π : Skew(A , σ ) ’ Z

448 VII. GALOIS COHOMOLOGY

the generalized pfa¬an of (A , σ ) (see (??)), we have π(s )2 = z 2 . It follows that

1 ’1

2 1+z π(s ) is a nonzero central idempotent of C(A , σ ). Set

E(s, z) = 1 + z ’1 π(s ) · C(A , σ ),

a central simple F -algebra with involution of the ¬rst kind of degree 2deg A’1 . We

have

C(A , σ ) = E(s, z) — E(s, ’z)

and it is shown in Garibaldi-Tignol-Wadsworth [?, Proposition 4.6] that

δ 1 (s, z)/≈ = E(s, z) ∈ 2 Br(F ).

In particular, the images under δ 1 of (s, z) and (s, ’z) are the two components of

C(A , σ ). By (??), it follows that E(s, z) E(s, ’z) = [A], hence the Brauer class

E(s, z) is uniquely determined by s ∈ Sym(A, σ)— up to a factor [A]. This is the

invariant of hermitian forms de¬ned by Bartels [?]. Explicitly, let D be a division

F -algebra with involution of the ¬rst kind and let h be a nonsingular hermitian or

skew-hermitian form on a D-vector space V such that the adjoint involution σ = σh

on A = EndD (V ) is orthogonal. Let S = {1, [D]} ‚ Br(F ). To every nonsingular

form h on V of the same type and discriminant as h, Bartels attaches an invariant

c(h, h ) in the factor group Br(F )/S as follows: since h and h are nonsingular and

of the same type, there exists s ∈ Sym(A, σ)— such that

h (x, y) = h s’1 (x), y for all x, y ∈ V .

We have NrdA (s) ∈ F —2 since h and h have the same discriminant. We may then

set

c(h, h ) = E(s, z) + S = E(s, ’z) + S ∈ Br(F )/S

where z ∈ F — is such that z 2 = NrdA (s).

The Tits class t(B, „ ) ∈ H 2 (F, µn[K] ) for (B, „ ) a central simple F -algebra with

unitary involution of degree n with center K was de¬ned by Qu´guiner [?, §3.5.2], [?,

e

§2.2], who called it the determinant class. (Actually, Qu´guiner™s determinant class

e

di¬ers from the Tits class by a factor which depends only on n.)

All the material in §?? is based on unpublished notes of Rost (to appear). See

Serre™s Bourbaki talk [?].

Finally, we note that getting information for special ¬elds F on the set H 1 (F, G),

for G an algebraic group, gives rise to many important questions which are not ad-

dressed here. Suppose that G is semisimple and simply connected. If F is a p-adic

¬eld, then H 1 (F, G) is trivial, as was shown by Kneser [?]. If F is a number ¬eld,

the “Hasse principle” due to Kneser, Springer, Harder and Chernousov shows that

the natural map H 1 (F, G) ’ v H 1 (Fv , G) is injective, where v runs over the real

places of F and Fv is the completion of F at v. We refer to Platonov-Rapinchuk

[?, Chap. 6] for a general survey. If F is a perfect ¬eld of cohomological dimension

at most 2 and G is of classical type, Bayer-Fluckiger and Parimala [?] have shown

that H 1 (F, G) is trivial, proving Serre™s “Conjecture II” [?, Chap. III, §3] for clas-

sical groups. Analogues of the Hasse principle for ¬elds of virtual cohomological

dimension 1 or 2 were obtained by Ducros [?], Scheiderer [?] and Bayer-Fluckiger-

Parimala [?].

CHAPTER VIII

Composition and Triality

The main topic of this chapter is composition algebras. Of special interest from

the algebraic group point of view are symmetric compositions. In dimension 8 there

are two such types: Okubo algebras, related to algebras of degree 3 with unitary in-

volutions (type A2 ), and para-Cayley algebras related to Cayley algebras (type G 2 ).

