algebra A is called split if it is isomorphic to a matrix algebra
Mn (F ) (or to EndF (V ) for some vector space V over F ).
Since the dimension of an algebra does not change under an extension of scalars,
it follows from the above theorem that the dimension of every central simple algebra
is a square: dimF A = n2 if AK Mn (K) for some extension K/F . The integer n is
called the degree of A and is denoted by deg A. The degree of the division algebra D
in condition (??) is called the index of A (or sometimes the Schur index of A) and
denoted by ind A. Alternately, the index of A can be de¬ned by the relation
deg A ind A = dimF M
where M is any simple left module over A. This relation readily follows from the
fact that if A Mr (D), then Dr is a simple left module over A.
We rephrase the implication (??) ’ (??) in Wedderburn™s theorem:
(1.2) Corollary. Every central simple F -algebra A has the form
A EndD (V )
for some (¬nite dimensional ) central division F -algebra D and some ¬nite-dimen-
sional right vector space V over D. The F -algebra D is uniquely determined by A
up to isomorphism, V is a simple left A-module and deg A = deg D dimD V .
In view of the uniqueness (up to isomorphism) of the division algebra D (or,
equivalently, of the simple left A-module M ), we may formulate the following de¬-
nition:
(1.3) De¬nition. Finite dimensional central simple algebras A, B over a ¬eld F
are called Brauer-equivalent if the F -algebras of endomorphisms of any simple left
A-module M and any simple left B-module N are isomorphic:
EndA (M ) EndB (N ).
Equivalently, A and B are Brauer-equivalent if and only if M (A) Mm (B)
for some integers , m.
Clearly, every central simple algebra is Brauer-equivalent to one and only one
division algebra (up to isomorphism). If A and B are Brauer-equivalent central
simple algebras, then ind A = ind B; moreover, A B if and only if deg A = deg B.
The tensor product endows the set of Brauer equivalence classes of central
simple algebras over F with the structure of an abelian group, denoted Br(F ) and
called the Brauer group of F . The unit element in this group is the class of F
which is also the class of all the matrix algebras over F . The inverse of the class of
a central simple algebra A is the class of the opposite algebra Aop , as part (??) of
Wedderburn™s theorem shows.
Uniqueness (up to isomorphism) of simple left modules over central simple
algebras leads to the following two fundamental results:
§1. CENTRAL SIMPLE ALGEBRAS 5


(1.4) Theorem (Skolem-Noether). Let A be a central simple F -algebra and let
B ‚ A be a simple subalgebra. Every F -algebra homomorphism ρ : B ’ A extends
to an inner automorphism of A: there exists a ∈ A— such that ρ(b) = aba’1 for all
b ∈ B. In particular, every F -algebra automorphism of A is inner.
References: Scharlau [?, Theorem 8.4.2], Draxl [?, §7] or Pierce [?, §12.6].
The centralizer CA B of a subalgebra B ‚ A is, by de¬nition, the set of elements
in A which commute with every element in B.
(1.5) Theorem (Double centralizer). Let A be a central simple F -algebra and let
B ‚ A be a simple subalgebra with center K ⊃ F . The centralizer C A B is a simple
subalgebra of A with center K which satis¬es
dimF A = dimF B · dimF CA B CA CA B = B.
and
If K = F , then multiplication in A de¬nes a canonical isomorphism A = B—F CA B.
References: Scharlau [?, Theorem 8.4.5], Draxl [?, §7] or Pierce [?, §12.7].
Let „¦ denote an algebraic closure of F . Under scalar extension to „¦, every
central simple F -algebra A of degree n becomes isomorphic to Mn („¦). We may
therefore ¬x an F -algebra embedding A ’ Mn („¦) and view every element a ∈ A
as a matrix in Mn („¦). Its characteristic polynomial has coe¬cients in F and is
independent of the embedding of A in Mn („¦) (see Scharlau [?, Ch. 8, §5], Draxl [?,
§22], Reiner [?, §9] or Pierce [?, §16.1]); it is called the reduced characteristic
polynomial of A and is denoted
PrdA,a (X) = X n ’ s1 (a)X n’1 + s2 (a)X n’2 ’ · · · + (’1)n sn (a).
(1.6)
The reduced trace and reduced norm of a are denoted TrdA (a) and NrdA (a) (or
simply Trd(a) and Nrd(a)):
TrdA (a) = s1 (a), NrdA (a) = sn (a).
We also write
(1.7) SrdA (a) = s2 (a).
(1.8) Proposition. The bilinear form TA : A — A ’ F de¬ned by
TA (x, y) = TrdA (xy) for x, y ∈ A
is nonsingular.
Proof : The result is easily checked in the split case and follows in the general case
by scalar extension to a splitting ¬eld. (See Reiner [?, Theorem 9.9]).
1.B. One-sided ideals in central simple algebras. A fundamental result
of the Wedderburn theory of central simple algebras is that all the ¬nitely generated
left (resp. right) modules over a central simple F -algebra A decompose into direct
sums of simple left (resp. right) modules (see Scharlau [?, p. 283]). Moreover, as
already pointed out in (??), the simple left (resp. right) modules are all isomorphic.
If A = Mr (D) for some integer r and some central division algebra D, then D r is
a simple left A-module (via matrix multiplication, writing the elements of D r as
column vectors). Therefore, every ¬nitely generated left A-module M is isomorphic
to a direct sum of copies of D r :
(Dr )s
M for some integer s,
6 I. INVOLUTIONS AND HERMITIAN FORMS


