to its Cli¬ord algebra. Trialitarian algebras also occur in the construction of Lie

algebras of type D4 . Some indications in this direction are given in the last section.

Exercises and notes can be found at the end of each chapter. Omitted proofs

sometimes occur as exercises. Moreover we included as exercises some results we

like, but which we did not wish to develop fully. In the notes we wanted to give com-

plements and to look at some results from a historical perspective. We have tried

our best to be useful; we cannot, however, give strong guarantees of completeness

or even fairness.

This book is the achievement of a joint (and very exciting) e¬ort of four very

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

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

Our work started out as an attempt to understand Tits™ de¬nition of the Cli¬ord

algebra of a generalized quadratic form, and ended up including many other topics

to which Tits made fundamental contributions, such as linear algebraic groups,

exceptional algebras, triality, . . . Not only was Jacques Tits a constant source of

inspiration through his work, but he also had a direct personal in¬‚uence, notably

through his threat ” early in the inception of our project ” to speak evil of

our work if it did not include the characteristic 2 case. Finally he also agreed to

bestow his blessings on our book sous forme de pr´face. For all that we thank him

e

wholeheartedly.

This book could not have been written without the help and the encourage-

ment of many friends. They are too numerous to be listed here individually, but

we hope they will recognize themselves and ¬nd here our warmest thanks. Richard

Elman deserves a special mention for his comment that the most useful book is

not the one to which nothing can be added, but the one which is published. This

no-nonsense statement helped us set limits to our endeavor. We were fortunate to

get useful advice on various points of the exposition from Ottmar Loos, Antonio

Paques, Parimala, Michel Racine, David Saltman, Jean-Pierre Serre and Sridharan.

We thank all of them for lending helping hands at the right time. A number of

people were nice enough to read and comment on drafts of parts of this book: Eva

Bayer-Fluckiger, Vladimir Chernousov, Ingrid Dejai¬e, Alberto Elduque, Darrell

Haile, Luc Haine, Pat Morandi, Holger Petersson, Ahmed Serhir, Tony Springer,

xii INTRODUCTION

Paul Swets and Oliver Villa. We know all of them had better things to do, and

we are grateful. Skip Garibaldi and Adrian Wadsworth actually summoned enough

grim self-discipline to read a draft of the whole book, detecting many shortcomings,

making shrewd comments on the organization of the book and polishing our bro-

ken English. Each deserves a medal. However, our capacity for making mistakes

certainly exceeds our friends™ sagacity. We shall gratefully welcome any comment

or correction.

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

simple algebras, involutions and quadratic forms” in April 1993 at the National

Taiwan University. He wants to thank Ming-chang Kang and the National Research

Council of China for this opportunity to test high doses of involutions on a very

patient audience, and Eng-Tjioe Tan for making his stay in Taiwan a most pleasant

experience. The lecture notes from this crash course served as a blueprint for the

¬rst chapters of this book.

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

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

Neuve in 1993, with support from the Fonds de D´veloppement Scienti¬que and the

e

Institut de Math´matique Pure et Appliqu´e of the Universit´ catholique de Lou-

e e e

vain, and Tignol™s stay in Z¨rich for the winter semester of 1995“96, with support

u

from the Eidgen¨ssische Technische Hochschule. Moreover, Merkurjev gratefully

o

acknowledges support from the Alexander von Humboldt foundation and the hos-

pitality of the Bielefeld university for the year 1995“96, and Jean-Pierre Tignol is

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

The four authors enthusiastically thank Herbert Rost (Markus™ father) for the

design of the cover page, in particular for his wonderful and colorful rendition of the

Dynkin diagram D4 . They also give special praise to Sergei Gelfand, Director of

Acquisitions of the American Mathematical Society, for his helpfulness and patience

in taking care of all our wishes for the publication.

Conventions and Notations

Maps. The image of an element x under a map f is generally denoted f (x);

the notation xf is also used however, notably for homomorphisms of left modules.

In that case, we also use the right-hand rule for mapping composition; for the image

f g

of x ∈ X under the composite map X ’ Y ’ Z we set either g —¦ f (x) or xf g and

’’

the composite is thus either g —¦ f or f g.

