ńņš. 1 |

Max-Albert Knus

Alexander Merkurjev

Markus Rost

Jean-Pierre Tignol

Author address:

ĀØ

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/

ĀØ

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/

DĀ“partement de mathĀ“matique, UniversitĀ“ 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

PrĀ“face

e .............................. vii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter I. Involutions and Hermitian Forms . . . . . . . . . . . . . 1

Ā§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

Ā§2. Involutions . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.A. Involutions of the ļ¬rst kind . . . . . . . . . . . . . . . 13

2.B. Involutions of the second kind . . . . . . . . . . . . . . 20

2.C. Examples . . . . . . . . . . . . . . . . . . . . . . . 23

2.D. Lie and Jordan structures . . . . . . . . . . . . . . . . 27

Ā§3. Existence of Involutions . . . . . . . . . . . . . . . . . . . 31

3.A. Existence of involutions of the ļ¬rst kind . . . . . . . . . . 32

3.B. Existence of involutions of the second kind . . . . . . . . 36

Ā§4. Hermitian Forms . . . . . . . . . . . . . . . . . . . . . . 41

4.A. Adjoint involutions . . . . . . . . . . . . . . . . . . . 42

4.B. Extension of involutions and transfer . . . . . . . . . . . 45

Ā§5. Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . 53

5.A. Standard identiļ¬cations . . . . . . . . . . . . . . . . . 53

5.B. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 56

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter II. Invariants of Involutions . . . . . . . . . . . . . . . . . 71

Ā§6. The Index . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.A. Isotropic ideals . . . . . . . . . . . . . . . . . . . . . 72

6.B. Hyperbolic involutions . . . . . . . . . . . . . . . . . 74

6.C. Odd-degree extensions . . . . . . . . . . . . . . . . . 79

Ā§7. The Discriminant . . . . . . . . . . . . . . . . . . . . . . 80

7.A. The discriminant of orthogonal involutions . . . . . . . . 80

7.B. The discriminant of quadratic pairs . . . . . . . . . . . . 83

Ā§8. The Cliļ¬ord Algebra . . . . . . . . . . . . . . . . . . . . . 87

8.A. The split case . . . . . . . . . . . . . . . . . . . . . 87

8.B. Deļ¬nition of the Cliļ¬ord algebra . . . . . . . . . . . . . 91

8.C. Lie algebra structures . . . . . . . . . . . . . . . . . . 95

iii

iv CONTENTS

8.D. The center of the Cliļ¬ord algebra . . . . . . . . . . . . 99

8.E. The Cliļ¬ord algebra of a hyperbolic quadratic pair . . . . . 106

Ā§9. The Cliļ¬ord Bimodule . . . . . . . . . . . . . . . . . . . . 107

9.A. The split case . . . . . . . . . . . . . . . . . . . . . 107

9.B. Deļ¬nition of the Cliļ¬ord bimodule . . . . . . . . . . . . 108

9.C. The fundamental relations . . . . . . . . . . . . . . . . 113

Ā§10. The Discriminant Algebra . . . . . . . . . . . . . . . . . . 114

10.A. The Ī»-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 Ī»-powers . . . . . . . . . . . . . 123

10.E. Deļ¬nition of the discriminant algebra . . . . . . . . . . . 126

10.F. The Brauer class of the discriminant algebra . . . . . . . . 130

Ā§11. Trace Form Invariants . . . . . . . . . . . . . . . . . . . . 132

11.A. Involutions of the ļ¬rst kind . . . . . . . . . . . . . . . 133

11.B. Involutions of the second kind . . . . . . . . . . . . . . 138

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Chapter III. Similitudes . . . . . . . . . . . . . . . . . . . . . . 153

Ā§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

Ā§13. Quadratic Pairs . . . . . . . . . . . . . . . . . . . . . . . 172

13.A. Relation with the Cliļ¬ord structures . . . . . . . . . . . 172

13.B. Cliļ¬ord groups . . . . . . . . . . . . . . . . . . . . . 176

13.C. Multipliers of similitudes . . . . . . . . . . . . . . . . 190

Ā§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

Ā§15. Exceptional Isomorphisms . . . . . . . . . . . . . . . . . . 205

15.A. B1 ā” C1 . . . . . . . . . . . . . . . . . . . . . . . . 207

15.B. A2 ā” D2 . . . . . . . . . . . . . . . . . . . . . . . . 210

1

15.C. B2 ā” C2 . . . . . . . . . . . . . . . . . . . . . . . . 216

15.D. A3 ā” D3 . . . . . . . . . . . . . . . . . . . . . . . . 220

Ā§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

Ā§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

Ā“

Ā§18. Etale and Galois Algebras . . . . . . . . . . . . . . . . . . 279

Ā“

18.A. Etale algebras . . . . . . . . . . . . . . . . . . . . . 280

18.B. Galois algebras . . . . . . . . . . . . . . . . . . . . . 287

18.C. Cubic Ā“tale algebras . . . . . . . . . . .

e . . . . . . . 296

Ā§19. Central Simple Algebras of Degree Three . . . . . . . . . . . . 302

19.A. Cyclic algebras . . . . . . . . . . . . . . . . . . . . . 302

19.B. Classiļ¬cation of involutions of the second kind . . . . . . . 304

Ā“

19.C. Etale subalgebras . . . . . . . . . . . . . . . . . . . . 307

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Chapter VI. Algebraic Groups . . . . . . . . . . . . . . . . . . . 323

Ā§20. Hopf Algebras and Group Schemes . . . . . . . . . . . . . . 324

20.A. Group schemes . . . . . . . . . . . . . . . . . . . . . 325

Ā§21. The Lie Algebra and Smoothness . . . . . . . . . . . . . . . 334

21.A. The Lie algebra of a group scheme . . . . . . . . . . . . 334

Ā§22. Factor Groups . . . . . . . . . . . . . . . . . . . . . . . 339

22.A. Group scheme homomorphisms . . . . . . . . . . . . . . 339

Ā§23. Automorphism Groups of Algebras . . . . . . . . . . . . . . 344

23.A. Involutions . . . . . . . . . . . . . . . . . . . . . . 345

23.B. Quadratic pairs . . . . . . . . . . . . . . . . . . . . 350

Ā§24. Root Systems . . . . . . . . . . . . . . . . . . . . . . . . 352

Ā§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

Ā§26. Semisimple Groups over an Arbitrary Field . . . . . . . . . . . 359

26.A. Basic classiļ¬cation results . . . . . . . . . . . . . . . . 362

26.B. Algebraic groups of small dimension . . . . . . . . . . . 372

Ā§27. Tits Algebras of Semisimple Groups . . . . . . . . . . . . . . 373

27.A. Deļ¬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

Ā§28. Cohomology of Proļ¬nite Groups . . . . . . . . . . . . . . . . 381

28.A. Cohomology sets . . . . . . . . . . . . . . . . . . . . 381

28.B. Cohomology sequences . . . . . . . . . . . . . . . . . 383

28.C. Twisting . . . . . . . . . . . . . . . . . . . . . . . 385

28.D. Torsors . . . . . . . . . . . . . . . . . . . . . . . . 386

Ā§29. Galois Cohomology of Algebraic Groups . . . . . . . . . . . . 389

29.A. Hilbertā™s Theorem 90 and Shapiroā™s lemma . . . . . . . . 390

29.B. Classiļ¬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

Ā§30. Galois Cohomology of Roots of Unity . . . . . . . . . . . . . 411

30.A. Cyclic algebras . . . . . . . . . . . . . . . . . . . . . 412

30.B. Twisted coeļ¬cients . . . . . . . . . . . . . . . . . . . 414

30.C. Cohomological invariants of algebras of degree three . . . . 418

Ā§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

Ā§32. Nonassociative Algebras . . . . . . . . . . . . . . . . . . . 447

Ā§33. Composition Algebras . . . . . . . . . . . . . . . . . . . . 451

33.A. Multiplicative quadratic forms . . . . . . . . . . . . . . 451

33.B. Unital composition algebras . . . . . . . . . . . . . . . 452

PrĀ“face

e

Quatre des meilleurs algĀ“bristes dā™aujourdā™hui (jā™aimerais dire, comme jadis,