hence
dimF M = rs dimF D = s deg A ind A.
More precisely, we may represent the elements in M by r — s-matrices with entries
in D:
M Mr,s (D)
so that the action of A = Mr (D) on M is the matrix multiplication.
(1.9) De¬nition. The reduced dimension of the left A-module M is de¬ned by
dimF M
rdimA M = .
deg A
The reduced dimension rdimA M will be simply denoted by rdim M when the al-
gebra A is clear from the context. Observe from the preceding relation that the re-
duced dimension of a ¬nitely generated left A-module is always a multiple of ind A.
Moreover, every left A-module M of reduced dimension s ind A is isomorphic to
Mr,s (D), hence the reduced dimension classi¬es left A-modules up to isomorphism.
The preceding discussion of course applies also to right A-modules; writing the
elements of Dr as row vectors, matrix multiplication also endows D r with a right
A-module structure, and D r is then a simple right A-module. Every right module
of reduced dimension s ind A over A = Mr (D) is isomorphic to Ms,r (D).
(1.10) Proposition. Every left module of ¬nite type M over a central simple F -
algebra A has a natural structure of right module over E = EndA (M ), so that
M is an A-E-bimodule. If M = {0}, the algebra E is central simple over F and
Brauer-equivalent to A; moreover,
deg E = rdimA M, rdimE M = deg A,
and
A = EndE (M ).
Conversely, if A and E are Brauer-equivalent central simple algebras over F , then
there is an A-E-bimodule M = {0} such that A = EndE (M ), E = EndA (M ),
rdimA (M ) = deg E and rdimE (M ) = deg A.
Proof : The ¬rst statement is clear. (Recall that endomorphisms of left modules
are written on the right of the arguments.) Suppose that A = Mr (D) for some
integer r and some central division algebra D. Then D r is a simple left A-module,
hence D EndA (Dr ) and M (Dr )s for some s. Therefore,
Ms EndA (Dr )
EndA (M ) Ms (D).
This shows that E is central simple and Brauer-equivalent to A. Moreover, deg E =
s deg D = rdimA M , hence
rs dim D
rdimE M = = r deg D = deg A.
s deg D
Since M is an A-E-bimodule, we have a natural embedding A ’ EndE (M ). Com-
puting the degree of EndE (M ) as we computed deg EndA (M ) above, we get
deg EndE (M ) = deg A,
hence this natural embedding is surjective.
§1. CENTRAL SIMPLE ALGEBRAS 7


For the converse, suppose that A and E are Brauer-equivalent central simple
F -algebras. We may assume that
A = Mr (D) and E = Ms (D)
for some central division F -algebra D and some integers r and s. Let M = Mr,s (D)
be the set of r — s-matrices over D. Matrix multiplication endows M with an A-
E-bimodule structure, so that we have natural embeddings
(1.11) A ’ EndE (M ) and E ’ EndA (M ).
Since dimF M = rs dimF D, it is readily computed that rdimE M = deg A and
rdimA M = deg E. The ¬rst part of the proposition then yields
deg EndA (M ) = rdimA M = deg E and deg EndE (M ) = rdimE M = deg A,
hence the natural embeddings (??) are surjective.

Ideals and subspaces. Suppose now that A = EndD (V ) for some central
division algebra D over F and some ¬nite dimensional right vector space V over D.
We aim to get an explicit description of the one-sided ideals in A in terms of
subspaces of V .
Let U ‚ V be a subspace. Composing every linear map from V to U with the
inclusion U ’ V , we identify HomD (V, U ) with a subspace of A = EndD (V ):
HomD (V, U ) = { f ∈ EndD (V ) | im f ‚ U }.
This space clearly is a right ideal in A, of reduced dimension
rdim HomD (V, U ) = dimD U deg D.
Similarly, composing every linear map from the quotient space V /U to V with
the canonical map V ’ V /U , we may identify HomD (V /U, V ) with a subspace of
A = EndD (V ):
HomD (V /U, V ) = { f ∈ EndD (V ) | ker f ⊃ U }.
This space is clearly a left ideal in A, of reduced dimension
rdim HomD (V /U, V ) = dimD (V /U ) deg D.
(1.12) Proposition. The map U ’ HomD (V, U ) de¬nes a one-to-one correspon-
dence between subspaces of dimension d in V and right ideals of reduced dimen-
sion d ind A in A = EndD (V ). Similarly, the map U ’ HomD (V /U, V ) de¬nes a
one-to-one correspondence between subspaces of dimension d in V and left ideals
of reduced dimension deg A ’ d ind A in A. Moreover, there are canonical isomor-
phisms of F -algebras:
EndA HomD (V, U ) EndD (U ) EndA HomD (V /U, V ) EndD (V /U ).
and
Proof : The last statement is clear: multiplication on the left de¬nes an F -algebra
homomorphism EndD (U ) ’ EndA HomD (V, U ) and multiplication on the right
de¬nes an F -algebra homomorphism
EndD (V /U ) ’ EndA HomD (V /U, V ) .
Since rdim HomD (V, U ) = dimD U deg D, we have
deg EndA HomD (V, U ) = dimD U deg D = deg EndD (U ),
8 I. INVOLUTIONS AND HERMITIAN FORMS