The existence of these two types is due to the existence of inequivalent outer actions

of the group Z/3Z on split simply connected simple groups of type D4 (“triality”

for Spin8 ), for which the ¬xed elements de¬ne groups of type A2 , resp. G2 . Triality

is de¬ned here through an explicit computation of the Cli¬ord algebra of the norm

of an 8-dimensional symmetric composition. As a step towards exceptional simple

Jordan algebras, we introduce in the last section twisted compositions, generaliz-

ing a construction of Springer. The corresponding group of automorphisms is the

semidirect product Spin8 S3 .

§32. Nonassociative Algebras

In this and the following chapter, by an F -algebra A we mean (unless further

speci¬ed) a ¬nite dimensional vector space over F equipped with an F -bilinear mul-

tiplication m : A — A ’ A. We shall use di¬erent notations for the multiplication:

m(x, y) = xy = x y = x y. We do not assume in general that the multiplication

has an identity. An algebra with identity 1 is unital. An ideal of A is a subspace M

such that ma ∈ M and am ∈ M for all m ∈ M , a ∈ A. The algebra A is simple

if the multiplication on A is not trivial (i.e., there are elements a, b of A such that

ab = 0) and 0, A are the only ideals of A. The multiplication algebra M (A) is the

subalgebra of EndF (A) generated by left and right multiplications with elements

of A. The centroid Z(A) is the centralizer of M (A) in EndF (A):

Z(A) = { f ∈ EndF (A) | f (ab) = f (a)b = af (b) for a, b ∈ A }

and A is central if F ·1 = Z(A). If Z(A) is a ¬eld, the algebra A is central over Z(A).

Observe that a commutative algebra may be central if it is not associative.

The algebra A is strictly power-associative if, for every R ∈ Alg F , the R-

subalgebra of AR generated by one element is associative. We then write an for

nth -power of a ∈ A, independently of the notation used for the multiplication of

A. Examples are associative algebras, Lie algebras (trivially), alternative algebras,

i.e., such that

x(xy) = (xx)y and (yx)x = y(xx)

for all x, y ∈ A, and Jordan algebras in characteristic di¬erent from 2 (see Chap-

ter ??). Let A be strictly power-associative and unital. Fixing a basis (ui )1¤i¤r

of A and taking indeterminates {x1 , . . . , xr } we have a generic element

x= xi ui ∈ A — F (x1 , . . . , xr )

449

450 VIII. COMPOSITION AND TRIALITY

and there is a unique monic polynomial

PA,x (X) = X m ’ s1 (x)X m’1 + · · · + (’1)m sm (x) · 1

of least degree which has x as a root. This is the generic minimal polynomial of A.

The coe¬cients si are homogeneous polynomials in the xi ™s, s1 = TA is the generic

trace, sm = NA the generic norm and m is the degree of A. It is convenient to

view F as an algebra of degree n for any n such that char F does not divide n; the

corresponding polynomial is PF,x (X) = (X · 1 ’ x)n . In view of McCrimmon [?,

Theorem 4, p. 535] we have

NA (X · 1 ’ x) = PA,x (X)

for a strictly power-associative algebra A. For any element a ∈ A we can special-

ize the generic minimal polynomial PA,x (X) to a polynomial PA,a (X) ∈ F [X] by

∼

writing a = i ai ui and substituting ai for xi . Let ± : A ’ A be an isomor-

’

phism of unital algebras. Uniqueness of the generic minimal polynomial implies

that PA ,±(x) = PA,x , in particular TA ±(x) = TA (x) and NA ±(x) = NA (x).

(32.1) Examples. (1) We have PA—B,(x,y) = PA,x · PB,y for a product algebra

A — B.

(2) For a central simple associative algebra A the generic minimal polynomial is

the reduced characteristic polynomial and for a commutative associative algebra it

is the characteristic polynomial.