e

gĀ“om`tres , au sens noble, mais hĀ“las dĀ“suet du terme) nous donnent ce beau

ee e e

Livre des Involutions, quā™ils me demandent de prĀ“facer.

e

Quel est le propos de lā™ouvrage et a quels lecteurs sā™adresse-t-il? Bien sĖr il y

` u

est souvent question dā™involutions, mais celles-ci sont loin dā™Ėtre omniprĀ“sentes et le

e e

titre est plus lā™expression dā™un Ā“tat dā™Ėme que lā™aļ¬rmation dā™un th`me central. En

e a e

fait, les questions envisagĀ“es sont multiples, relevant toutes de domaines importants

e

des mathĀ“matiques contemporaines ; sans vouloir Ėtre exhaustif (ceci nā™est pas une

e e

introduction), on peut citer :

- les formes quadratiques et les alg`bres de Cliļ¬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 algĀ“briques, principalement semi-simples,

e

- la cohomologie galoisienne.

Pour ce qui est du public concernĀ“, la lecture ou la consultation du livre sera

e

proļ¬table a un large Ā“ventail de mathĀ“maticiens. Le non-initiĀ“ y trouvera une

` e e e

introduction claire aux concepts fondamentaux des domaines en question ; exposĀ“s e

le plus souvent en fonction dā™applications concr`tes, ces notions de base sont prĀ“-

e e

sentĀ“es de faĀøon vivante et dĀ“pouillĀ“e, sans gĀ“nĀ“ralitĀ“s gratuites (les auteurs ne sont

e c e e ee e

pas adeptes de grandes thĀ“ories abstraites). Le lecteur dĀ“j` informĀ“, ou croyant

e ea e

lā™Ėtre, pourra rĀ“apprendre (ou dĀ“couvrir) quelques beaux thĀ“or`mes jadis bien

e e e ee

connus mais un peu oubliĀ“s dans la littĀ“rature rĀ“cente, ou au contraire, voir

e e e

des rĀ“sultats qui lui sont en principe familiers exposĀ“s sous un jour nouveau et

e e

Ā“clairant (je pense par exemple a lā™introduction des alg`bres trialitaires au dernier

e ` e

chapitre). Enļ¬n, les spĀ“cialistes et les chercheurs auront a leur disposition une

e `

rĀ“fĀ“rence prĀ“cieuse, parfois unique, pour des dĀ“veloppements rĀ“cents, souvents dĖs

ee e e e u

aux auteurs eux-mĖmes, et dont certains sont exposĀ“s ici pour la premi`re fois

e e e

(cā™est par exemple le cas pour plusieurs rĀ“sultats sur les invariants cohomologiques,

e

donnĀ“s a la ļ¬n du chapitre 7).

e`

MalgrĀ“ la grande variĀ“tĀ“ des th`mes considĀ“rĀ“s et les individualitĀ“s tr`s mar-

e ee e ee e e

quĀ“es des quatre auteurs, ce Livre des Involutions a une unitĀ“ remarquable. Le

e e

ciment un peu fragile des involutions nā™est certes pas seul a lā™expliquer. Il y a

`

aussi, bien sĖr, les interconnections multiples entre les sujets traitĀ“s ; mais plus

u e

dĀ“terminante encore est lā™importance primordiale accordĀ“e a des structures fortes,

e e`

se prĖtant par exemple a des thĀ“or`mes de classiļ¬cation substantiels. Ce nā™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 MajestĀ“ E8 e

vii

Ā“

viii PREFACE

fasse ici ļ¬gure de parent pauvre), les alg`bres de composition, . . . , reĀøoivent autant

e c

dā™attention.

On lā™a compris, ce Livre est tout a la fois un livre de lecture passionnant et

`

un ouvrage de rĀ“fĀ“rence dā™une extrĖme richesse. Je suis reconnaissant aux auteurs

ee e

de lā™honneur quā™ils mā™ont fait en me demandant de le prĀ“facer, et plus encore de

e

mā™avoir permis de le dĀ“couvrir et dā™apprendre a mā™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 suļ¬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 classiļ¬cation

over Q. This is the historical origin of our subject, however our motivation has a

diļ¬erent source. The basic objects are still central simple algebras, i.e., āformsā

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 reļ¬‚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 Cliļ¬ord algebra (for orthogonal involutions) or the

Ī»-powers and the discriminant algebra (for unitary involutions). These algebras are

exactly the āTits algebrasā, deļ¬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 āalgebraā counterpart is, so

far as we know, systematically approached here for the ļ¬rst time.

In the ļ¬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 ļ¬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

ļ¬elds. Quadratic pairs, introduced at the end of the chapter, give a corresponding

interpretation for quadratic forms in characteristic 2.

In Chapter ?? we deļ¬ne several invariants of involutions; the index is deļ¬ned for

every type of involution. For quadratic pairs additional invariants are the discrim-

inant, the (even) Cliļ¬ord algebra and the Cliļ¬ord module; for unitary involutions

we introduce the discriminant algebra. The deļ¬nition of the discriminant algebra

ix

x INTRODUCTION

is prepared for by the construction of the Ī»-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

deļ¬nition is based on scalar extension: after specifying the deļ¬nitions ārationallyā

(i.e., over an arbitrary base ļ¬eld), the main properties are proven by working over

a splitting ļ¬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 deļ¬ned over the base ļ¬eld. A main source of inspiration for Chapters ?? and ??

is the paper [?] of Tits on āFormes quadratiques, groupes orthogonaux et alg`bres e

de Cliļ¬ord.ā

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 deļ¬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 ļ¬rst part of Chapter ?? by prop-

erties of the Cliļ¬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 classiļ¬cation of these algebras. In preparation for this, we recall

general results on Ā“tale and Galois algebras.

e

The aim of Chapter ?? is to give the classiļ¬cation of semisimple algebraic groups

over arbitrary ļ¬elds. We use the functorial approach to algebraic groups, although

we quote without proof some basic results on algebraic groups over algebraically

closed ļ¬elds. In the central section we describe in detail Weilā™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 deļ¬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 āsymmetricā compositions. In dimension 8 these

are of two types, corresponding to algebraic groups of type A2 or type G2 . Triality

INTRODUCTION xi

is deļ¬ned through the Cliļ¬ord algebra of symmetric 8-dimensional compositions.

