modules.

12. Let (A, σ) be a central simple algebra over F with a symplectic involution σ.

Show that the map

Symd(A, σ)— /∼ = H 1 F, Sp(A, σ) ’ H 1 F, SL1 (A) = F — / Nrd(A— )

induced by the inclusion Sp(A, σ) ’ SL1 (A) takes a ∈ Sym(A, σ)— to its

pfa¬an norm NrpA (a) modulo Nrd(A— ).

13. Let A be a central simple algebra over F . For any c ∈ F — write Xc for the set

of all x ∈ A— such that Nrd(x) = c. Prove that

sep

(a) Xc is a SL1 (Asep )-torsor.

(b) Any SL1 (Asep )-torsor is isomorphic to Xc for some c.

(c) Xc Xd if and only if cd’1 ∈ Nrd(A— ).

14. Describe H 1 F, Spin(V, q) in terms of twisted forms of tensors.

15. Let (A, σ, f ) be a central simple F -algebra with quadratic pair of even degree 2n

over an arbitrary ¬eld F . Let Z be the center of the Cli¬ord algebra C(A, σ, f )

and let „¦(A, σ, f ) be the extended Cli¬ord group.

(a) Show that the connecting map

δ 1 : H 1 F, PGO+ (A, σ, f ) ’ H 2 F, RZ/F (Gm,Z ) = Br(Z)

EXERCISES 445

in the cohomology sequence associated to

χ

1 ’ RZ/F (Gm,Z ) ’ „¦(A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1

’

’1

maps the 4-tuple (A , σ , f , •) to C(A , σ , f )—Z Z C(A, σ, f ) , where

the tensor product is taken with respect to •.

(b) Show that the multiplication homomorphism

Spin(A, σ, f ) — RZ/F (Gm,Z ) ’ „¦(A, σ, f )

induces an isomorphism

„¦(A, σ, f ) Spin(A, σ, f ) — RZ/F (Gm,Z ) /C

where C is isomorphic to the center of Spin(A, σ, f ). Similarly, show that

GO+ (A, σ, f ) O+ (A, σ, f ) — Gm /µ2

where µ2 is embedded diagonally in the product.

(c) Assume that n is even. Let ± : „¦(A, σ, f ) ’ GO+ (A, σ, f ) be the ho-

momorphism which, under the isomorphism in (??), is the vector rep-

resentation χ on Spin(A, σ, f ) and the norm map on RZ/F (Gm,Z ). By

relating via ± the exact sequence in (??) to a similar exact sequence for

GO+ (A, σ, f ), show that for all 4-tuple (A , σ , f , •) representing an ele-

ment of H 1 F, PGO+ (A, σ, f ) ,

’1

= [A ][A]’1

NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) in Br(F ).

In particular, NZ/F C(A, σ, f ) = [A].

Similarly, using the homomorphism „¦(A, σ, f ) ’ RZ/F (Gm,Z ) which is

trivial on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that

’1 2

C(A , σ , f ) —Z Z C(A, σ, f ) = 1.

2

In particular, C(A, σ, f ) = 1. (Compare with (??).)

(d) Assume that n is odd. Let G = O+ (A, σ, f ) — RZ/F (Gm,Z ) /µ2 . Using

the homomorphism ± : „¦(A, σ, f ) ’ G which is the vector representation

χ on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that for all

4-tuple (A , σ , f , •) representing an element of H 1 F, PGO+ (A, σ, f ) ,

’1 2

= [AZ ][AZ ]’1

C(A , σ , f ) —Z Z C(A, σ, f ) in Br(Z).

2

In particular, C(A, σ, f ) = [AZ ].

Using the character of „¦(A, σ, f ) which is trivial on Spin(A, σ, f ) and is

the norm on RZ/F (Gm,Z ), show that

’1

NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) = 1.

In particular, NZ/F C(A, σ, f ) = 1. (Compare with (??).)

16. (Qu´guiner [?]) Let (B, „ ) be a central simple F -algebra with unitary involution

e

of degree n. Let K be the center of B and let „ = Int(u) —¦ „ for some unit u ∈

Sym(B, „ ). Assume that char F does not divide n. Show that the Tits classes

t(B, „ ) and t(B, „ ) in H 2 (F, µn[K] ) are related by t(B, „ ) = t(B, „ ) + ζK ∪

NrdB (u) where ζK is the nontrivial element of H 1 (F, Z[K] ) and NrdB (u) =

NrdB (u) · F —n ∈ F — /F —n = H 1 (F, µn ). (Compare with (??).)

446 VII. GALOIS COHOMOLOGY

Notes

§??. The concept of a nonabelian cohomology set H 1 (“, A) has its origin in the

theory of principal homogeneous spaces (or torsors) due to Grothendieck [?], see

also Frenkel [?] and Serre [?]. The ¬rst steps in the theory of principal homogeneous

spaces attached to an algebraic group (in fact a commutative group variety) are

found in Weil [?].

Galois descent was implicitly used by Chˆtelet [?], in the case where A is an

a

elliptic curve (see also [?]). An explicit formulation (and proof) of Galois descent

in algebraic geometry was ¬rst given by Weil [?]. The idea of twisting the action

of the Galois group using automorphisms appears also in this paper, see Weil™s

commentaries in [?, pp. 543“544].