(3) For a central simple algebra with involution we have a generic minimal poly-

nomial on the Jordan algebra of symmetric elements depending on the type of

involution:

An : If J = H(B, „ ), where (B, „ ) is central simple of degree n + 1 with a

unitary involution over a quadratic ´tale F -algebra K, PJ,a (X) is the restriction of

e

the reduced characteristic polynomial of B to H(B, „ ). The coe¬cients of PJ,a (X),

a priori in K, actually lie in F since they are invariant under ι. The degree of J is

the degree of B.

Bn and Dn : For J = H(A, σ), A central simple over F with an orthogonal

involution of degree 2n + 1, or 2n, PJ,a (X) is the reduced characteristic polynomial,

so that the degree of J is the degree of A.

Cn : For J = H(A, σ), A central simple of degree 2n over F with a symplec-

tic involution, PJ,a (X) is the polynomial Prpσ,a of (??). Here the degree of J

1

is 2 deg(A).

We now describe an invariance property of the coe¬cients si (x). Let s ∈ S(A— )

be a polynomial function on A, let d : A ’ A be an F -linear transformation, and

let F [µ] be the F -algebra of dual numbers. We say that s is Lie invariant under d

if

s a + µd(a) = s(a)

holds in A[µ] = A — F [µ] for all a ∈ A. The following result is due to Tits [?]:

(32.2) Proposition. The coe¬cients si (x) of the generic minimal polynomial of

a strictly power-associative F -algebra A are Lie invariant under all derivations d

of A.

§32. NONASSOCIATIVE ALGEBRAS 451

Proof : Let F be an arbitrary ¬eld extension of F . The extensions of the forms

si and d to AF will be denoted by the same symbols si and d. We de¬ne forms

{a, b}i and µi (a, b) by

(a + µb)i = ai + µ{a, b}i and si (a + µb) = si (a) + µµi (a, b).

It is easy to see (for example by induction) that d(ai ) = {a, d(a)}i for any deriva-

tion d. We obtain

0 = PA[µ],a+µb (a + µb)

n

n

(’1)i si (a) + µµi (a, b) an’i + µ{a, b}n’i ,

= a + µ{a, b}n +

i=1

where n is the degree of the generic minimal polynomial, so that

n n

i

(’1)i µi (a, b)an’i = 0.

(1) {a, b}n + (’1) si (a){a, b}n’i +

i=1 i=1

On the other hand we have

n

(’1)i si (a){a, d(a)}n’i = 0.

(2) d PA,a (a) = {a, d(a)}n +

i=1

Setting b = d(a) in (??) and subtracting (??) gives

n

(’1)i µi a, d(a) an’i = 0.

i=1

If a is generic over F , it does not satisfy any polynomial identity of degree n ’ 1.

Thus µi a, d(a) = 0. This is the Lie invariance of the si under the derivation d.

(32.3) Corollary. The identity s1 (a·b) = s1 (b·a) holds for any associative algebra

and the identity s1 a q (b q c) = s1 (a q b) q c holds for any Jordan algebra over a

¬eld of characteristic not 2.

Proof : The maps da (b) = a · b ’ b · a, resp. db,c (a) = a q (b q c) ’ (a q b) q c are

derivations of the corresponding algebras (see for example Schafer [?, p. 92] for the

last claim).

An algebra A is separable if A — F is a direct sum of simple ideals for every ¬eld

extension F of F . The following criterion (??) for separability is quite useful; it

applies to associative algebras and Jordan algebras in view of Corollary (??) and to

alternative algebras (see McCrimmon [?, Theorem 2.8]). For alternative algebras

of degree 2 and 3, which are the cases we shall consider, the lemma also follows

from (??) and Proposition (??). We ¬rst give a de¬nition: a symmetric bilinear

form T on an algebra A is called associative or invariant if

T (xy, z) = T (x, yz) for x, y, z ∈ A.

(32.4) Lemma (Dieudonn´). Let A be a strictly power-associative algebra with

e

generic trace TA . If the bilinear form T : (x, y) ’ TA (xy) is symmetric, nonsingular

and associative, then A is separable.