so the homomorphism EndD (U ) ’ EndA HomD (V, U ) is an isomorphism. Simi-
larly, the homomorphism EndD (V /U ) ’ EndA HomD (V /U, V ) is an isomorphism
by dimension count.
For the ¬rst part, it su¬ces to show that every right (resp. left) ideal in A has
the form HomD (V, U ) (resp. HomD (V /U, V )) for some subspace U ‚ V . This is
proved for instance in Baer [?, §5.2].
(1.13) Corollary. For every left (resp. right) ideal I ‚ A there exists an idempo-
tent e ∈ A such that I = Ae (resp. I = eA). Multiplication on the right (resp. left)
induces a surjective homomorphism of right (resp. left) EndA (I)-modules:
ρ : I ’ EndA (I)
which yields an isomorphism: eAe EndA (I).
Proof : If I = HomD (V /U, V ) (resp. HomD (V, U )), choose a complementary sub-
space U in V , so that V = U • U , and take for e the projection on U parallel to U
(resp. the projection on U parallel to U ). We then have I = Ae (resp. I = eA).
For simplicity of notation, we prove the rest only in the case of a left ideal I.
Then EndA (I) acts on I on the right. For x ∈ I, de¬ne ρ(x) ∈ EndA (I) by
y ρ(x) = yx.
For f ∈ EndA (I) we have
f
(yx)f = yxf = y ρ(x ) ,
hence
ρ(xf ) = ρ(x) —¦ f,
which means that ρ is a homomorphism of right EndA (I)-modules. In order to see
that ρ is onto, pick an idempotent e ∈ A such that I = Ae. For every y ∈ I we
have y = ye; it follows that every f ∈ EndA (I) is of the form f = ρ(ef ), since for
every y ∈ I,
f
y f = (ye)f = yef = y ρ(e ) .
Therefore, ρ is surjective.
To complete the proof, we show that the restriction of ρ to eAe is an isomor-

phism eAe ’ EndA (I). It is readily veri¬ed that this restriction is an F -algebra

homomorphism. Moreover, for every x ∈ I one has ρ(x) = ρ(ex) since y = ye for
every y ∈ I. Therefore, the restriction of ρ to eAe is also surjective onto End A (I).
Finally, if ρ(ex) = 0, then in particular
eρ(ex) = ex = 0,
so ρ is injective on eAe.
Annihilators. For every left ideal I in a central simple algebra A over a ¬eld F ,
the annihilator I 0 is de¬ned by
I 0 = { x ∈ A | Ix = {0} }.
This set is clearly a right ideal. Similarly, for every right ideal I, the annihilator I 0
is de¬ned by
I 0 = { x ∈ A | xI = {0} };
it is a left ideal in A.
§1. CENTRAL SIMPLE ALGEBRAS 9


(1.14) Proposition. For every left or right ideal I ‚ A, rdim I + rdim I 0 = deg A
and I 00 = I.
Proof : Let A = EndD (V ). For any subspace U ‚ V it follows from the de¬nition
of the annihilator that
HomD (V, U )0 = HomD (V /U, V ) HomD (V /U, V )0 = HomD (V, U ).
and
Since every left (resp. right) ideal I ‚ A has the form I = HomD (V /U, V ) (resp.
I = HomD (V, U )), the proposition follows.
Now, let J ‚ A be a right ideal of reduced dimension k and let B ‚ A be the
idealizer of J:
B = { a ∈ A | aJ ‚ J }.
This set is a subalgebra of A containing J as a two-sided ideal. It follows from the
de¬nition of J 0 that J 0 b ‚ J 0 for all b ∈ B and that J 0 ‚ B. Therefore, (??) shows
that the map ρ : B ’ EndA (J 0 ) de¬ned by multiplication on the right is surjective.

Its kernel is J 00 = J, hence it induces an isomorphism B/J ’ EndA (J 0 ).