As a step towards exceptional simple Jordan algebras, we introduce twisted compo-

sitions, which are deļ¬ned over cubic Ā“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 diļ¬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, deļ¬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 deļ¬ning cohomological invariants for exceptional simple

Jordan algebras.

The last chapter deals with trialitarian actions on simple adjoint groups of

type D4 . To complete Weilā™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 Ā“tale algebra. The trialitarian condition relates the algebra

e

to its Cliļ¬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) eļ¬ort of four very

diļ¬erent people. We are aware that the result is still quite heterogeneous; however,

we ļ¬‚atter ourselves that the diļ¬erences in style may be viewed as a positive feature.

Our work started out as an attempt to understand Titsā™ deļ¬nition of the Cliļ¬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 inļ¬‚uence, notably

through his threat ā” early in the inception of our project ā” 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 prĀ“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 ļ¬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 Dejaiļ¬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 friendsā™ sagacity. We shall gratefully welcome any comment

or correction.

Jean-Pierre Tignol had the privilege to give a series of lectures on āCentral

simple algebras, involutions and quadratic formsā 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

ļ¬rst chapters of this book.

Our project immensely beneļ¬ted by reciprocal visits among the authors. We

should like to mention with particular gratitude Merkurjevā™s stay in Louvain-la-

Neuve in 1993, with support from the Fonds de DĀ“veloppement Scientiļ¬que and the

e

Institut de MathĀ“matique Pure et AppliquĀ“e of the UniversitĀ“ catholique de Lou-

e e e

vain, and Tignolā™s stay in ZĀØrich for the winter semester of 1995ā“96, with support

u

from the EidgenĀØssische 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 1995ā“96, and Jean-Pierre Tignol is

grateful to the National Fund for Scientiļ¬c Research of Belgium for partial support.

The four authors enthusiastically thank Herbert Rost (Markusā™ 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 ā X under the composite map X ā’ Y ā’ Z we set either g ā—¦ f (x) or xf g and

ā’ā’

the composite is thus either g ā—¦ 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 deļ¬nes a natural homomorphism:

R ā’ EndEndR (M ) (M ).

Note therefore that if S ā‚ 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 ) ā’

EndR (R M ) is an anti-isomorphism.

The characteristic polynomial and its coeļ¬cients. Let F denote an ar-

bitrary ļ¬eld. The characteristic polynomial of a matrix m ā Mn (F ) (or an endo-

morphism m of an n-dimensional F -vector space) is denoted

Pm (X) = X n ā’ s1 (m)X nā’1 + s2 (m)X nā’2 ā’ Ā· Ā· Ā· + (ā’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 coeļ¬cients of the characteristic polyno-

mial:

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

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

Proof : It suļ¬ces to prove these relations for generic matrices m = (xij )1ā¤i,jā¤n ,

m = (xij )1ā¤i,jā¤n whose entries are indeterminates over Z; the general case follows

by specialization. If Ī»1 , . . . , Ī»n are the eigenvalues of the generic matrix m (in

xiii

xiv CONVENTIONS AND NOTATIONS

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

1ā¤iā¤n

s2 (m) = 1ā¤i<jā¤n Ī»i Ī»j , hence

s1 (m)2 ā’ 2s2 (m) = Ī»2 = s1 (m2 ),

i

1ā¤iā¤n

proving the ļ¬rst relation. The second relation follows by linearization, since 2 is

not a zero-divisor in Z[xij , xij | 1 ā¤ i, j ā¤ n].

If L is an associative and commutative F -algebra of dimension n and ā 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 ā’ s1 ( )X nā’1 + s2 ( )X nā’2 ā’ Ā· Ā· Ā· + (ā’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 deļ¬ne a generic polynomial for central

simple algebras, called the reduced characteristic polynomial : see (??). Generaliza-

tions to certain nonassociative algebras are given in Ā§ ??.

Bilinear forms. A bilinear form b : V Ć— V ā’ F on a ļ¬nite dimensional vector

space V over an arbitrary ļ¬eld F is called symmetric if b(x, y) = b(y, x) for all

x, y ā V , skew-symmetric if b(x, y) = ā’b(y, x) for all x, y ā V and alternating

if b(x, x) = 0 for all x ā 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 (ļ¬nite dimensional) vector

space V , the induced map

Ė V ā’ V ā— = HomF (V, F )

b:

is deļ¬ned by Ė b(x)(y) = b(x, y) for x, y ā V . The bilinear form b is nonsingular (or

regular , or nondegenerate) if Ė is bijective. (It suļ¬ces to require that Ė be injective,

b b

i.e., that the only vector x ā V such that b(x, y) = 0 for all y ā V is x = 0, since

we are dealing with ļ¬nite dimensional vector spaces over ļ¬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.

1ā¤i,jā¤n

In that case, the square class of this determinant is called the determinant of b :

Ā· F Ć—2 ā F Ć— /F Ć—2 .

det b = det b(ei , ej ) 1ā¤i,jā¤n

The discriminant of b is the signed determinant:

disc b = (ā’1)n(nā’1)/2 det b ā F Ć— /F Ć—2 where n = dim V .

For Ī±1 , . . . , Ī±n ā F , the bilinear form Ī±1 , . . . , Ī±n on F n is deļ¬ned by

Ī±1 , . . . , Ī±n (x1 , . . . , xn ), (y1 , . . . , yn ) = Ī±1 x1 y1 + Ā· Ā· Ā· + Ī±n xn yn .

We also deļ¬ne the n-fold Pļ¬ster bilinear form Ī±1 , . . . , Ī±n by

Ī±1 , . . . , Ī±n = 1, ā’Ī±1 ā— Ā· Ā· Ā· ā— 1, ā’Ī±n .

CONVENTIONS AND NOTATIONS xv

If b : V Ć— V ā’ F is a symmetric bilinear form, we denote by qb : V ā’ F the

associated quadratic map, deļ¬ned by

qb (x) = b(x, x) for x ā V .

Quadratic forms. If q : V ā’ F is a quadratic map on a ļ¬nite dimensional

vector space over an arbitrary ļ¬eld F , the associated symmetric bilinear form b q is

called the polar of q; it is deļ¬ned by

bq (x, y) = q(x + y) ā’ q(x) ā’ q(y) for x, y ā V ,

hence bq (x, x) = 2q(x) for all x ā 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 ā„ = { x ā V | bq (x, y) = 0 for y ā V }. The quadratic map q is called

nonsingular (or regular , or nondegenerate) if either V ā„ = {0} or dim V ā„ = 1 and

q(V ā„ ) = {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 x2iā’1 x2i (if n is even) or to x0 +

(nā’1)/2

x2iā’1 x2i (if n is odd).

i=1

The determinant and the discriminant of a nonsingular quadratic form q of

dimension n over a ļ¬eld F are deļ¬ned as follows: let M be a matrix representing q

in the sense that

q(X) = X Ā· M Ā· 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 ā’ 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 ā’ N t ; therefore, M + M t is uniquely

determined by q.