452 VIII. COMPOSITION AND TRIALITY

Proof : This is a special case of a theorem attributed to Dieudonn´, see for example

e

Schafer [?, p. 24]. Let I be an ideal. The orthogonal complement I ⊥ of I (with

respect to the bilinear form T ) is an ideal since T is associative. For x, y ∈ J = I©I ⊥

and z ∈ A, we have T (xy, z) = T (x, yz) = 0, hence J 2 = 0 and elements of J

are nilpotent. Nilpotent elements have generic trace 0 (see Jacobson [?, p. 226,

Cor. 1(2)]); thus T (x, z) = TA (xz) = 0 for all z ∈ A and x ∈ J. This implies J = 0

and A = I • I ⊥ . It then follows that A (and A — F for all ¬eld extensions F /F ) is

a direct sum of simple ideals, hence separable.

A converse of Lemma (??) also holds for associative algebras, alternative alge-

bras and Jordan algebras; a proof can be obtained by using Theorems (??) and (??).

Alternative algebras. The structure of ¬nite dimensional separable alterna-

tive algebras is similar to that of ¬nite dimensional separable associative algebras:

(32.5) Theorem. (1) Any separable alternative F -algebra is the product of simple

alternative algebras whose centers are separable ¬eld extensions of F .

(2) A central simple separable alternative algebra is either associative central simple

or is a Cayley algebra.

Reference: A reference for (??) is Schafer [?, p. 58]; (??) is a result due to Zorn,

see for example Schafer [?, p. 56]. We shall only use Theorem (??) for algebras of

degree 3. A description of Cayley algebras is given in the next section.

For nonassociative algebras the associator

(x, y, z) = (xy)z ’ x(yz)

is a useful notion. Alternative algebras are de¬ned by the identities

(x, x, y) = 0 = (x, y, y).

Linearizing we obtain

(32.6) (x, y, z) + (y, x, z) = 0 = (x, y, z) + (x, z, y),

i.e., in an alternative algebra the associator is an alternating function of the three

variables. The following result is essential for the study of alternative algebras:

(32.7) Theorem (E. Artin). Any subalgebra of an alternative algebra A generated

by two elements is associative.

Reference: See for example Schafer [?, p. 29] or Zorn [?].

Thus we have NA (xy) = NA (x)NA (y) and TA (xy) = TA (yx) for x, y ∈ A, A a

alternative algebra, since both are true for an associative algebra (see Jacobson [?,

Theorem 3, p. 235]). The symmetric bilinear form T (x, y) = TA (xy) is the bilinear

trace form of A.

In the next two sections separable alternative F -algebras of degree 2 and 3

are studied in detail. We set Sepalt n (m) for the groupoid of separable alternative

F -algebras of dimension n and degree m with isomorphisms as morphisms.

§33. COMPOSITION ALGEBRAS 453

§33. Composition Algebras

33.A. Multiplicative quadratic forms. Let C be an F -algebra with multi-

plication (x, y) ’ x y (but not necessarily with identity). We say that a quadratic

form q on C is multiplicative if

(33.1) q(x y) = q(x)q(y)

for all x, y ∈ C. Let bq (x, y) = q(x + y) ’ q(x) ’ q(y) be the polar of q and let

C ⊥ = { z ∈ C | bq (z, C) = 0 }.

(33.2) Proposition. The space C ⊥ is an ideal in C.

Proof : This is clear if q = 0. So let x ∈ C be such that q(x) = 0. Linearizing (??)

we have

bq (x y, x z) = q(x)bq (y, z).

Thus x y ∈ C ⊥ implies y ∈ C ⊥ . It follows that the kernel of the composed map

(of F -spaces)

p

φx : C ’x C ’ C/C ⊥ ,

’’

where x (y) = x y and p is the projection, is contained in C ⊥ . By dimension

count it must be equal to C ⊥ , so x C ⊥ ‚ C ⊥ and similarly C ⊥ x ‚ C ⊥ . Since

C ⊥ — L = (C — L)⊥ for any ¬eld extension L/F , the claim now follows from the

next lemma.