For every right ideal I ‚ A containing J, let
˜
I = ρ(I © B).
˜
(1.15) Proposition. The map I ’ I de¬nes a one-to-one correspondence between
right ideals of reduced dimension r in A which contain J and right ideals of reduced
dimension r ’ k in EndA (J 0 ). If A = EndD (V ) and J = HomD (V, U ) for some
subspace U ‚ V of dimension r/ ind A, then for I = HomD (V, W ) with W ⊃ U , we
have under the natural isomorphism EndA (J 0 ) = EndD (V /U ) of (??) that
˜
I = HomD (V /U, W/U ).
Proof : In view of (??), the second part implies the ¬rst, since the map W ’ W/U
de¬nes a one-to-one correspondence between subspaces of dimension r/ ind A in V
which contain U and subspaces of dimension (r ’ k)/ ind A in V /U .
Suppose that A = EndD (V ) and J = HomD (V, U ), hence J 0 = HomD (V /U, V )
and B = { f ∈ A | f (U ) ‚ U }. Every f ∈ B induces a linear map f ∈ EndD (V /U ),
and the homomorphism ρ : B ’ EndA (J 0 ) = EndD (V /U ) maps f to f since for
g ∈ J 0 we have
g ρ(f ) = g —¦ f = g —¦ f .
For I = HomD (V, W ) with W ⊃ U , it follows that
˜
I = { f | f ∈ I and f (U ) ‚ U } ‚ HomD (V /U, W/U ).
The converse inclusion is clear, since using bases of U , W and V it is easily seen
that every linear map h ∈ HomD (V /U, W/U ) is of the form h = f for some f ∈
HomD (V, W ) such that f (U ) ‚ U .
1.C. Severi-Brauer varieties. Let A be a central simple algebra of degree n
over a ¬eld F and let r be an integer, 1 ¤ r ¤ n. Consider the Grassmannian
Gr(rn, A) of rn-dimensional subspaces in A. The Pl¨cker embedding identi¬es
u
Gr(rn, A) with a closed subvariety of the projective space on the rn-th exterior
power of A (see Harris [?, Example 6.6, p. 64]):
rn
Gr(rn, A) ‚ P( A).
10 I. INVOLUTIONS AND HERMITIAN FORMS


The rn-dimensional subspace U ‚ A corresponding to a non-zero rn-vector u1 §
rn
· · · § urn ∈ A is
U = { x ∈ A | u1 § · · · § urn § x = 0 } = u1 F + · · · + urn F.
Among the rn-dimensional subspaces in A, the right ideals of reduced dimension r
are the subspaces which are preserved under multiplication on the right by the
elements of A. Such ideals may fail to exist: for instance, if A is a division algebra,
it does not contain any nontrivial ideal; on the other hand, if A Mn (F ), then it
contains right ideals of every reduced dimension r = 0, . . . , n. Since every central
simple F -algebra becomes isomorphic to a matrix algebra over some scalar extension
of F , this situation is best understood from an algebraic geometry viewpoint: it is
comparable to the case of varieties de¬ned over some base ¬eld F which have no
rational point over F but acquire points over suitable extensions of F .
To make this viewpoint precise, consider an arbitrary basis (ei )1¤i¤n2 of A.
rn
The rn-dimensional subspace represented by an rn-vector u1 § · · · § urn ∈ A
is a right ideal of reduced dimension r if and only if it is preserved under right
multiplication by e1 , . . . , en2 , i.e.,
for i = 1, . . . , n2 ,
u1 ei § · · · § urn ei ∈ u1 § · · · § urn F
or, equivalently,
u1 ei § · · · § urn ei § uj = 0 for i = 1, . . . , n2 and j = 1, . . . , rn.
This condition translates to a set of equations on the coordinates of the rn-vector
u1 § · · · § urn , hence the right ideals of reduced dimension r in A form a closed
subvariety of Gr(rn, A).
(1.16) De¬nition. The (generalized ) Severi-Brauer variety SBr (A) is the vari-
ety of right ideals of reduced dimension r in A. It is a closed subvariety of the
Grassmannian:
SBr (A) ‚ Gr(rn, A).
For r = 1, we write simply SB(A) = SB1 (A). This is the (usual) Severi-Brauer
variety of A, ¬rst de¬ned by F. Chˆtelet [?].
a
(1.17) Proposition. The Severi-Brauer variety SBr (A) has a rational point over
an extension K of F if and only if the index ind AK divides r. In particular, SB(A)
has a rational point over K if and only if K splits A.
Proof : From the de¬nition, it follows that SBr (A) has a rational point over K if
and only if AK contains a right ideal of reduced dimension r. Since the reduced
dimension of any ¬nitely generated right AK -module is a multiple of ind AK , it
follows that ind AK divides r if SBr (A) has a rational point over K. Conversely,
suppose r = m ind AK for some integer m and let AK Mt (D) for some division
algebra D and some integer t. The set of matrices in Mt (D) whose t ’ m last rows
are zero is a right ideal of reduced dimension r, hence SBr (A) has a rational point
over K.
The following theorem shows that Severi-Brauer varieties are twisted forms of
Grassmannians:
(1.18) Theorem. For A = EndF (V ), there is a natural isomorphism
SBr (A) Gr(r, V ).
§1. CENTRAL SIMPLE ALGEBRAS 11