If char F = 2 we set

+ M t ) Ā· F Ć—2 ā F Ć— /F Ć—2

1

det q = det 2 (M

and

disc q = (ā’1)n(nā’1)/2 det q ā F Ć— /F Ć—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) Ā· F Ć—2 ā F Ć— /F Ć—2

(0.4)

where y ā F n is a nonzero vector such that (M + M t ) Ā· y = 0. Such a vector y is

uniquely determined up to a scalar factor, since M + M t has rank n ā’ 1, hence the

deļ¬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 )ā’1 M + ā„˜(F ) ā F/ā„˜(F )

and

m(mā’1)

disc q = + det q ā F/ā„˜(F )

2

xvi CONVENTIONS AND NOTATIONS

where m = n/2 and ā„˜(F ) = { x + x2 | x ā F }. (More generally, for ļ¬elds of

characteristic p = 0, ā„˜ is deļ¬ned as ā„˜(x) = x + xp , x ā F .) The following lemma

shows that the deļ¬nition of det q does not depend on the choice of M :

(0.5) Lemma. Suppose char F = 2. Let M, N ā Mn (F ) and W = M + M t . If W

is invertible, then

2

s2 W ā’1 (M + N + N t ) = s2 (W ā’1 M ) + s1 (W ā’1 N ) + s1 (W ā’1 N ) .

Proof : The second relation in (??) yields

s2 W ā’1 M + W ā’1 (N + N t ) =

s2 (W ā’1 M ) + s2 W ā’1 (N + N t ) + s1 (W ā’1 M )s1 W ā’1 (N + N t )

+ s1 W ā’1 M W ā’1 (N + N t ) .

In order to prove the lemma, we show below:

2

s2 W ā’1 (N + N t ) = s1 (W ā’1 N )

(0.6)

s1 (W ā’1 M )s1 W ā’1 (N + N t ) = 0

(0.7)

s1 W ā’1 M W ā’1 (N + N t ) = s1 (W ā’1 N ).

(0.8)

Since a matrix and its transpose have the same characteristic polynomial, the traces

of W ā’1 N and (W ā’1 N )t = N t W ā’1 are the same, hence

s1 (W ā’1 N t ) = s1 (N t W ā’1 ) = s1 (W ā’1 N ).

Therefore, s1 W ā’1 (N + N t ) = 0, and (??) follows.

Similarly, we have

s1 (W ā’1 M W ā’1 N t ) = s1 (N W ā’1 M t W ā’1 ) = s1 (W ā’1 M t W ā’1 N ),

hence the left side of (??) is

s1 (W ā’1 M W ā’1 N ) + s1 (W ā’1 M t W ā’1 N ) = s1 W ā’1 (M + M t )W ā’1 N .

Since M + M t = W , (??) follows.

The second relation in (??) shows that the left side of (??) is

s2 (W ā’1 N ) + s2 (W ā’1 N t ) + s1 (W ā’1 N )s1 (W ā’1 N t ) + s1 (W ā’1 N W ā’1 N t ).

Since W ā’1 N and W ā’1 (W ā’1 N )t W (= W ā’1 N t ) have the same characteristic poly-

nomial, we have si (W ā’1 N ) = si (W ā’1 N t ) for i = 1, 2, hence the ļ¬rst two terms

cancel and the third is equal to s1 (W ā’1 N )2 . In order to prove (??), it therefore

suļ¬ces to show

s1 (W ā’1 N W ā’1 N t ) = 0.

Since W = M + M t , we have W ā’1 = W ā’1 M W ā’1 + W ā’1 M t W ā’1 , hence

s1 (W ā’1 N W ā’1 N t ) = s1 (W ā’1 M W ā’1 N W ā’1 N t ) + s1 (W ā’1 M t W ā’1 N W ā’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 ā’1 M W ā’1 N W ā’1 N t )t = N W ā’1 N t W ā’1 M t W ā’1 , hence

s1 (W ā’1 M W ā’1 N W ā’1 N t ) = s1 (N W ā’1 N t )(W ā’1 M t W ā’1 )

= s1 (W ā’1 M t W ā’1 N W ā’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 veriļ¬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 deļ¬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 Ā§?? for the relation between the discriminant

and the even Cliļ¬ord algebra.

Let Ī±1 , . . . , Ī±n ā F . If char F = 2 we denote by Ī±1 , . . . , Ī±n the diagonal

quadratic form

Ī± 1 , . . . , Ī± n = Ī± 1 x2 + Ā· Ā· Ā· + Ī± n x2

1 n

which is the quadratic form associated to the bilinear form Ī±1 , . . . , Ī±n . We also

deļ¬ne the n-fold Pļ¬ster quadratic form Ī±1 , . . . , Ī±n by

Ī±1 , . . . , Ī±n = 1, ā’Ī±1 ā— Ā· Ā· Ā· ā— 1, ā’Ī±n

where ā— = ā—F is the tensor product over F . If char F = 2, the quadratic forms

[Ī±1 , Ī±2 ] and [Ī±1 ] are deļ¬ned by

2 2

[Ī±1 ] = Ī±1 X 2 ,

[Ī±1 , Ī±2 ] = Ī±1 X1 + X1 X2 + Ī±2 X2 and

and the n-fold Pļ¬ster quadratic form Ī±1 , . . . , Ī±n ]] by

Ī±1 , . . . , Ī±n ]] = Ī±1 , . . . , Ī±nā’1 ā— [1, Ī±n ].

(See Baeza [?, p. 5] or Knus [?, p. 50] for the deļ¬nition of the tensor product of a

bilinear form and a quadratic form.) For instance,

Ī±1 , Ī±2 ]] = (x2 + x1 x2 + Ī±2 x2 ) + Ī±1 (x2 + x3 x4 + Ī±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 ļ¬nite dimensional vector space over a ļ¬eld F of arbitrary char-

acteristic. A bilinear form b : V Ć— V ā’ F is called nonsingular if the induced

map

Ė V ā’ V ā— = HomF (V, F )

b:

deļ¬ned by

Ė

b(x)(y) = b(x, y) for x, y ā V

is an isomorphism of vector spaces. For any f ā EndF (V ) we may then deļ¬ne

Ļb (f ) ā EndF (V ) by

Ļb (f ) = Ėā’1 ā—¦ f t ā—¦ Ė

b b

where f t ā EndF (V ā— ) is the transpose of f , deļ¬ned by mapping Ļ• ā V ā— to Ļ• ā—¦ f .

Alternately, Ļb (f ) may be deļ¬ned by the following property:

(ā—) b x, f (y) = b Ļb (f )(x), y for x, y ā V .

The map Ļb : EndF (V ) ā’ 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 Ļb 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 Ļb induces a one-to-one correspondence between equiv-

alence classes of nonsingular bilinear forms on V modulo multiplication by a factor

in F Ć— 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 (ā—) it follows that for Ī± ā F Ć— the adjoint anti-automorphism

ĻĪ±b with respect to the multiple Ī±b of b is the same as the adjoint anti-automor-

phism Ļb . Therefore, the map b ā’ Ļb induces a well-deļ¬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 = Ėā’1 ā—¦ b ā GL(V ) satisļ¬es

Ė

b

b (x, y) = b v(x), y for x, y ā V .

1

2 I. INVOLUTIONS AND HERMITIAN FORMS

From this relation, it follows that the adjoint anti-automorphisms Ļb , Ļb are related

by

Ļb (f ) = v ā—¦ Ļb (f ) ā—¦ v ā’1 for f ā EndF (V ),

or equivalently

Ļb = Int(v) ā—¦ Ļb ,

where Int(v) denotes the inner automorphism of EndF (V ) induced by v:

Int(v)(f ) = v ā—¦ f ā—¦ v ā’1 for f ā EndF (V ).