No Galois cohomology appears in the paper [?] on principal homogeneous spaces

mentioned above. The fact that Weil™s group of classes of principal homogeneous

spaces for a commutative group variety A over a ¬eld F stands in bijection with

the Galois 1-cohomology set H 1 (F, A) was noticed by Serre; details are given in

Lang and Tate [?], see also Tate™s Bourbaki talk [?].

The ¬rst systematic treatment of Galois descent, including nonabelian cases

(linear groups, in particular PGLn with application to the Brauer group), appeared

in Serre™s book “Corps locaux” [?], which was based on a course at the Coll`ge de

e

France in 1958/59. Twisted forms of algebraic structures viewed as tensors are

mentioned as examples. Applications to quadratic forms are given in Springer [?].

Another early application is the realization by Weil [?], following an observation of

“un amateur de cocycles tr`s connu”33 , of Siegel™s idea that classical groups can be

e

described as automorphism groups of algebras with involution (Weil [?, pp. 548“

549]).

Since then this simple but very useful formalism found many applications. See

the latest revised and completed edition of the Lecture Notes of Serre [?] and his

Bourbaki talk [?] for more information and numerous references. A far-reaching

generalization of nonabelian Galois cohomology, which goes beyond Galois exten-

sions and applies in the setting of schemes, was given by Grothendieck [?].

Our presentation in this section owes much to Serre™s Lecture Notes [?] and to

the paper [?] of Borel and Serre. The technique of changing base points by twisting

coe¬cients in cohomology, which we use systematically, was ¬rst developed there.

Note that the term “co-induced module” is used by Serre [?] and by Brown [?] for

the modules which we call “induced”, following Serre [?].

§??. Lemma (??), the so-called “Shapiro lemma”, was independently proved

by Eckmann [?, Theorem 4], D. K. Faddeev [?], and Arnold Shapiro. Shapiro™s

proof appears in Hochschild-Nakayama [?, Lemma 1.1].

Besides algebras and quadratic forms, Severi-Brauer varieties also have a nice

interpretation in terms of Galois cohomology: the group scheme PGL n occurs not

only as the automorphism group of a split central simple algebra of degree n, but

also as the automorphism group of the projective space Pn’1 . The Severi-Brauer

variety SB(A) attached to a central simple algebra A is a twisted form of the

projective space, given by the cocycle of A (see Artin [?]).

For any quadratic space (V, q) of even dimension 2n, the Cli¬ord functor de¬nes

a homomorphism C : PGO(V, q) ’ Autalg C0 (V, q) (see (??)). The induced map

in cohomology C 1 : H 1 F, PGO(V, q) ’ H 1 F, Autalg C0 (V, q) associates to

33 also referred to as “Mr. P. (the famous winner of many cocycle races)”

NOTES 447

every central simple F -algebra with quadratic pair of degree 2n a separable F -

algebra of dimension 22n’1 ; this is the de¬nition of the Cli¬ord algebra of a central

simple algebra with quadratic pair by Galois descent.

§??. Although the cyclic algebra construction is classical, the case considered

here, where L is an arbitrary Galois Z/nZ-algebra, is not so common in the lit-

erature. It can be found however in Albert [?, Chapter VII]. Note that if L is a

¬eld, its Galois Z/nZ-algebra structure designates a generator of the Galois group

Gal(L/F ).

The exact sequence (??) was observed by Arason-Elman [?, Appendix] and by

Serre [?, Chapter I, §2, Exercise 2]. (This exercise is not in the 1973 edition.) The

special case where M = µ2 (Fsep ) (Corollary (??)) plays a crucial rˆle in Arason [?].

o

The cohomological invariants f1 , g2 , f3 for central simple F -algebras with

unitary involution of degree 3 are discussed in Haile-Knus-Rost-Tignol [?, Corol-

lary 32]. It is also shown in [?] that these invariants are not independent and that

the invariant g2 (B, „ ) gives information on the ´tale F -subalgebras of B. To state

e

precise results, recall from (??) that cubic ´tale F -algebras with discriminant ∆

e

1

are classi¬ed by the orbit set H (F, A3[∆] )/S2 . Suppose char F = 2, 3 and let

F (ω) = F [X]/(X 2 + X + 1), so that µ3 = A3[F (ω)] . Let (B, „ ) be a central simple

F -algebra with unitary involution of degree 3 and let L be a cubic ´tale F -algebra

e

1

with discriminant ∆. Let K be the center of B and let cL ∈ H (F, A3[∆] ) be a co-

homology class representing L. The algebra B contains a subalgebra isomorphic to

L if and only if g2 (B, „ ) = cL ∪d for some d ∈ H 1 (F, A3[K—F (ω)—∆] ). (Compare with

Proposition (??).) If this condition holds, then B also contains an ´tale subalgebra

e

L with associated cohomology class d (hence with discriminant K — F (ω) — ∆).

Moreover, there exists an involution „ such that Sym(B, „ ) contains L and L .

See [?, Proposition 31].