(33.3) Lemma. Let q : V ’ F be a nontrivial quadratic form. There exists a ¬eld

extension L/F such that V — L is generated as an L-linear space by anisotropic

vectors.

Proof : Let n = dimF V and let L = F (t1 , . . . , tn ). Taking n generic vectors in

V — L gives a set of anisotropic generators of V — L.

Let

R(C) = { z ∈ C ⊥ | q(z) = 0 }.

(33.4) Proposition. If (C, q) is a multiplicative quadratic form, then either C ⊥ =

R(C) or char F = 2 and C = C ⊥ .

Proof : We show that q|C ⊥ = 0 implies that char F = 2 and C = C ⊥ . If char F = 2,

then q(x) = 1 bq (x, x) = 0 for x ∈ C ⊥ , hence q|C ⊥ = 0 already implies char F =

2

2. To show that C = C ⊥ we may assume that F is algebraically closed. Since

char F = 2 the set R(C) is a linear subspace of C ⊥ ; by replacing C by C/R(C)

we may assume that R(C) = 0. Then q : C ⊥ ’ F is injective and semilinear with

∼

respect to the isomorphism F ’ F , x ’ x2 . It follows that dimF C ⊥ = 1; let

’

u ∈ C ⊥ be a generator, so that q(u) = 0. For x ∈ C we have x u ∈ C ⊥ by

Proposition (??) and we de¬ne a linear form f : C ’ F by x u = f (x)u. Since

q(x)q(u) = q(x u) = q f (x)u = f (x)2 q(u)

q(x) = f (x)2 and the polar bq (x, y) is identically zero. This implies C = C ⊥ , hence

the claim.

(33.5) Example. Let (C, q) be multiplicative and regular of odd rank (de¬ned

on p. ??) over a ¬eld of characteristic 2. Since dimF C ⊥ = 1 and R(C) = 0,

Proposition (??) implies that C = C ⊥ and C is of dimension 1.

454 VIII. COMPOSITION AND TRIALITY

(33.6) Corollary. The set R(C) is always an ideal of C and q induces a multi-

plicative form q on C = C/R(C) such that either

(1) (C, q) is regular, or

(2) char F = 2 and C is a purely inseparable ¬eld extension of F of exponent 1 of

dimension 2n for some n and q(x) = x2 .

Proof : If R(C) = C ⊥ , R(C) is an ideal in C by Proposition (??) and the polar of q

is nonsingular. Then (??) follows from Corollary (??) except when dimF C = 1 in

characteristic 2. If R(C) = C ⊥ , then char F = 2 and C = C ⊥ by Proposition (??).

It follows that the polar bq (x, y) is identically zero, q : C ’ F is a homomorphism

and R(C) is again an ideal. For the description of the induced form q : C ’ F

we follow Kaplansky [?, p. 95]: the map q : C ’ F is an injective homomorphism,

thus C is a commutative associative integral domain. Moreover, for x such that

q(x) = 0, x2 /q(x) is an identity element 1 with q(1) = 1 and C is even a ¬eld. Since

q(» · 1) = »2 · 1 for all » ∈ F , we have

q(x2 ) = q q(x) · 1

(33.7)

for all x ∈ C. Let C0 = q(C), let x0 = q(x), and let be the induced multiplication.

0

It follows from (??) that

x0 x0 = q(x) · 1.

0

If dimF C = 1 we have the part of assertion (??) in characteristic 2 which was

left over. If dimF C > 1, then C is a purely inseparable ¬eld extension of F of

exponent 1, as claimed in (??).

(33.8) Remark. In case (??) of (??) C has dimension 1, 2, 4 or 8 in view of the

later Corollary (??).

33.B. Unital composition algebras. Let C be an F -algebra with identity

and multiplication (x, y) ’ x y and let n be a regular multiplicative quadratic form

on C. We call the triple (C, , n) a composition algebra. In view of Example (??),

the form 1 is the unique regular multiplicative quadratic form of odd dimension.