Therefore, if Ļb = Ļb , then v ā F Ć— and b, b are scalar multiples of each other.

Moreover, if b is a ļ¬xed nonsingular bilinear form on V with adjoint anti-

automorphism Ļb , then for any linear anti-automorphism Ļ of EndF (V ), the com-

ā’1

posite Ļb ā—¦ Ļ is an F -linear automorphism of EndF (V ). Since these automor-

phisms are inner, by the Skolem-Noether theorem (see (??) below), there exists

ā’1

u ā GL(V ) such that Ļb ā—¦ Ļ = Int(u). Then Ļ is the adjoint anti-automorphism

with respect to the bilinear form b deļ¬ned by

b (x, y) = b u(x), y .

Thus, the ļ¬rst part of the theorem is proved.

Observe also that if b is a nonsingular bilinear form on V with adjoint anti-

automorphism Ļb , then the bilinear form b deļ¬ned by

b (x, y) = b(y, x) for x, y ā V

ā’1 2

has adjoint anti-automorphism Ļb = Ļb . Therefore, Ļb = Id if and only if b and b

are scalar multiples of each other; since the scalar factor Īµ such that b = Īµb clearly

satisļ¬es Īµ2 = 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 Ļb 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 ļ¬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 ļ¬elds. We ļ¬rst review basic notions concerning

central simple algebras. The ļ¬rst section also discusses Severi-Brauer varieties, for

use in Ā§??. In Ā§?? we present the basic deļ¬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 ļ¬rst two types leave the center invariant; they are called involutions

of the ļ¬rst kind. Unitary involutions are also called involutions of the second kind ;

they restrict to a nontrivial automorphism of the center. Necessary and suļ¬cient

conditions for the existence of an involution on a central simple algebra are given

in Ā§??.

The theorem above, relating bilinear forms on a vector space to involutions

on the endomorphism algebra, is generalized in Ā§??, where hermitian forms over

simple algebras are investigated. Relations between an analogue of the Scharlau

Ā§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 deļ¬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 ļ¬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 diļ¬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 rĖle in the deļ¬nition of twisted

o

forms of orthogonal groups in Chapter ??.

Ā§1. Central Simple Algebras

Unless otherwise mentioned, all the algebras we consider in this work are ļ¬nite-

dimensional with 1. For any algebra A over a ļ¬eld F and any ļ¬eld extension K/F ,

we write AK for the K-algebra obtained from A by extending scalars to K:

AK = A ā—F K.

We also deļ¬ne the opposite algebra Aop by

Aop = { aop | a ā A },

with the operations deļ¬ned as follows:

aop + bop = (a + b)op , aop bop = (ba)op , Ī± Ā· aop = (Ī± Ā· a)op

for a, b ā A and Ī± ā F .

A central simple algebra over a ļ¬eld F is a (ļ¬nite dimensional) algebra A = {0}

with center F (= F Ā· 1) which has no two-sided ideals except {0} and A. An algebra

A = {0} is a division algebra (or a skew ļ¬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 ļ¬eld F , the following

conditions are equivalent:

(1) A is central simple.

(2) The canonical map A ā—F Aop ā’ EndF (A) which associates to a ā— bop the linear

map x ā’ axb is an isomorphism.

(3) There is a ļ¬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 ā„¦ is an algebraically closed ļ¬eld containing F ,

Aā„¦ Mn (ā„¦) for some n.

4 I. INVOLUTIONS AND HERMITIAN FORMS

(5) There is a ļ¬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 [?, Ā§3].

The ļ¬elds K for which condition (??) holds are called splitting ļ¬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 deļ¬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 (??) ā’ (??) in Wedderburnā™s theorem:

(1.2) Corollary. Every central simple F -algebra A has the form

A EndD (V )

for some (ļ¬nite dimensional ) central division F -algebra D and some ļ¬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 deļ¬-

nition:

(1.3) Deļ¬nition. Finite dimensional central simple algebras A, B over a ļ¬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

Wedderburnā™s theorem shows.

Uniqueness (up to isomorphism) of simple left modules over central simple

algebras leads to the following two fundamental results:

Ā§1. CENTRAL SIMPLE ALGEBRAS 5

(1.4) Theorem (Skolem-Noether). Let A be a central simple F -algebra and let

B ā‚ A be a simple subalgebra. Every F -algebra homomorphism Ļ : B ā’ A extends

to an inner automorphism of A: there exists a ā AĆ— such that Ļ(b) = abaā’1 for all

b ā B. In particular, every F -algebra automorphism of A is inner.

References: Scharlau [?, Theorem 8.4.2], Draxl [?, Ā§7] or Pierce [?, Ā§12.6].

The centralizer CA B of a subalgebra B ā‚ A is, by deļ¬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 ā‚ A be a simple subalgebra with center K ā F . The centralizer C A B is a simple

subalgebra of A with center K which satisļ¬es

dimF A = dimF B Ā· dimF CA B CA CA B = B.

and

If K = F , then multiplication in A deļ¬nes a canonical isomorphism A = Bā—F CA B.

References: Scharlau [?, Theorem 8.4.5], Draxl [?, Ā§7] or Pierce [?, Ā§12.7].

Let ā„¦ denote an algebraic closure of F . Under scalar extension to ā„¦, every

central simple F -algebra A of degree n becomes isomorphic to Mn (ā„¦). We may

therefore ļ¬x an F -algebra embedding A ā’ Mn (ā„¦) and view every element a ā A

as a matrix in Mn (ā„¦). Its characteristic polynomial has coeļ¬cients in F and is

independent of the embedding of A in Mn (ā„¦) (see Scharlau [?, Ch. 8, Ā§5], Draxl [?,

Ā§22], Reiner [?, Ā§9] or Pierce [?, Ā§16.1]); it is called the reduced characteristic

polynomial of A and is denoted

PrdA,a (X) = X n ā’ s1 (a)X nā’1 + s2 (a)X nā’2 ā’ Ā· Ā· Ā· + (ā’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 Ć— A ā’ F deļ¬ned by

TA (x, y) = TrdA (xy) for x, y ā 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 ļ¬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 ļ¬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 ļ¬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 Ć— 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) Deļ¬nition. The reduced dimension of the left A-module M is deļ¬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 ļ¬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 classiļ¬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 ļ¬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 ļ¬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 ā’ 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.

Ā§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 Ć— s-matrices over D. Matrix multiplication endows M with an A-

E-bimodule structure, so that we have natural embeddings

(1.11) A ā’ EndE (M ) and E ā’ EndA (M ).

Since dimF M = rs dimF D, it is readily computed that rdimE M = deg A and

rdimA M = deg E. The ļ¬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 ļ¬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 ā‚ V be a subspace. Composing every linear map from V to U with the

inclusion U ā’ V , we identify HomD (V, U ) with a subspace of A = EndD (V ):

HomD (V, U ) = { f ā EndD (V ) | im f ā‚ 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 ā’ V /U , we may identify HomD (V /U, V ) with a subspace of

A = EndD (V ):

HomD (V /U, V ) = { f ā EndD (V ) | ker f ā 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 ā’ HomD (V, U ) deļ¬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 ā’ HomD (V /U, V ) deļ¬nes a

one-to-one correspondence between subspaces of dimension d in V and left ideals

of reduced dimension deg A ā’ 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 deļ¬nes an F -algebra

homomorphism EndD (U ) ā’ EndA HomD (V, U ) and multiplication on the right

deļ¬nes an F -algebra homomorphism

EndD (V /U ) ā’ 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 ) ā’ EndA HomD (V, U ) is an isomorphism. Simi-

larly, the homomorphism EndD (V /U ) ā’ EndA HomD (V /U, V ) is an isomorphism

by dimension count.