§??. Let (A, σ) be a central simple algebra with orthogonal involution of even

degree over a ¬eld F of characteristic di¬erent from 2. The connecting homomor-

phism

δ 1 : H 1 F, O+ (A, σ) ’ H 2 (F, µ2 ) = 2 Br(F )

in the cohomology sequence associated to the exact sequence

1 ’ µ2 ’ Spin(A, σ) ’ O+ (A, σ) ’ 1

is described in Garibaldi-Tignol-Wadsworth [?]. Recall from (??) the bijection

H 1 F, O+ (A, σ) SSym(A, σ)— /≈.

For (s, z) ∈ SSym(A, σ)— , consider the algebra A = M2 (A) EndA (A2 ) with the

involution σ adjoint to the hermitian form 1, ’s’1 , i.e.,

’σ(c)s’1

a b σ(a)

σ = for a, b, c, d ∈ A.

sσ(d)s’1

c d ’sσ(b)

01

∈ A . We have s ∈ Skew(A , σ ) and NrdA (s ) = NrdA (s) = z 2 .

Let s =

s0

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,

hence, by Lemma (??), C is separable.

(??) ’ (??) Let

X 2 ’ TC (x)X + NC (x) · 1

be the generic minimal polynomial of C. We de¬ne π(x) = TC (x) ’ x and we put

n = NC ; then

x π(x) = π(x) x = n(x) · 1 ∈ F · 1

follows from x2 ’ x + π(x) x + n(x) · 1 = 0. Since bn (x, y) = T (x, y) and C is

separable, n is nonsingular. The fact that π is an involution follows as in the proof

of (??) ’ (??).

(??) ’ (??) The existence of an involution with the properties given in (??)

implies that C admits a generic minimal polynomial as given in (??). Since C is

alternative we have

x (x y) = n(x)y = (y x) x

Using that the associator (x, y, z) is an alternating function we obtain

n(x y) = (x y) (x y) = (x y) (y x)

= (x y) y x ’ (x y, y, x) = n(x)n(y) ’ (x, x y, y)

= n(x)n(y) ’ x (x y) y+x (x y) y

= n(x)n(y) ’ n(x)n(y) + n(x)n(y) = n(x)n(y)

so that n is multiplicative.

The fact that n and π are uniquely determined by (C, ) follows from the

uniqueness of the generic minimal polynomial.

Let Comp m be the groupoid of composition algebras of dimension m with iso-

morphisms as morphisms and let Comp + be the groupoid of unital composition

m

algebras with isomorphisms as morphisms.

(33.15) Corollary. The identity map C ’ C induces an isomorphism of groupoids

Comp+ ≡ Sepalt m (2) for m ≥ 2.

m

33.C. Hurwitz algebras. Let (B, π) be a unital F -algebra of dimension m

with an involution π such that

x + π(x) ∈ F · 1 and x π(x) = π(x) x ∈ F · 1

for all x ∈ B. Assume further that the quadratic form n(x) = x π(x) is nonsingular.

Let » ∈ F — . The Cayley-Dickson algebra CD(B, ») associated to (B, π) and » is

the vector space

CD(B, ») = B • vB

where v is a new symbol, endowed with the multiplication

(a + vb) (a + vb ) = a a + »b π(b) + v π(a) b + a b,

§33. COMPOSITION ALGEBRAS 457

for a, a , b and b ∈ A. In particular CD(B, ») contains B as a subalgebra and

v 2 = ».

Further we set

n(a + vb) = n(a) ’ »n(b) and π(a + vb) = π(a) ’ vb.

(33.16) Lemma. The algebra C = CD(B, ») is an algebra with identity 1 + v0

and π is an involution such that

TC (x) = x + π(x) ∈ F · 1, NC (x) = n(x) = x π(x) = π(x) x ∈ F · 1.

The algebra B is contained in CD(B, ») as a subalgebra and

(1) C is alternative if and only if B associative,

(2) C is associative if and only if B is commutative,

(3) C is commutative if and only if B = F .

Proof : The fact that C = CD(B, ») is an algebra follows immediately from the

de¬nition of C. Identifying v with v1 we have vB = v B and we view v as an

element of C. We leave the “if” directions as an exercise. The assertions about T C

and NC are easy to check, so that elements of C satisfy

x2 ’ TC (x)x + NC (x)1 = 0

and C is of degree 2. Thus, if C is alternative, n = NC is multiplicative by

Proposition (??). We have

n (a + v b) (c + v d) = n a c + »d b + v (c b + a d)

= n(a c + »d b) ’ »n(c b + a d),

on the other hand,

n (a + v b) (c + v d) = n(a + v b)n(c + v d)

= n(a) ’ »n(b) n(c) ’ »n(d) .

Comparing both expressions and using once more that n is multiplicative, we obtain

bn (a c, »d b) + n(v)bn (c b, a d) = 0

or, since n(v) = ’»,

bn (a c, d b) = bn (a d, c b)

so that

bn (a c) b, d = bn a (c b), d

for all a, b, c, d ∈ B by (??). Thus we obtain (a c) b = a (c b) and B is

associative. If C is associative, we have (v a) b = v (a b) = v(b a) and

b a = a b. Therefore B is commutative. Claim (??) is evident.

The passage from B to CD(B, ») is sometimes called a Cayley-Dickson process.