In particular, for r = 1,
P(V ).
SB(A)
Proof : Let V — = HomF (V, F ) be the dual of V . Under the natural isomorphism
A = EndF (V ) V —F V — , multiplication is given by
(v — φ) · (w — ψ) = (v — ψ)φ(w).
By (??), the right ideals of reduced dimension r in A are of the form HomF (V, U ) =
U — V — where U is an r-dimensional subspace in V .
We will show that the correspondence U ” U — V — between r-dimensional
subspaces in V and right ideals of reduced dimension r in A induces an isomorphism
of varieties Gr(r, V ) SBr (A).
For any vector space W of dimension n, there is a morphism Gr(r, V ) ’
Gr(rn, V —W ) which maps an r-dimensional subspace U ‚ V to U —W ‚ V —W . In
the particular case where W = V — we thus get a morphism ¦ : Gr(r, V ) ’ SBr (A)
which maps U to U — V — .
In order to show that ¦ is an isomorphism, we consider the following a¬ne
covering of Gr(r, V ): for each subspace S ‚ V of dimension n ’ r, we denote by US
the set of complementary subspaces:
US = { U ‚ V | U • S = V }.
The set US is an a¬ne open subset of Gr(r, V ); more precisely, if U0 is a ¬xed
complementary subspace of S, there is an isomorphism:

HomF (U0 , S) ’ US

which maps f ∈ HomF (U0 , S) to U = { x + f (x) | x ∈ U0 } (see Harris [?, p. 65]).
Similarly, we may also consider US—V — ‚ Gr(rn, A). The image of the restriction
of ¦ to US is
{ U — V — ‚ V — V — | (U — V — ) • (S — V — ) = V — V — } = US—V — © SBr (A).
Moreover, there is a commutative diagram:
¦|U
’ ’S
US ’’’ US—V —
¦ ¦
¦ ¦

φ
HomF (U0 , S) ’ ’ ’ HomF (U0 — V — , S — V — )
’’
where φ(f ) = f — IdV — . Since φ is linear and injective, it is an isomorphism of
varieties between HomF (U0 , S) and its image. Therefore, the restriction of ¦ to US

is an isomorphism ¦|US : US ’ US—V — © SBr (A). Since the open sets US form a

covering of Gr(r, V ), it follows that ¦ is an isomorphism.

Although Severi-Brauer varieties are de¬ned in terms of right ideals, they can
also be used to derive information on left ideals. Indeed, if J is a left ideal in a
central simple algebra A, then the set
J op = { j op ∈ Aop | j ∈ J }
is a right ideal in the opposite algebra Aop . Therefore, the variety of left ideals
of reduced dimension r in A can be identi¬ed with SBr (Aop ). We combine this
observation with the annihilator construction (see §??) to get the following result:
12 I. INVOLUTIONS AND HERMITIAN FORMS


(1.19) Proposition. For any central simple algebra A of degree n, there is a
canonical isomorphism

± : SBr (A) ’ SBn’r (Aop )

which maps a right ideal I ‚ A of reduced dimension r to (I 0 )op .
Proof : In order to prove that ± is an isomorphism, we may extend scalars to a
splitting ¬eld of A. We may therefore assume that A = EndF (V ) for some n-
dimensional vector space V . Then Aop = EndF (V — ) under the identi¬cation f op =
f t for f ∈ EndF (V ). By (??), we may further identify
SBn’r (Aop ) = Gr(n ’ r, V — ).
SBr (A) = Gr(r, V ),
Under these identi¬cations, the map ± : Gr(r, V ) ’ Gr(n ’ r, V — ) carries every
r-dimensional subspace U ‚ V to U 0 = { • ∈ V — | •(U ) = {0} }.
To show that ± is an isomorphism of varieties, we restrict it to the a¬ne open
sets US de¬ned in the proof of Theorem (??): let S be an (n ’ r)-dimensional
subspace in V and
US = { U ‚ V | U • S = V } ‚ Gr(r, V ).
Let U0 ‚ V be such that U0 • S = V , so that US HomF (U0 , S). We also have
0 0 — 0 0
U0 • S = V , US 0 HomF (U0 , S ), and the map ± restricts to ±|US : US ’ US 0 .
It therefore induces a map ± which makes the following diagram commute:
±|U
’ ’S
US ’’’ US 0
¦ ¦
¦ ¦