As a general rule, module homomorphisms are written on the opposite side of

the scalars. (Right modules are usually preferred.) Thus, if M is a module over a

ring R, it is also a module (on the opposite side) over EndR (M ), and the R-module

structure de¬nes a natural homomorphism:

R ’ EndEndR (M ) (M ).

Note therefore that if S ‚ EndR (M ) is a subring, and if we endow M with its

natural S-module structure, then EndS (M ) is the opposite of the centralizer of S

in EndR (M ):

op

EndS (M ) = CEndR (M ) S .

Of course, if R is commutative, every right R-module MR may also be regarded as a

left R-module R M , and every endomorphism of MR also is an endomorphism of R M .

Note however that with the convention above, the canonical map EndR (MR ) ’

EndR (R M ) is an anti-isomorphism.

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

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

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

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

(0.1)

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

tr(m) = s1 (m), det(m) = sn (m).

We recall the following relations between coe¬cients of the characteristic polyno-

mial:

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

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

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

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

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

xiii

xiv CONVENTIONS AND NOTATIONS

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

1¤i¤n

s2 (m) = 1¤i<j¤n »i »j , hence

s1 (m)2 ’ 2s2 (m) = »2 = s1 (m2 ),

i

1¤i¤n

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

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

If L is an associative and commutative F -algebra of dimension n and ∈ L,

the characteristic polynomial of multiplication by , viewed as an F -endomorphism

of L, is called the generic polynomial of and is denoted

PL, (X) = X n ’ s1 ( )X n’1 + s2 ( )X n’2 ’ · · · + (’1)n sn ( ).

The trace and norm of are denoted TL/F ( ) and NL/F ( ) (or simply T ( ), N ( )):

TL/F ( ) = s1 ( ), NL/F ( ) = sn ( ).

We also denote

(0.3) SL/F ( ) = S( ) = s2 ( ).

The characteristic polynomial is also used to de¬ne a generic polynomial for central

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

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

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

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

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

if b(x, x) = 0 for all x ∈ V . Thus, the notions of skew-symmetric and alternating

(resp. symmetric) form coincide if char F = 2 (resp. char F = 2). Alternating forms

are skew-symmetric in every characteristic.

If b is a symmetric or alternating bilinear form on a (¬nite dimensional) vector

space V , the induced map

ˆ V ’ V — = HomF (V, F )

b:

is de¬ned by ˆ b(x)(y) = b(x, y) for x, y ∈ V . The bilinear form b is nonsingular (or

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

b b

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

we are dealing with ¬nite dimensional vector spaces over ¬elds.) Alternately, b is

nonsingular if and only if the determinant of its Gram matrix with respect to an

arbitrary basis of V is nonzero:

det b(ei , ej ) = 0.

1¤i,j¤n

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

· F —2 ∈ F — /F —2 .

det b = det b(ei , ej ) 1¤i,j¤n

The discriminant of b is the signed determinant:

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

For ±1 , . . . , ±n ∈ F , the bilinear form ±1 , . . . , ±n on F n is de¬ned by

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

We also de¬ne the n-fold P¬ster bilinear form ±1 , . . . , ±n by

±1 , . . . , ±n = 1, ’±1 — · · · — 1, ’±n .

CONVENTIONS AND NOTATIONS xv

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

associated quadratic map, de¬ned by

qb (x) = b(x, x) for x ∈ V .

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

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

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

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

hence bq (x, x) = 2q(x) for all x ∈ V . Thus, the quadratic map qbq associated to bq

is qbq = 2q. Similarly, for every symmetric bilinear form b on V , we have bqb = 2b.

Let V ⊥ = { x ∈ V | bq (x, y) = 0 for y ∈ V }. The quadratic map q is called

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

q(V ⊥ ) = {0}. The latter case occurs only if char F = 2 and V is odd-dimensional.

Equivalently, a quadratic form of dimension n is nonsingular if and only if it is

n/2 2

equivalent over an algebraic closure to i=1 x2i’1 x2i (if n is even) or to x0 +

(n’1)/2

x2i’1 x2i (if n is odd).

i=1

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

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

in the sense that

q(X) = X · M · X t

where X = (x1 , . . . , xn ) and t denotes the transpose of matrices; the condition that

q is nonsingular implies that M + M t is invertible if n is even or char F = 2, and

has rank n ’ 1 if n is odd and char F = 2. The matrix M is uniquely determined by

q up to the addition of a matrix of the form N ’ N t ; therefore, M + M t is uniquely

determined by q.