A quadratic ´tale algebra K satis¬es the conditions of Lemma (??) and the corre-

e

sponding Cayley-Dickson algebra Q = CD(K, ») is a quaternion algebra over F for

any » ∈ F — . Repeating the process leads to an alternative algebra CD(Q, µ). A

Cayley algebra is a unital F -algebra isomorphic to an algebra of the type CD(Q, µ)

for some quaternion algebra Q over F and some µ ∈ F — .

In view of Lemma (??) and Proposition (??), the Cayley-Dickson process ap-

plied to a Cayley algebra does not yield a composition algebra. We now come to

the well-known classi¬cation of unital composition algebras:

458 VIII. COMPOSITION AND TRIALITY

(33.17) Theorem. Composition algebras with identity element over F are ei-

ther F , quadratic ´tale F -algebras, quaternion algebras over F , or Cayley algebras

e

over F .

Proof : As already observed, all algebras in the list are unital composition algebras.

Conversely, let C be a composition algebra with identity element over F . If C = F ,

let c ∈ C be such that {1, c} generates a nonsingular quadratic subspace of (C, n):

choose c ∈ 1⊥ if char F = 2 and c such that bn (1, c) = 1 if char F = 2. Then

B = F · 1 • F · c is a quadratic ´tale subalgebra of C. Thus we may assume that C

e

contains a unital composition algebra with nonsingular norm and it su¬ces to show

that if B = C, then C contains a Cayley-Dickson process B+vB. If B = C, we have

C = B • B ⊥ , B ⊥ is nonsingular and there exists v ∈ B ⊥ such that n(v) = ’» = 0.

We claim that B • v B is a subalgebra of C obtained by a Cayley-Dickson process,

i.e., that

(v a) b = v (b a), a (v b) = v (a b)

and

(v a) (v b) = »b a

for a, b ∈ B. We only check the ¬rst formula. The proofs of the others are similar.

We have v = ’v, since bn (v, 1) = 0, and 0 = bn (v, a)·1 = v a+a v = ’v a+a v,

thus v a = a v = ’a v for a ∈ B. Further

bn (v b) a, z = bn (v b, z a) = bn (b v, z a) = ’bn (b a, z v).

The last equality follows from formula (??), putting x = b, u = z, y = a, z = v,

and using that bn (v, a) = 0 for a ∈ B. On the other hand we have

’bn (b a, z v) = ’bn (b a) v, z = bn v (a b), z

where the last equality follows from the fact that v a = ’a v for all a ∈ B. This

holds for all z ∈ C, hence (v b) a = v (a b) as claimed. The formulas for the

norm and the involution are easy and we do not check them.

The classi¬cation of composition algebras with identity is known as the The-

orem of Hurwitz and the algebras occurring in Theorem (??) are called Hurwitz

algebras.

From now on we set Comp + = Hurw m for m = 1, 2, 4, and 8. If Sm , A1 ,

m

resp. G2 , are the groupoids of ´tale algebras of dimension m, quaternion algebras,

e

resp. Cayley algebras over F , then Hurw 2 = S2 , Hurw 4 = A1 , and Hurw 8 = G2 .

Hurwitz algebras are related to P¬ster forms. Let PQ m be the groupoid of

P¬ster quadratic forms of dimension m with isometries as morphisms.

(33.18) Proposition. (1) Norms of Hurwitz algebras are 0-, 1-, 2-, or 3-fold

P¬ster quadratic forms and conversely, all 0-, 1-, 2- or 3-fold P¬ster quadratic

forms occur as norms of Hurwitz algebras.

(2) For any Hurwitz algebra (C, NC ) the space

(C, NC )0 = { x ∈ C | TC (x) = 0 },

where TC is the trace, is regular.

Proof : (??) This is clear for quadratic ´tale algebras. The higher cases follow from

e

the Cayley-Dickson construction.

§33. COMPOSITION ALGEBRAS 459

Similarly, (??) is true for quadratic ´tale algebras, hence for Hurwitz algebras

e

of higher dimension by the Cayley-Dickson construction.

(33.19) Theorem. Let C, C be Hurwitz algebras. The following claims are equiv-

alent:

(1) The algebras C and C are isomorphic.

(2) The norms NC and NC are isometric.

(3) The norms NC and NC are similar.

Proof : (??) ’ (??) follows from the uniqueness of the generic minimal polynomial

∼

and (??) ’ (??) is obvious. Let now ± : (C, NC ) ’ (C , NC ) be a similitude with

’

factor ». Since NC ±(1C ) = »N (1C ) = », » is represented by NC . Since NC is

a P¬ster quadratic form, »NC is isometric to NC (Baeza [?, p. 95, Theorem 2.4]).

Thus we may assume that ± is an isometry. Let dimF C ≥ 2 and let B1 be a

quadratic ´tale subalgebra of C such that its norm is of the form [1, b] = X 2 +

e

2

XY +bY with respect to the basis (1, u) for some u ∈ B1 . Let ±(1) = e, ±(u) = w,

and let e be the left multiplication with e. Then u = e (w) generates a quadratic

∼

´tale subalgebra B1 of C and β = e —¦ ± is an isometry NC ’ NC which restricts

e ’

∼

to an isomorphism B1 ’ B1 . Thus we may assume that the isometry ± restricts

’

to an isomorphism on a pair of quadratic ´tale algebras B1 and B1 . Then ± is

e

⊥

∼ ⊥∼

an isometry NB1 ’ NB1 , hence induces an isometry B1 ’ B1 . If B1 = C,

’ ’

⊥

choose v ∈ B1 such that N (v) = 0 and put v = ±(v). By the Cayley-Dickson

construction (??) we may de¬ne an isomorphism

∼

±0 : B 2 = B 1 • v B 1 ’ B 2 = B 1 • v

’ B1

by putting ±0 (a + v b) = ±(a) + v ±(b) (which is not necessarily equal to

±(a + v b)!). Assume that B2 = C. Since ±0 is an isometry, it can be extended by

∼

Witt™s Theorem to an isometry C ’ C . We now conclude by repeating the last

’

step.