±
HomF (U0 , S) ’ ’ ’ HomF (U0 , S 0 ).
0
’’
We now proceed to show that ± is an isomorphism of (a¬ne) varieties.
0
Every linear form in U0 restricts to a linear form on S, and since V = U0 •S we
0
S — . Similarly, S 0 U0 , so HomF (U0 , S 0 )
— 0
thus get a natural isomorphism U0
HomF (S — , U0 ). Under this identi¬cation, a direct calculation shows that the map ±


carries f ∈ HomF (U0 , S) to ’f t ∈ HomF (S — , U0 ) = HomF (U0 , S 0 ). It is therefore
— 0

an isomorphism of varieties. Since the open sets US cover Gr(r, V ), it follows that
± is an isomorphism.
If V is a vector space of dimension n over a ¬eld F and U ‚ V is a subspace of
dimension k, then for r = k, . . . , n the Grassmannian Gr(r ’ k, V /U ) embeds into
Gr(r, V ) by mapping every subspace W ‚ V /U to the subspace W ⊃ U such that
W/U = W . The image of Gr(r ’ k, V /U ) in Gr(r, V ) is the sub-Grassmannian of
r-dimensional subspaces in V which contain U (see Harris [?, p. 66]). There is an
analogous notion for Severi-Brauer varieties:
(1.20) Proposition. Let A be a central simple F -algebra and let J ‚ A be a right
ideal of reduced dimension k (i.e., a rational point of SBk (A)). The one-to-one
correspondence between right ideals of reduced dimension r in A which contain J
and right ideals of reduced dimension r ’ k in EndA (J 0 ) set up in (??) de¬nes an
embedding:
SBr’k EndA (J 0 ) ’ SBr (A).
The image of SBr’k EndA (J 0 ) in SBr (A) is the variety of right ideals of reduced
dimension r in A which contain J.
§2. INVOLUTIONS 13


Proof : It su¬ces to prove the proposition over a scalar extension. We may therefore
assume that A is split, i.e., that A = EndF (V ). Let then J = HomF (V, U ) for some
subspace U ‚ V of dimension k. We have J 0 = HomF (V /U, V ) and (??) shows
that there is a canonical isomorphism EndA (J 0 ) = EndF (V /U ). Theorem (??)
then yields canonical isomorphisms SBr (A) = Gr(r, V ) and SBr’k EndA (J 0 ) =
Gr(r ’ k, V /U ). Moreover, from (??) it follows that the map SBr’k EndA (J 0 ) ’
SBr (A) corresponds under these identi¬cations to the embedding Gr(r’k, V /U ) ’
Gr(r, V ) described above.

§2. Involutions
An involution on a central simple algebra A over a ¬eld F is a map σ : A ’ A
subject to the following conditions:
(a) σ(x + y) = σ(x) + σ(y) for x, y ∈ A.
(b) σ(xy) = σ(y)σ(x) for x, y ∈ A.
(c) σ 2 (x) = x for x ∈ A.
Note that the map σ is not required to be F -linear. However, it is easily checked
that the center F (= F · 1) is preserved under σ. The restriction of σ to F is
therefore an automorphism which is either the identity or of order 2. Involutions
which leave the center elementwise invariant are called involutions of the ¬rst kind.
Involutions whose restriction to the center is an automorphism of order 2 are called
involutions of the second kind.
This section presents the basic de¬nitions and properties of central simple alge-
bras with involution. Involutions of the ¬rst kind are considered ¬rst. As observed
in the introduction to this chapter, they are adjoint to nonsingular symmetric or
skew-symmetric bilinear forms in the split case. Involutions of the ¬rst kind are
correspondingly divided into two types: the orthogonal and the symplectic types.
We show in (??) how to characterize these types by properties of the symmetric ele-
ments. Involutions of the second kind, also called unitary, are treated next. Various
examples are provided in (??)“(??).

2.A. Involutions of the ¬rst kind. Throughout this subsection, A denotes
a central simple algebra over a ¬eld F of arbitrary characteristic, and σ is an
involution of the ¬rst kind on A. Our basic object of study is the couple (A, σ); from
this point of view, a homomorphism of algebras with involution f : (A, σ) ’ (A , σ )
is an F -algebra homomorphism f : A ’ A such that σ —¦ f = f —¦ σ. Our main tool
is the extension of scalars: if L is any ¬eld containing F , the involution σ extends
to an involution of the ¬rst kind σL = σ — IdL on AL = A —F L. In particular, if
L is a splitting ¬eld of A, we may identify AL = EndL (V ) for some vector space V
over L of dimension n = deg A. As observed in the introduction to this chapter,
the involution σL is then the adjoint involution σb with respect to some nonsingular
symmetric or skew-symmetric bilinear form b on V . By means of a basis of V , we
may further identify V with Ln , hence also A with Mn (L). For any matrix m, let
mt denote the transpose of m. If g ∈ GLn (L) denotes the Gram matrix of b with
respect to the chosen basis, then
b(x, y) = xt · g · y
where x, y are considered as column matrices and g t = g if b is symmetric, g t = ’g
if b is skew-symmetric. The involution σL is then identi¬ed with the involution σg
14 I. INVOLUTIONS AND HERMITIAN FORMS