If char F = 2 we set

+ M t ) · F —2 ∈ F — /F —2

1

det q = det 2 (M

and

disc q = (’1)n(n’1)/2 det q ∈ F — /F —2 .

Thus, the determinant (resp. the discriminant) of a quadratic form is the determi-

nant (resp. the discriminant) of its polar form divided by 2n .

If char F = 2 and n is odd we set

det q = disc q = q(y) · F —2 ∈ F — /F —2

(0.4)

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

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

de¬nition above does not depend on the choice of y.

If char F = 2 and n is even we set

det q = s2 (M + M t )’1 M + „˜(F ) ∈ F/„˜(F )

and

m(m’1)

disc q = + det q ∈ F/„˜(F )

2

xvi CONVENTIONS AND NOTATIONS

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

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

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

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

is invertible, then

2

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

Proof : The second relation in (??) yields

s2 W ’1 M + W ’1 (N + N t ) =

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

+ s1 W ’1 M W ’1 (N + N t ) .

In order to prove the lemma, we show below:

2

s2 W ’1 (N + N t ) = s1 (W ’1 N )

(0.6)

s1 (W ’1 M )s1 W ’1 (N + N t ) = 0

(0.7)

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

(0.8)

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

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

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

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

Similarly, we have

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

hence the left side of (??) is

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

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

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

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

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

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

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

su¬ces to show

s1 (W ’1 N W ’1 N t ) = 0.

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

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

and (??) follows if we show that the two terms on the right side are equal. Since

W t = W we have (W ’1 M W ’1 N W ’1 N t )t = N W ’1 N t W ’1 M t W ’1 , hence

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

= s1 (W ’1 M t W ’1 N W ’1 N t ).

CONVENTIONS AND NOTATIONS xvii

Quadratic forms are called equivalent if they can be transformed into each other

by invertible linear changes of variables. The various quadratic forms representing a

quadratic map with respect to various bases are thus equivalent. It is easily veri¬ed

that the determinant det q (hence also the discriminant disc q) is an invariant of the

equivalence class of the quadratic form q; the determinant and the discriminant are

therefore also de¬ned for quadratic maps. The discriminant of a quadratic form (or

map) of even dimension in characteristic 2 is also known as the pseudodiscriminant

or the Arf invariant of the form. See §?? for the relation between the discriminant

and the even Cli¬ord algebra.

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

quadratic form

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

1 n

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

de¬ne the n-fold P¬ster quadratic form ±1 , . . . , ±n by

±1 , . . . , ±n = 1, ’±1 — · · · — 1, ’±n

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

[±1 , ±2 ] and [±1 ] are de¬ned by

2 2

[±1 ] = ±1 X 2 ,

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

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

±1 , . . . , ±n ]] = ±1 , . . . , ±n’1 — [1, ±n ].

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

bilinear form and a quadratic form.) For instance,

±1 , ±2 ]] = (x2 + x1 x2 + ±2 x2 ) + ±1 (x2 + x3 x4 + ±2 x2 ).

1 2 3 4

xviii CONVENTIONS AND NOTATIONS

CHAPTER I

Involutions and Hermitian Forms

Our perspective in this work is that involutions on central simple algebras

are twisted forms of symmetric or alternating bilinear forms up to a scalar factor.

To motivate this point of view, we consider the basic, classical situation of linear

algebra.

Let V be a ¬nite dimensional vector space over a ¬eld F of arbitrary char-

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

map

ˆ V ’ V — = HomF (V, F )

b:

de¬ned by

ˆ

b(x)(y) = b(x, y) for x, y ∈ V

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

σb (f ) ∈ EndF (V ) by

σb (f ) = ˆ’1 —¦ f t —¦ ˆ

b b

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

Alternately, σb (f ) may be de¬ned by the following property:

(—) b x, f (y) = b σb (f )(x), y for x, y ∈ V .