(33.20) Corollary. There is a natural bijection between the isomorphism classes

of Hurw m and the isomorphism classes of PQ m for m = 1, 2, 4, and 8.

Proof : By (??) and (??).

The following “Skolem-Noether” type of result is an immediate consequence of

the proof of the implication (??) ’ (??) of (??):

(33.21) Corollary. Let C1 , C2 be separable subalgebras of a Hurwitz algebra C.

∼

Any isomorphism φ : C1 ’ C2 extends to an isomorphism or an anti-isomorphism

’

of C.

(33.22) Remark. It follows from the proof of Theorem (??) that an isometry

of a quadratic or quaternion algebra which maps 1 to 1 is an isomorphism or an

anti-isomorphism (“A1 ≡ B1 ”). This is not true for Cayley algebras (“G2 ≡ B3 ”).

(33.23) Proposition. If the norm of a Hurwitz algebra is isotropic, it is hyper-

bolic.

Proof : This is true in general for P¬ster quadratic forms (Baeza [?, Corollary 3.2,

p. 105]), but we still give a proof, since it is an easy consequence of the Cayley-

Dickson process. We may assume that dimF C ≥ 2. If the norm of a Hurwitz

algebra C is isotropic, it contains a hyperbolic plane and we may assume that 1C

460 VIII. COMPOSITION AND TRIALITY

lies in this plane. Hence C contains the split separable F -algebra B = F — F . But

then any B • vB obtained by the Cayley-Dickson process is a quaternion algebra

with zero divisors, hence a matrix algebra, and its norm is hyperbolic. Applying

once more the Cayley-Dickson process if necessary shows that the norm must be

hyperbolic if dimF C = 8.

It follows from Theorem (??) and Proposition (??) that in each possible dimen-

sion there is only one isomorphism class of Hurwitz algebras with isotropic norms.

For Cayley algebras a model is the Cayley algebra

Cs = CD M2 (F ), ’1 .

We call it the split Cayley algebra. Its norm is the hyperbolic space of dimen-

sion 8. The group of F -automorphisms of the split Cayley algebra Cs over F is an

exceptional simple split group G of type G2 (see Theorem (??)).

(33.24) Proposition. Let G be a simple split algebraic group of type G 2 . Cayley

algebras over a ¬eld F are classi¬ed by the pointed set H 1 (F, G).

Proof : Since all Cayley algebras over a separable closure Fs of F are split, any

Cayley algebra over F is a form of the split algebra Cs . Thus we are in the situation

of (??), hence the claim.

If the characteristic of F is di¬erent from 2, norms of Hurwitz algebras corre-

spond to n-fold (bilinear) P¬ster forms for n = 0, 1, 2, and 3. We recall that for

any n-fold P¬ster form qn = a1 , . . . , an the element fn (qn ) = (±1 ) ∪ · · · ∪ (±n ) ∈

H n (F, µ2 ) is an invariant of the isometry class of qn and classi¬es the form. (see

Theorem (??)). Thus in characteristic not 2, the cohomological invariant fi (NC )

of the norm NC of a Hurwitz algebra C of dimension 2i ≥ 2 is an invariant of the

algebra. We denote it by fi (C) ∈ H i (F ).

(33.25) Corollary. Let C, C be Hurwitz algebras of dimension 2i , i ≥ 1. The

following conditions are equivalent:

(1) The algebras C and C are isomorphic.

(2) fi (C) = fi (C ).

Proof : By Theorem (??) and Theorem (??).

(33.26) Remark. There is also a cohomological invariant for P¬ster quadratic

forms in characteristic 2 (see for example Serre [?]). For this invariant, Theo-

rem (??) holds, hence, accordingly, Corollary (??) also.

33.D. Composition algebras without identity. We recall here some gen-

eral facts about composition algebras without identity, as well as consequences of

previous results for such algebras.

The norm n of a composition algebra is determined by the multiplication even

if C does not have an identity:

(33.27) Proposition. (1) Let (C, , n) be a composition algebra with multiplica-

tion (x, y) ’ x y, not necessarily with identity. Then there exists a multiplication

(x, y) ’ x y on C such that (C, , n) is a unital composition algebra with respect

to the new multiplication.

(2) Let (C, , n), (C , , n ) be composition algebras (not necessarily with identity).

∼ ∼

Any isomorphism of algebras ± : (C, ) ’ (C , ) is an isometry (C, n) ’ (C , n ).

’ ’

§34. SYMMETRIC COMPOSITIONS 461

Proof : (??) (Kaplansky [?]) Let a ∈ C be such that n(a) = 0 and let u = n(a)’1 a2 ,

so that n(u) = 1. The linear maps u : x ’ u x and ru : x ’ x u are isometries,

hence bijective. We claim that v = u2 is an identity for the multiplication

’1 ’1

x y = (ru x) ( u y).