de¬ned by
σg (m) = g ’1 · mt · g for m ∈ Mn (L).
For future reference, we summarize our conclusions:
(2.1) Proposition. Let (A, σ) be a central simple F -algebra of degree n with in-
volution of the ¬rst kind and let L be a splitting ¬eld of A. Let V be an L-vector
space of dimension n. There is a nonsingular symmetric or skew-symmetric bilin-
ear form b on V and an invertible matrix g ∈ GLn (L) such that g t = g if b is
symmetric and g t = ’g if b is skew-symmetric, and
(AL , σL ) EndL (V ), σb Mn (L), σg .
As a ¬rst application, we have the following result:
(2.2) Corollary. For all a ∈ A, the elements a and σ(a) have the same reduced
characteristic polynomial. In particular, TrdA σ(a) = TrdA (a) and NrdA σ(a) =
NrdA (a).
Proof : For all m ∈ Mn (L), g ∈ GLn (L), the matrix g ’1 · mt · g has the same
characteristic polynomial as m.
Of course, in (??), neither the form b nor the matrix g (nor even the splitting
¬eld L) is determined uniquely by the involution σ; some of their properties re¬‚ect
properties of σ, however. As a ¬rst example, we show in (??) below that two types
of involutions of the ¬rst kind can be distinguished which correspond to symmetric
and to alternating1 forms. This distinction is made on the basis of properties of
symmetric elements which we de¬ne next.
In a central simple F -algebra A with involution of the ¬rst kind σ, the sets of
symmetric, skew-symmetric, symmetrized and alternating elements in A are de¬ned
as follows:
Sym(A, σ) = { a ∈ A | σ(a) = a },
Skew(A, σ) = { a ∈ A | σ(a) = ’a },
Symd(A, σ) = { a + σ(a) | a ∈ A },
Alt(A, σ) = { a ’ σ(a) | a ∈ A }.
If char F = 2, then Symd(A, σ) = Sym(A, σ), Alt(A, σ) = Skew(A, σ) and A =
1
Sym(A, σ) • Skew(A, σ) since every element a ∈ A decomposes as a = 2 a +
1
σ(a) + 2 a ’ σ(a) . If char F = 2, then Symd(A, σ) = Alt(A, σ) ‚ Skew(A, σ) =
Sym(A, σ), and (??) below shows that this inclusion is strict.
(2.3) Lemma. Let n = deg A; then dim Sym(A, σ) + dim Alt(A, σ) = n2 . More-
over, Alt(A, σ) is the orthogonal space of Sym(A, σ) for the bilinear form T A on A
induced by the reduced trace:
Alt(A, σ) = { a ∈ A | TrdA (as) = 0 for s ∈ Sym(A, σ) }.
Similarly, dim Skew(A, σ)+dim Symd(A, σ) = n2 , and Symd(A, σ) is the orthogonal
space of Skew(A, σ) for the bilinear form TA .

1 If
char F = 2, every skew-symmetric bilinear form is alternating; if char F = 2, the notions
of symmetric and skew-symmetric bilinear forms coincide, but the notion of alternating form is
more restrictive.
§2. INVOLUTIONS 15