For the ļ¬rst part, it suļ¬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 ā‚ V . This is

proved for instance in Baer [?, Ā§5.2].

(1.13) Corollary. For every left (resp. right) ideal I ā‚ A there exists an idempo-

tent e ā 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:

Ļ : I ā’ 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 ā• 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 ā I, deļ¬ne Ļ(x) ā EndA (I) by

y Ļ(x) = yx.

For f ā EndA (I) we have

f

(yx)f = yxf = y Ļ(x ) ,

hence

Ļ(xf ) = Ļ(x) ā—¦ f,

which means that Ļ is a homomorphism of right EndA (I)-modules. In order to see

that Ļ is onto, pick an idempotent e ā A such that I = Ae. For every y ā I we

have y = ye; it follows that every f ā EndA (I) is of the form f = Ļ(ef ), since for

every y ā I,

f

y f = (ye)f = yef = y Ļ(e ) .

Therefore, Ļ is surjective.

To complete the proof, we show that the restriction of Ļ to eAe is an isomor-

ā¼

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

ā’

homomorphism. Moreover, for every x ā I one has Ļ(x) = Ļ(ex) since y = ye for

every y ā I. Therefore, the restriction of Ļ to eAe is also surjective onto End A (I).

Finally, if Ļ(ex) = 0, then in particular

eĻ(ex) = ex = 0,

so Ļ is injective on eAe.

Annihilators. For every left ideal I in a central simple algebra A over a ļ¬eld F ,

the annihilator I 0 is deļ¬ned by

I 0 = { x ā A | Ix = {0} }.

This set is clearly a right ideal. Similarly, for every right ideal I, the annihilator I 0

is deļ¬ned by

I 0 = { x ā A | xI = {0} };

it is a left ideal in A.

Ā§1. CENTRAL SIMPLE ALGEBRAS 9

(1.14) Proposition. For every left or right ideal I ā‚ A, rdim I + rdim I 0 = deg A

and I 00 = I.

Proof : Let A = EndD (V ). For any subspace U ā‚ V it follows from the deļ¬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 ā‚ A has the form I = HomD (V /U, V ) (resp.

I = HomD (V, U )), the proposition follows.

Now, let J ā‚ A be a right ideal of reduced dimension k and let B ā‚ A be the

idealizer of J:

B = { a ā A | aJ ā‚ J }.

This set is a subalgebra of A containing J as a two-sided ideal. It follows from the

deļ¬nition of J 0 that J 0 b ā‚ J 0 for all b ā B and that J 0 ā‚ B. Therefore, (??) shows

that the map Ļ : B ā’ EndA (J 0 ) deļ¬ned by multiplication on the right is surjective.

ā¼

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

ā’

For every right ideal I ā‚ A containing J, let

Ė

I = Ļ(I ā© B).

Ė

(1.15) Proposition. The map I ā’ I deļ¬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 ā’ k in EndA (J 0 ). If A = EndD (V ) and J = HomD (V, U ) for some

subspace U ā‚ V of dimension r/ ind A, then for I = HomD (V, W ) with W ā U , we

have under the natural isomorphism EndA (J 0 ) = EndD (V /U ) of (??) that

Ė

I = HomD (V /U, W/U ).

Proof : In view of (??), the second part implies the ļ¬rst, since the map W ā’ W/U

deļ¬nes a one-to-one correspondence between subspaces of dimension r/ ind A in V

which contain U and subspaces of dimension (r ā’ 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 ā A | f (U ) ā‚ U }. Every f ā B induces a linear map f ā EndD (V /U ),

and the homomorphism Ļ : B ā’ EndA (J 0 ) = EndD (V /U ) maps f to f since for

g ā J 0 we have

g Ļ(f ) = g ā—¦ f = g ā—¦ f .

For I = HomD (V, W ) with W ā U , it follows that

Ė

I = { f | f ā I and f (U ) ā‚ U } ā‚ 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 ā HomD (V /U, W/U ) is of the form h = f for some f ā

HomD (V, W ) such that f (U ) ā‚ U .

1.C. Severi-Brauer varieties. Let A be a central simple algebra of degree n

over a ļ¬eld F and let r be an integer, 1 ā¤ r ā¤ n. Consider the Grassmannian

Gr(rn, A) of rn-dimensional subspaces in A. The PlĀØcker embedding identiļ¬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) ā‚ P( A).

10 I. INVOLUTIONS AND HERMITIAN FORMS

The rn-dimensional subspace U ā‚ A corresponding to a non-zero rn-vector u1 ā§

rn

Ā· Ā· Ā· ā§ urn ā A is

U = { x ā A | u1 ā§ Ā· Ā· Ā· ā§ urn ā§ x = 0 } = u1 F + Ā· Ā· Ā· + 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 deļ¬ned over some base ļ¬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 )1ā¤iā¤n2 of A.

rn

The rn-dimensional subspace represented by an rn-vector u1 ā§ Ā· Ā· Ā· ā§ urn ā 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 ā§ Ā· Ā· Ā· ā§ urn ei ā u1 ā§ Ā· Ā· Ā· ā§ urn F

or, equivalently,

u1 ei ā§ Ā· Ā· Ā· ā§ urn ei ā§ 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 ā§ Ā· Ā· Ā· ā§ urn , hence the right ideals of reduced dimension r in A form a closed

subvariety of Gr(rn, A).

(1.16) Deļ¬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) ā‚ Gr(rn, A).

For r = 1, we write simply SB(A) = SB1 (A). This is the (usual) Severi-Brauer

variety of A, ļ¬rst deļ¬ned by F. ChĖtelet [?].

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 deļ¬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 ļ¬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 ā’ 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 ).

Ā§1. CENTRAL SIMPLE ALGEBRAS 11

In particular, for r = 1,

P(V ).

SB(A)

Proof : Let V ā— = HomF (V, F ) be the dual of V . Under the natural isomorphism

A = EndF (V ) V ā—F V ā— , multiplication is given by

(v ā— Ļ) Ā· (w ā— Ļ) = (v ā— Ļ)Ļ(w).

By (??), the right ideals of reduced dimension r in A are of the form HomF (V, U ) =

U ā— V ā— where U is an r-dimensional subspace in V .

We will show that the correspondence U ā” U ā— V ā— 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 ) ā’

Gr(rn, V ā—W ) which maps an r-dimensional subspace U ā‚ V to U ā—W ā‚ V ā—W . In

the particular case where W = V ā— we thus get a morphism Ī¦ : Gr(r, V ) ā’ SBr (A)

which maps U to U ā— V ā— .

In order to show that Ī¦ is an isomorphism, we consider the following aļ¬ne

covering of Gr(r, V ): for each subspace S ā‚ V of dimension n ā’ r, we denote by US

the set of complementary subspaces:

US = { U ā‚ V | U ā• S = V }.