We have ru v = ’1 v = u, hence x

’1 ’1 ’1 ’1

v= (ru x) ( u v) = ru (ru x) = x and

u

similarly v x = x. Furthermore,

’1 ’1 ’1 ’1

n(x y) = n (ru x) ( u y) = n(ru x)n( u y) = n(x)n(y).

(??) (Petersson [?]) The claim follows from the uniqueness of the degree two

generic minimal polynomial if ± is an isomorphism of unital algebras. In particular

n is uniquely determined by if there is a multiplication with identity. Assume now

that C and C do not have identity elements. We use ± to transport n to C, so that

we have one multiplication on C which admits two multiplicative norms n, n .

If there exists some a ∈ C with n(a) = n (a) = 1, we modify the multiplication

as in (??) to obtain a multiplication with 1 which admits n and n , so n = n .

To ¬nd a, let u ∈ C be such that n(u) = 1 (such an element exists by (??)). The

map u : x ’ u x is then an isometry of (C, n), and in particular it is bijective.

This implies that n (u) = 0. The element a such u a = u has the desired property

n(a) = n (a) = 1.

(33.28) Corollary. The possible dimensions for composition algebras (not neces-

sarily unital ) are 1, 2, 4 or 8.

Proof : The claim follows from Theorem (??) for unital algebras and hence from

Proposition (??) in general.

(33.29) Corollary. Associating a unital composition algebra (C, , n) to a compo-

sition algebra (C, , n) de¬nes a natural map to the isomorphism classes of Hurw m

from the isomorphism classes of Comp m .

Proof : By Proposition (??) we have a unital multiplication on (C, n) which, by

Theorem (??), is determined up to isomorphism.

(33.30) Remark. As observed in Remark (??) an isometry of unital composition

algebras which maps 1 to 1 is not necessarily an isomorphism, however isometric

unital composition algebras are isomorphic. This is not necessarily the case for

algebras without identity (see Remark (??)).

§34. Symmetric Compositions

In this section we discuss a special class of composition algebras without iden-

tity, independently considered by Petersson [?], Okubo [?], Faulkner [?] and re-

cently by Elduque-Myung [?], Elduque-P´rez [?]. Let (S, n) be a ¬nite dimensional

e

F -algebra with a quadratic form n : S ’ F . Let bn (x, y) = n(x+y)’n(x)’n(y) for

x, y ∈ S and let (x, y) ’ x y be the multiplication of S. We recall that the quad-

ratic form n is called associative or invariant with respect to the multiplication

if

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

holds for all x, y, z ∈ S.

(34.1) Lemma. Assume that n is nonsingular. The following conditions are equiv-

alent:

462 VIII. COMPOSITION AND TRIALITY

(1) n is multiplicative and associative.

(2) n satis¬es the relations x (y x) = n(x)y = (x y) x for x, y ∈ S.

Proof : (Okubo-Osborn [?, Lemma II.2.3]) Assume (??). Linearizing n(x y) =

n(x)n(y), we obtain

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

Since n is associative, this yields

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

hence (??), n being nonsingular.

Conversely, if (??) holds, linearizing gives

(34.2) x (y z) + z (y x) = bn (x, z)y = (x y) z + (z y) x.

By substituting x y for x in the ¬rst equation and y z for z in the second equation,

we obtain

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

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

hence bn (x y, z)y = bn (x, y z)y and bn (x, y) is associative. Finally, we have

n(x y)y = (x y) [y (x y)] = (x y) n(y)x = n(y)n(x)y

and the form n is multiplicative. If 2 = 0, we can also argue as follows:

n(x y) = 2 bn (x y, x y) = 1 bn x, y (x y) = 2 bn (x, x)n(y) = n(x)n(y).

1 1

2

We call a composition algebra with an associative norm a symmetric compo-

sition algebra and denote the full subcategory of Comp m consisting of symmetric

composition algebras by Scomp m . A symmetric composition algebra is cubic, be-

cause

x (x x) = (x x) x = n(x)x,

however it is not power-associative in general, since

(34.3) (x x) (x x) = bn (x, x x)x ’ x x (x x)

by (??) and

x x (x x) = n(x)x x.

A complete list of power-associative symmetric composition algebras is given in Ex-

ercise ?? of this chapter.

The ¬eld F is a symmetric composition algebra with identity. However it can

be shown that a symmetric composition algebra of dimension ≥ 2 never admits an

identity.

§34. SYMMETRIC COMPOSITIONS 463

34.A. Para-Hurwitz algebras. Let (C, , n) be a Hurwitz algebra. The

multiplication

(x, y) ’ x y = x y

also permits composition and it follows from bn (x y, z) = bn (x, z y) (see (??)) that

the norm n is associative with respect to (but not with respect to if C = F ).

Thus (C, , n) is a symmetric composition algebra. We say that (C, , n) is the

para-Hurwitz algebra associated with (C, , n) (resp. the para-quadratic algebra, the

para-quaternion algebra or the para-Cayley algebra). We denote the corresponding

full subcategories of Scomp by Hurw , resp. S2 , A1 , and G2 .