Proof : The ¬rst relation comes from the fact that Alt(A, σ) is the image of the
linear endomorphism Id ’ σ of A, whose kernel is Sym(A, σ). If a ∈ Alt(A, σ), then
a = x ’ σ(x) for some x ∈ A, hence for s ∈ Sym(A, σ),
TrdA (as) = TrdA (xs) ’ TrdA σ(x)s = TrdA (xs) ’ TrdA σ(sx) .
Corollary (??) shows that the right side vanishes, hence the inclusion
Alt(A, σ) ‚ { a ∈ A | TrdA (as) = 0 for s ∈ Sym(A, σ) }.
Dimension count shows that this inclusion is an equality since TA is nonsingular
(see (??)).
The statements involving Symd(A, σ) readily follow, either by mimicking the
arguments above, or by using the fact that in characteristic di¬erent from 2,
Symd(A, σ) = Sym(A, σ) and Alt(A, σ) = Skew(A, σ), and, in characteristic 2,
Symd(A, σ) = Alt(A, σ) and Skew(A, σ) = Sym(A, σ).
We next determine the dimensions of Sym(A, σ) and Skew(A, σ) (and therefore
also of Symd(A, σ) and Alt(A, σ)).
Consider ¬rst the split case, assuming that A = EndF (V ) for some vector
space V over F . As observed in the introduction to this chapter, every involution
of the ¬rst kind σ on A is the adjoint involution with respect to a nonsingular
symmetric or skew-symmetric bilinear form b on V which is uniquely determined
by σ up to a factor in F — .
(2.4) Lemma. Let σ = σb be the adjoint involution on A = EndF (V ) with respect
to the nonsingular symmetric or skew-symmetric bilinear form b on V , and let
n = dimF V .
(1) If b is symmetric, then dimF Sym(A, σ) = n(n + 1)/2.
(2) If b is skew-symmetric, then dimF Skew(A, σ) = n(n + 1)/2.
(3) If char F = 2, then b is alternating if and only if tr(f ) = 0 for all f ∈
Sym(A, σ). In this case, n is necessarily even.
Proof : As in (??), we use a basis of V to identify (A, σ) with Mn (F ), σg , where
g ∈ GLn (F ) satis¬es g t = g if b is symmetric and g t = ’g if b is skew-symmetric.
For m ∈ Mn (F ), the relation gm = (gm)t is equivalent to σg (m) = m if g t = g and
to σg (m) = ’m if g t = ’g. Therefore,
Sym(A, σ) if b is symmetric,
g ’1 · Sym Mn (F ), t =
Skew(A, σ) if b is skew-symmetric.
The ¬rst two parts then follow from the fact that the space Sym Mn (F ), t of n — n
symmetric matrices (with respect to the transpose) has dimension n(n + 1)/2.
Suppose now that char F = 2. If b is not alternating, then b(v, v) = 0 for some
v ∈ V . Consider the map f : V ’ V de¬ned by
f (x) = vb(v, x)b(v, v)’1 for x ∈ V .
Since b is symmetric we have
b f (x), y = b(v, y)b(v, x)b(v, v)’1 = b x, f (y) for x, y ∈ V ,
hence σ(f ) = f . Since f is an idempotent in A, its trace is the dimension of its
image:
tr(f ) = dim im f = 1.
16 I. INVOLUTIONS AND HERMITIAN FORMS


Therefore, if the trace of every symmetric element in A is zero, then b is alternating.
Conversely, suppose b is alternating; it follows that n is even, since every al-
ternating form on a vector space of odd dimension is singular. Let (ei )1¤i¤n be
a symplectic basis of V , in the sense that b(e2i’1 , e2i ) = 1, b(e2i , e2i+1 ) = 0 and
b(ei , ej ) = 0 if |i ’ j| > 1. Let f ∈ Sym(A, σ); for j = 1, . . . , n let
n
f (ej ) = ei aij for some aij ∈ F ,
i=1
n
so that tr(f ) = aii . For i = 1, . . . , n/2 we have
i=1

b f (e2i’1 ), e2i = a2i’1,2i’1 and b e2i’1 , f (e2i ) = a2i,2i ;
since σ(f ) = f , it follows that a2i’1,2i’1 = a2i,2i for i = 1, . . . , n/2, hence
n/2
tr(f ) = 2 a2i,2i = 0.
i=1



We now return to the general case where A is an arbitrary central simple F -
algebra and σ is an involution of the ¬rst kind on A. Let n = deg A and let L be a
splitting ¬eld of A. Consider an isomorphism as in (??):
(AL , σL ) EndL (V ), σb .
This isomorphism carries Sym(AL , σL ) = Sym(A, σ)—F L to Sym EndL (V ), σb and
Skew(AL , σL ) to Skew EndL (V ), σb . Since extension of scalars does not change
dimensions, (??) shows
(a) dimF Sym(A, σ) = n(n + 1)/2 if b is symmetric;
(b) dimF Skew(A, σ) = n(n + 1)/2 if b is skew-symmetric.
These two cases coincide if char F = 2 but are mutually exclusive if char F = 2;
indeed, in this case A = Sym(A, σ)•Skew(A, σ), hence the dimensions of Sym(A, σ)
and Skew(A, σ) add up to n2 .
Since the reduced trace of A corresponds to the trace of endomorphisms under
the isomorphism AL EndL (V ), we have TrdA (s) = 0 for all s ∈ Sym(A, σ) if
and only if tr(f ) = 0 for all f ∈ Sym EndL (V ), σb , and Lemma (??) shows that,
when char F = 2, this condition holds if and only if b is alternating. Therefore, in
arbitrary characteristic, the property of b being symmetric or skew-symmetric or
alternating depends only on the involution and not on the choice of L nor of b. We
may thus set the following de¬nition:
(2.5) De¬nition. An involution σ of the ¬rst kind is said to be of symplectic type
(or simply symplectic) if for any splitting ¬eld L and any isomorphism (AL , σL )
EndL (V ), σb , the bilinear form b is alternating; otherwise it is called of orthogonal
type (or simply orthogonal ). In the latter case, for any splitting ¬eld L and any
isomorphism (AL , σL ) EndL (V ), σb , the bilinear form b is symmetric (and
nonalternating).
The preceding discussion yields an alternate characterization of orthogonal and
symplectic involutions:

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