Thus it su¬ces to consider composition algebras with nonsingular forms in even

dimension ≥ 2. We then have the following equivalent properties:

(33.9) Proposition. Let (C, ) be a unital F -algebra with dimF C ≥ 2. The fol-

lowing properties are equivalent:

(1) There exists a nonsingular multiplicative quadratic form n on C.

(2) C is alternative separable of degree 2.

(3) C is alternative and has an involution π : x ’ x such that

x + x ∈ F · 1, n(x) = x x ∈ F · 1,

and n is a nonsingular quadratic form on C.

Moreover, the quadratic form n in (??) and the involution π in (??) are uniquely

determined by (C, ).

Proof : (??) ’ (??) Let (C, , n) be a composition algebra. To show that C is

alternative we reproduce the proof of van der Blij and Springer [?], which is valid

for any characteristic. Let

bn (x, y) = n(x + y) ’ n(x) ’ n(y)

§33. COMPOSITION ALGEBRAS 455

be the polar of n. The following formulas are deduced from n(x y) = n(x)n(y) by

linearization:

bn (x y, x z) = n(x)bn (y, z)

bn (x y, u y) = n(y)bn (x, u)

and

(33.10) bn (x z, u y) + bn (x y, u z) = bn (x, u)bn (y, z).

We have n(1) = 1. By putting z = x and y = 1 in (??), we obtain

bn x2 ’ bn (1, x)x + n(x) · 1, u = 0

for all u ∈ C. Since n is nonsingular any x ∈ C satisfy the quadratic equation

x2 ’ bn (1, x)x + n(x) · 1 = 0.

(33.11)

Hence C is of degree 2 and C is strictly power-associative. Furthermore b n (1, x)

is the trace TC (x) and n is the norm NC of C (as an algebra of degree 2). Let

x = TC (x) · 1 ’ x. It follows from (??) that

x x = x x = n(x) · 1

and it is straightforward to check that

x=x and 1 = 1.

Hence x ’ x is bijective. We claim that

(33.12) bn (x y, z) = bn (y, x z) = bn (x, z y).

The ¬rst formula follows from

bn (x y, z) + bn (y, x z) = bn (x, 1)bn (z, y) = TC (x)bn (z, y),

which is a special case of (??), and the proof of the second is similar. We further

need the formulas

x (x y) = n(x)y and (y x) x = n(x)y.

For the proof of the ¬rst one, we have

bn x (x y), z = bn (x y, x z) = bn n(x)y, z for z ∈ C.

The proof of the other one is similar. It follows that x (x y) = (x x) y.

Therefore

x (x y) = x TC (x)y ’ x y = TC (x)x ’ x x y = (x x) y

and similarly (y x) x = y (x x). This shows that C is an alternative algebra.

To check that the bilinear trace form T (x, y) = TC (x y) is nonsingular, we ¬rst

verify that π satis¬es π(x y) = π(y) π(x), so that π is an involution of C. By

linearizing the generic polynomial (??) we obtain

(33.13) x y + y x ’ TC (y)x + TC (x)y + bn (x, y)1 = 0.

On the other hand, putting u = z = 1 in (??) we obtain

bn (x, y) = TC (x)TC (y) ’ TC (x y)

(which shows that T (x, y) = TC (x y) is a symmetric bilinear form). By substituting

this in (??), we ¬nd that

TC (x) ’ x TC (y) ’ y = TC (y x) ’ y x,

456 VIII. COMPOSITION AND TRIALITY

thus π(x y) = π(y) π(x). It now follows that

TC (x y) = x y + x y = y x + x y = x y + y x = bn (x, y),

hence the bilinear form T is nonsingular if n is nonsingular. Furthermore TC (x y) =

bn (x, y) and (??) imply that

(33.14) T (x y, z) = T (x, y z) for x, y, z ∈ C,