. 1
( : 75)



. . >>

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

. 1
( : 75)



. . >>