The map σb : EndF (V ) ’ EndF (V ) is then an anti-automorphism of EndF (V )

which is known as the adjoint anti-automorphism with respect to the nonsingular

bilinear form b. The map σb clearly is F -linear.

The basic result which motivates our approach and which will be generalized

in (??) is the following:

Theorem. The map which associates to each nonsingular bilinear form b on V its

adjoint anti-automorphism σb induces a one-to-one correspondence between equiv-

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

in F — and linear anti-automorphisms of EndF (V ). Under this correspondence, F -

linear involutions on EndF (V ) (i.e., anti-automorphisms of period 2) correspond

to nonsingular bilinear forms which are either symmetric or skew-symmetric.

Proof : From relation (—) it follows that for ± ∈ F — the adjoint anti-automorphism

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

phism σb . Therefore, the map b ’ σb induces a well-de¬ned map from the set

of nonsingular bilinear forms on V up to a scalar factor to the set of F -linear

anti-automorphisms of End(V ).

To show that this map is one-to-one, note that if b, b are nonsingular bilinear

forms on V , then the map v = ˆ’1 —¦ b ∈ GL(V ) satis¬es

ˆ

b

b (x, y) = b v(x), y for x, y ∈ V .

1

2 I. INVOLUTIONS AND HERMITIAN FORMS

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

by

σb (f ) = v —¦ σb (f ) —¦ v ’1 for f ∈ EndF (V ),

or equivalently

σb = Int(v) —¦ σb ,

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

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

Therefore, if σb = σb , then v ∈ F — and b, b are scalar multiples of each other.

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

automorphism σb , then for any linear anti-automorphism σ of EndF (V ), the com-

’1

posite σb —¦ σ is an F -linear automorphism of EndF (V ). Since these automor-

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

’1

u ∈ GL(V ) such that σb —¦ σ = Int(u). Then σ is the adjoint anti-automorphism

with respect to the bilinear form b de¬ned by

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

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

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

automorphism σb , then the bilinear form b de¬ned by

b (x, y) = b(y, x) for x, y ∈ V

’1 2

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

are scalar multiples of each other; since the scalar factor µ such that b = µb clearly

satis¬es µ2 = 1, this condition holds if and only if b is symmetric or skew-symmetric.

This shows that F -linear involutions correspond to symmetric or skew-sym-

metric bilinear forms under the bijection above.

The involution σb associated to a nonsingular symmetric or skew-symmetric

bilinear form b under the correspondence of the theorem is called the adjoint in-

volution with respect to b. Our aim in this ¬rst chapter is to give an analogous

interpretation of involutions on arbitrary central simple algebras in terms of hermit-

ian forms on vector spaces over skew ¬elds. We ¬rst review basic notions concerning

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

use in §??. In §?? we present the basic de¬nitions concerning involutions on cen-

tral simple algebras. We distinguish three types of involutions, according to the

type of pairing they are adjoint to over an algebraic closure: involutions which are

adjoint to symmetric (resp. alternating) bilinear forms are called orthogonal (resp.

symplectic); those which are adjoint to hermitian forms are called unitary. Invo-

lutions of the ¬rst two types leave the center invariant; they are called involutions

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

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

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

in §??.

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

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

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

§1. CENTRAL SIMPLE ALGEBRAS 3

transfer for hermitian forms and extensions of involutions are also discussed in this

section.

When F has characteristic 2, it is important to distinguish between bilinear

and quadratic forms. Every quadratic form de¬nes (by polarization) an alternating

form, but not conversely since a given alternating form is the polar of various quad-

ratic forms. The quadratic pairs introduced in the ¬nal section may be regarded

as twisted analogues of quadratic forms up to a scalar factor in the same way that

involutions may be thought of as twisted analogues of nonsingular symmetric or

skew-symmetric bilinear forms. If the characteristic is di¬erent from 2, every or-

thogonal involution determines a unique quadratic pair since a quadratic form is

uniquely determined by its polar bilinear form. By contrast, in characteristic 2 the

involution associated to a quadratic pair is symplectic since the polar of a quadratic

form is alternating, and the quadratic pair is not uniquely determined by its asso-

ciated involution. Quadratic pairs play a central rˆle in the de¬nition of twisted