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Therefore, letting Z be the center of the Cli¬ord algebra C(A , σ ) and

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 is based on unpublished notes of Rost (to appear).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,

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