Observe that the unital composition algebra associated with (C, ) by the con-

struction given in the proof of Proposition (??) is the Hurwitz algebra (C, ) if we

set a = 1.

(34.4) Proposition. Let (C1 , , n1 ) and (C2 , , n2 ) be Hurwitz algebras and let

∼

± : C1 ’ C 2

’

be an isomorphism of vector spaces such that ±(1C1 ) = 1C2 . Then ± is an isomor-

∼ ∼

phism (C1 , ) ’ (C2 , ) of algebras if and only if it is an isomorphism (C1 , ) ’

’ ’

(C2 , ) of para-Hurwitz algebras. Moreover

∼

(1) Any isomorphism of algebras (C1 , ) ’ (C2 , ) is an isomorphism of the cor-

’

responding para-Hurwitz algebras.

∼

(2) If dim C1 ≥ 4, then an isomorphism (C1 , ) ’ (C2 , ) of para-Hurwitz algebras

’

is an isomorphism of the corresponding Hurwitz algebras.

Proof : Let ± : C1 ’ C2 be an isomorphism of algebras. By uniqueness of the

quadratic generic polynomial we have ±(x) = ±(x) and ± is an isomorphism of

para-Hurwitz algebras. Conversely, an isomorphism of para-Hurwitz algebras is an

isometry by Proposition (??) (or since x (y x) = n(x)y), and we have TC2 ±(x) =

TC1 (x), since TC1 (x) = bC1 (1, x) and ±(1C1 ) = 1C2 . As above it follows that

±(x) = ±(x) and ± is an isomorphism of Hurwitz algebras.

Claim (??) obviously follows from the ¬rst part and claim (??) will also follow

from the ¬rst part if we show that ±(1C1 ) = 1C2 . We use Okubo-Osborn [?, p. 1238]:

we have 1 x = ’x for x ∈ 1⊥ and the claim follows if we show that there exists

exactly one element u ∈ C1 such that u x = ’x for x ∈ u⊥ . Let u be such an

element. Since by Corollary (??.??), 1⊥ is nondegenerate, the maximal dimension

1

of a subspace of 1⊥ on which the form is trivial is 2 (dimF C1 ’ 2). If dimF C1 ≥ 4,

there exists some x ∈ 1⊥ © u⊥ with n1 (x) = 0. For this element x we have

n1 (x)1 = x (1 x) = x (’x) = x (u x) = n1 (x)u,

so that, as claimed, 1 = u.

For quadratic algebras the following nice result holds:

(34.5) Proposition. Let C1 , C2 be quadratic algebras and assume that there exists

∼

an isomorphism of para-quadratic algebras ± : (C1 , ) ’ (C2 , ), which is not an

’

isomorphism of algebras. Then u = ±(1) ∈ F · 1 is such that u3 = 1 and β(x) =

∼ ∼

±(x)u2 is an algebra isomorphism C1 ’ C2 . Conversely, if β : C1 ’ C2 is an

’ ’

isomorphism of algebras, then, for any u ∈ C2 such that u3 = 1, the map ± de¬ned

∼

by ±(x) = β(x)u is an isomorphism (C1 , ) ’ (C2 , ) of para-quadratic algebras.

’

∼

In particular an isomorphism C1 ’ C2 of para-Hurwitz algebras which is not an

’

464 VIII. COMPOSITION AND TRIALITY

F [X]/(X 2 + X + 1), i.e., C1 is

isomorphism of algebras can only occur if C1

isomorphic to F (u) with u3 = 1.

Proof : The proof of (??) shows that u = ±(1) ∈ F . We show that u3 = 1. We have

u u = u2 = u. Thus multiplying by u and conjugating gives u3 = uu = n2 (u) =

n2 ±(1) = n1 (1) = 1 by Proposition (??). It then follows that C2 F (u). The

condition ±(x) ±(y) = ±(x y) with y = 1 gives ±(x)u = ±(x) and, replacing x

by xy,

±(xy)u = ±(xy) = ±(x)±(y).

By conjugating and multiplying both sides with u4 = u we obtain

[±(x)u2 ][±(y)u2 ] = ±(xy)u2 ,

so that the map β : C1 ’ C2 de¬ned by β(x) = ±(x)u2 is an isomorphism of

∼

C2 = F (u) with u3 = 1 and if β : C1 ’ C2 is

algebras. Conversely, if C1 ’

an isomorphism, then ± : C1 ’ C2 de¬ned by ±(x) = β(x)u is an isomorphism

∼

(C1 , ) ’ (C2 , ).

’

Observe that ru : x ’ x u is an automorphism of F (u), of order 3. In

fact we have AutF F (u), = S3 , generated by the conjugation and ru . This is in

contrast with the quadratic algebra F (u), · for which AutF F (u) = Z/2Z.

(34.6) Corollary. The map P : (C, ) ’ (C, ) is an equivalence Hurw m ≡ Hurw m

of groupoids if m = 4, 8, and P is bijective on isomorphism classes if m = 2.

In view of Corollary (??) we call a n-dimensional para-Hurwitz composition

algebra of type A1 if n = 4 and of type G2 if n = 8.