The set US is an aļ¬ne open subset of Gr(r, V ); more precisely, if U0 is a ļ¬xed

complementary subspace of S, there is an isomorphism:

ā¼

HomF (U0 , S) ā’ US

ā’

which maps f ā HomF (U0 , S) to U = { x + f (x) | x ā U0 } (see Harris [?, p. 65]).

Similarly, we may also consider USā—V ā— ā‚ Gr(rn, A). The image of the restriction

of Ī¦ to US is

{ U ā— V ā— ā‚ V ā— V ā— | (U ā— V ā— ) ā• (S ā— V ā— ) = V ā— V ā— } = USā—V ā— ā© SBr (A).

Moreover, there is a commutative diagram:

Ī¦|U

ā’ ā’S

US ā’ā’ā’ USā—V ā—

ļ£¦ ļ£¦

ļ£¦ ļ£¦

Ļ

HomF (U0 , S) ā’ ā’ ā’ HomF (U0 ā— V ā— , S ā— V ā— )

ā’ā’

where Ļ(f ) = f ā— IdV ā— . Since Ļ is linear and injective, it is an isomorphism of

varieties between HomF (U0 , S) and its image. Therefore, the restriction of Ī¦ to US

ā¼

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

ā’

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

Although Severi-Brauer varieties are deļ¬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 ā Aop | j ā J }

is a right ideal in the opposite algebra Aop . Therefore, the variety of left ideals

of reduced dimension r in A can be identiļ¬ed with SBr (Aop ). We combine this

observation with the annihilator construction (see Ā§??) 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

ā¼

Ī± : SBr (A) ā’ SBnā’r (Aop )

ā’

which maps a right ideal I ā‚ A of reduced dimension r to (I 0 )op .

Proof : In order to prove that Ī± is an isomorphism, we may extend scalars to a

splitting ļ¬eld of A. We may therefore assume that A = EndF (V ) for some n-

dimensional vector space V . Then Aop = EndF (V ā— ) under the identiļ¬cation f op =

f t for f ā EndF (V ). By (??), we may further identify

SBnā’r (Aop ) = Gr(n ā’ r, V ā— ).

SBr (A) = Gr(r, V ),

Under these identiļ¬cations, the map Ī± : Gr(r, V ) ā’ Gr(n ā’ r, V ā— ) carries every

r-dimensional subspace U ā‚ V to U 0 = { Ļ• ā V ā— | Ļ•(U ) = {0} }.

To show that Ī± is an isomorphism of varieties, we restrict it to the aļ¬ne open

sets US deļ¬ned in the proof of Theorem (??): let S be an (n ā’ r)-dimensional

subspace in V and

US = { U ā‚ V | U ā• S = V } ā‚ Gr(r, V ).

Let U0 ā‚ V be such that U0 ā• S = V , so that US HomF (U0 , S). We also have

0 0 ā— 0 0

U0 ā• S = V , US 0 HomF (U0 , S ), and the map Ī± restricts to Ī±|US : US ā’ US 0 .

It therefore induces a map Ī± which makes the following diagram commute:

Ī±|U

ā’ ā’S

US ā’ā’ā’ US 0

ļ£¦ ļ£¦

ļ£¦ ļ£¦

Ī±

HomF (U0 , S) ā’ ā’ ā’ HomF (U0 , S 0 ).

0

ā’ā’

We now proceed to show that Ī± is an isomorphism of (aļ¬ne) varieties.

0

Every linear form in U0 restricts to a linear form on S, and since V = U0 ā•S we

0

S ā— . Similarly, S 0 U0 , so HomF (U0 , S 0 )

ā— 0

thus get a natural isomorphism U0

HomF (S ā— , U0 ). Under this identiļ¬cation, a direct calculation shows that the map Ī±

ā—

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

ā— 0

an isomorphism of varieties. Since the open sets US cover Gr(r, V ), it follows that

Ī± is an isomorphism.

If V is a vector space of dimension n over a ļ¬eld F and U ā‚ V is a subspace of

dimension k, then for r = k, . . . , n the Grassmannian Gr(r ā’ k, V /U ) embeds into

Gr(r, V ) by mapping every subspace W ā‚ V /U to the subspace W ā U such that

W/U = W . The image of Gr(r ā’ 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 ā‚ 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 ā’ k in EndA (J 0 ) set up in (??) deļ¬nes an

embedding:

SBrā’k EndA (J 0 ) ā’ SBr (A).

The image of SBrā’k EndA (J 0 ) in SBr (A) is the variety of right ideals of reduced

dimension r in A which contain J.

Ā§2. INVOLUTIONS 13

Proof : It suļ¬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 ā‚ 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 SBrā’k EndA (J 0 ) =

Gr(r ā’ k, V /U ). Moreover, from (??) it follows that the map SBrā’k EndA (J 0 ) ā’

SBr (A) corresponds under these identiļ¬cations to the embedding Gr(rā’k, V /U ) ā’

Gr(r, V ) described above.

Ā§2. Involutions

An involution on a central simple algebra A over a ļ¬eld F is a map Ļ : A ā’ A

subject to the following conditions:

(a) Ļ(x + y) = Ļ(x) + Ļ(y) for x, y ā A.

(b) Ļ(xy) = Ļ(y)Ļ(x) for x, y ā A.

(c) Ļ 2 (x) = x for x ā A.

Note that the map Ļ is not required to be F -linear. However, it is easily checked

that the center F (= F Ā· 1) is preserved under Ļ. The restriction of Ļ 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 ļ¬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 deļ¬nitions and properties of central simple alge-

bras with involution. Involutions of the ļ¬rst kind are considered ļ¬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 ļ¬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 (??)ā“(??).

2.A. Involutions of the ļ¬rst kind. Throughout this subsection, A denotes

a central simple algebra over a ļ¬eld F of arbitrary characteristic, and Ļ is an

involution of the ļ¬rst kind on A. Our basic object of study is the couple (A, Ļ); from

this point of view, a homomorphism of algebras with involution f : (A, Ļ) ā’ (A , Ļ )

is an F -algebra homomorphism f : A ā’ A such that Ļ ā—¦ f = f ā—¦ Ļ. Our main tool

is the extension of scalars: if L is any ļ¬eld containing F , the involution Ļ extends

to an involution of the ļ¬rst kind ĻL = Ļ ā— IdL on AL = A ā—F L. In particular, if

L is a splitting ļ¬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 ĻL is then the adjoint involution Ļb 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 ā GLn (L) denotes the Gram matrix of b with

respect to the chosen basis, then

b(x, y) = xt Ā· g Ā· y

where x, y are considered as column matrices and g t = g if b is symmetric, g t = ā’g

if b is skew-symmetric. The involution ĻL is then identiļ¬ed with the involution Ļg

14 I. INVOLUTIONS AND HERMITIAN FORMS

deļ¬ned by

Ļg (m) = g ā’1 Ā· mt Ā· g for m ā Mn (L).

For future reference, we summarize our conclusions:

(2.1) Proposition. Let (A, Ļ) be a central simple F -algebra of degree n with in-

volution of the ļ¬rst kind and let L be a splitting ļ¬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 ā GLn (L) such that g t = g if b is

symmetric and g t = ā’g if b is skew-symmetric, and

(AL , ĻL ) EndL (V ), Ļb Mn (L), Ļg .

As a ļ¬rst application, we have the following result:

(2.2) Corollary. For all a ā A, the elements a and Ļ(a) have the same reduced

characteristic polynomial. In particular, TrdA Ļ(a) = TrdA (a) and NrdA Ļ(a) =

NrdA (a).

Proof : For all m ā Mn (L), g ā GLn (L), the matrix g ā’1 Ā· mt Ā· g has the same

characteristic polynomial as m.

Of course, in (??), neither the form b nor the matrix g (nor even the splitting

ļ¬eld L) is determined uniquely by the involution Ļ; some of their properties reļ¬‚ect

properties of Ļ, however. As a ļ¬rst example, we show in (??) below that two types

of involutions of the ļ¬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 deļ¬ne next.

In a central simple F -algebra A with involution of the ļ¬rst kind Ļ, the sets of

symmetric, skew-symmetric, symmetrized and alternating elements in A are deļ¬ned

as follows:

Sym(A, Ļ) = { a ā A | Ļ(a) = a },

Skew(A, Ļ) = { a ā A | Ļ(a) = ā’a },

Symd(A, Ļ) = { a + Ļ(a) | a ā A },

Alt(A, Ļ) = { a ā’ Ļ(a) | a ā A }.

If char F = 2, then Symd(A, Ļ) = Sym(A, Ļ), Alt(A, Ļ) = Skew(A, Ļ) and A =

1

Sym(A, Ļ) ā• Skew(A, Ļ) since every element a ā A decomposes as a = 2 a +

1

Ļ(a) + 2 a ā’ Ļ(a) . If char F = 2, then Symd(A, Ļ) = Alt(A, Ļ) ā‚ Skew(A, Ļ) =

Sym(A, Ļ), and (??) below shows that this inclusion is strict.

(2.3) Lemma. Let n = deg A; then dim Sym(A, Ļ) + dim Alt(A, Ļ) = n2 . More-

over, Alt(A, Ļ) is the orthogonal space of Sym(A, Ļ) for the bilinear form T A on A

induced by the reduced trace:

Alt(A, Ļ) = { a ā A | TrdA (as) = 0 for s ā Sym(A, Ļ) }.

Similarly, dim Skew(A, Ļ)+dim Symd(A, Ļ) = n2 , and Symd(A, Ļ) is the orthogonal

space of Skew(A, Ļ) 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.

Ā§2. INVOLUTIONS 15

Proof : The ļ¬rst relation comes from the fact that Alt(A, Ļ) is the image of the

linear endomorphism Id ā’ Ļ of A, whose kernel is Sym(A, Ļ). If a ā Alt(A, Ļ), then

a = x ā’ Ļ(x) for some x ā A, hence for s ā Sym(A, Ļ),

TrdA (as) = TrdA (xs) ā’ TrdA Ļ(x)s = TrdA (xs) ā’ TrdA Ļ(sx) .

Corollary (??) shows that the right side vanishes, hence the inclusion

Alt(A, Ļ) ā‚ { a ā A | TrdA (as) = 0 for s ā Sym(A, Ļ) }.

Dimension count shows that this inclusion is an equality since TA is nonsingular

(see (??)).

The statements involving Symd(A, Ļ) readily follow, either by mimicking the

arguments above, or by using the fact that in characteristic diļ¬erent from 2,

Symd(A, Ļ) = Sym(A, Ļ) and Alt(A, Ļ) = Skew(A, Ļ), and, in characteristic 2,

Symd(A, Ļ) = Alt(A, Ļ) and Skew(A, Ļ) = Sym(A, Ļ).

We next determine the dimensions of Sym(A, Ļ) and Skew(A, Ļ) (and therefore

also of Symd(A, Ļ) and Alt(A, Ļ)).

Consider ļ¬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 ļ¬rst kind Ļ 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 Ļ up to a factor in F Ć— .

(2.4) Lemma. Let Ļ = Ļb 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, Ļ) = n(n + 1)/2.

(2) If b is skew-symmetric, then dimF Skew(A, Ļ) = n(n + 1)/2.

(3) If char F = 2, then b is alternating if and only if tr(f ) = 0 for all f ā

Sym(A, Ļ). In this case, n is necessarily even.

Proof : As in (??), we use a basis of V to identify (A, Ļ) with Mn (F ), Ļg , where

g ā GLn (F ) satisļ¬es g t = g if b is symmetric and g t = ā’g if b is skew-symmetric.

For m ā Mn (F ), the relation gm = (gm)t is equivalent to Ļg (m) = m if g t = g and

to Ļg (m) = ā’m if g t = ā’g. Therefore,

Sym(A, Ļ) if b is symmetric,

g ā’1 Ā· Sym Mn (F ), t =

Skew(A, Ļ) if b is skew-symmetric.

The ļ¬rst two parts then follow from the fact that the space Sym Mn (F ), t of n Ć— 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 ā V . Consider the map f : V ā’ V deļ¬ned by

f (x) = vb(v, x)b(v, v)ā’1 for x ā V .

Since b is symmetric we have

b f (x), y = b(v, y)b(v, x)b(v, v)ā’1 = b x, f (y) for x, y ā V ,

hence Ļ(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 )1ā¤iā¤n be

a symplectic basis of V , in the sense that b(e2iā’1 , e2i ) = 1, b(e2i , e2i+1 ) = 0 and

b(ei , ej ) = 0 if |i ā’ j| > 1. Let f ā Sym(A, Ļ); for j = 1, . . . , n let

n

f (ej ) = ei aij for some aij ā F ,

i=1

n

so that tr(f ) = aii . For i = 1, . . . , n/2 we have

i=1

b f (e2iā’1 ), e2i = a2iā’1,2iā’1 and b e2iā’1 , f (e2i ) = a2i,2i ;

since Ļ(f ) = f , it follows that a2iā’1,2iā’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 Ļ is an involution of the ļ¬rst kind on A. Let n = deg A and let L be a

splitting ļ¬eld of A. Consider an isomorphism as in (??):

(AL , ĻL ) EndL (V ), Ļb .

This isomorphism carries Sym(AL , ĻL ) = Sym(A, Ļ)ā—F L to Sym EndL (V ), Ļb and

Skew(AL , ĻL ) to Skew EndL (V ), Ļb . Since extension of scalars does not change

dimensions, (??) shows

(a) dimF Sym(A, Ļ) = n(n + 1)/2 if b is symmetric;

(b) dimF Skew(A, Ļ) = 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, Ļ)ā•Skew(A, Ļ), hence the dimensions of Sym(A, Ļ)

and Skew(A, Ļ) 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 ā Sym(A, Ļ) if

and only if tr(f ) = 0 for all f ā Sym EndL (V ), Ļb , 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 deļ¬nition:

(2.5) Deļ¬nition. An involution Ļ of the ļ¬rst kind is said to be of symplectic type

(or simply symplectic) if for any splitting ļ¬eld L and any isomorphism (AL , ĻL )

EndL (V ), Ļb , the bilinear form b is alternating; otherwise it is called of orthogonal

type (or simply orthogonal ). In the latter case, for any splitting ļ¬eld L and any

isomorphism (AL , ĻL ) EndL (V ), Ļb , the bilinear form b is symmetric (and

nonalternating).

The preceding discussion yields an alternate characterization of orthogonal and

symplectic involutions:

ńņš. 1 |