(34.7) Remark. It follows from Corollary (??) that

AutF (C, ) = AutF (C, )

for any Hurwitz algebra C of dimension ≥ 4. Thus the classi¬cation of twisted

forms of para-Hurwitz algebras is equivalent to the classi¬cation of Hurwitz alge-

bras in dimensions ≥ 4. In particular any twisted form of a para-Hurwitz algebra of

dimension ≥ 4 is again a para-Hurwitz algebra. The situation is di¬erent in dimen-

sion 2: There exist forms of para-quadratic algebras which are not para-quadratic

algebras (see Theorem (??)).

The identity 1 of a Hurwitz algebra C plays a special role also for the associated

para-Hurwitz algebra: it is an idempotent and satis¬es 1 x = x 1 = ’x for all

x ∈ C such that bnC (x, 1) = 0. Let (S, , n) be a symmetric composition algebra.

An idempotent e of S (i.e., an element such that e e = e) is called a para-unit if

e x = x e = ’x for x ∈ S, bn (e, x) = 0.

(34.8) Lemma. A symmetric composition algebra is para-Hurwitz if and only if

it admits a para-unit.

Proof : If (S, ) is para-Hurwitz, then 1 ∈ S is a para-unit. Conversely, for any

para-unit e in a symmetric composition algebra (S, , n), we have n(e) = 1 and

x y = (e x) (y e)

de¬nes a multiplication with identity element e on S. We have x y = x y where

x = bn (e, x)e ’ x.

§34. SYMMETRIC COMPOSITIONS 465

34.B. Petersson algebras. Let (C, , n) be a Hurwitz algebra and let • be

an F -automorphism of C such that •3 = 1. Following Petersson [?] we de¬ne a

new multiplication on C by

x y = •(x) •2 (y).

This algebra, denoted C• , is a composition algebra for the same norm n and we

call it a Petersson algebra. It is straightforward to check that

(x y) x = n(x)y = x (y x)

so that Petersson algebras are symmetric composition algebras. Observe that • is

automatically an automorphism of (C, ). For • = 1, (C, ) is para-Hurwitz.

Conversely, symmetric composition algebras with nontrivial idempotents are

Petersson algebras (Petersson [?, Satz 2.1], or Elduque-P´rez [?, Theorem 2.5]):

e

(34.9) Proposition. Let (S, , n) be a symmetric composition algebra and let e ∈

S be a nontrivial idempotent.

(1) The product x y = (e x) (y e) gives S the structure of a Hurwitz algebra

with identity e, norm n, and conjugation x ’ x = bn (x, e)e ’ x.

(2) The map

•(x) = e (e x) = bn (e, x)e ’ x e = x e

is an automorphism of (S, ) (and (S, )) of order ¤ 3 and (S, ) = S• is a Petersson

algebra with respect to •.

Proof : (??) is easy and left as an exercise.

(??) Replacing x by e x and z by e in the identity (??):

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

gives

x y = bn (e, x)y ’ e y (e x)

hence

(x y) e = y bn (x, e)e ’ e x = e (y e) (x e) e = (y e) (x e).

Thus • is an automorphism of (S, ), •3 (x) = x = x, x •2 (y) and

y = •(x)

(S, ) = S• as claimed.

In general a symmetric composition may not contain an idempotent. However:

(34.10) Lemma. Let (S, , n) be a symmetric composition algebra.

(1) If the cubic form bn (x x, x) is isotropic on S, then (S, ) contains an idempo-

tent. In particular there always exists a ¬eld extension L/F of degree 3 such that

(S, )L contains an idempotent e.

(2) For any nontrivial idempotent e ∈ S we have n(e) = 1.

Proof : (??) It su¬ces to ¬nd f = 0 with f f = »f , » ∈ F — so that e = f »’1

then is an idempotent. Let x = 0 be such that bn (x x, x) = 0. We have

(x x) (x x) = ’n(x)(x x)

by (??), so we take f = x x if n(x) = 0. If n(x) = 0, we may also assume that

x x = 0: if x x = 0 we replace x by x x and use again (??). Since x is isotropic

466 VIII. COMPOSITION AND TRIALITY

and n is nonsingular, there exists some y ∈ S such that n(y) = 0 and bn (x, y) = ’1.

A straightforward computation using (??) shows that

(x y + y x) (x y + y x) = (x y + y x) + 3bn (y, y x)x,

and

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

is an idempotent and is nonzero since

e x = (y x) x = bn (x, y)x = ’x.

(??) Since e = (e e) e = n(e)e, we have n(e) = 1.

(34.11) Remark. Lemma (??.??) is in fact a special case of Theorem (??) and

its proof is copied from the proof of implication (??) ’ (??) of (??).

Assume that char F = 3 and that F contains a primitive cube root of unity ω.

The existence of an automorphism of order 3 on a Hurwitz algebra C is equivalent

with the existence of a Z/3Z-grading:

(34.12) Lemma. Suppose that F contains a primitive cube root of unity ω.

(1) If • is an automorphism of C of order 3, then C (or C• ) admits a decomposition

C = C • = S0 • S1 • S2 ,

with

Si = { x ∈ C | •(x) = ω i x }

and such that

(a) Si Sj ‚ Si+j (resp. Si Sj ‚ Si+j ), with subscripts taken modulo 3,

(b) bn (Si , Sj ) = 0 unless i + j ≡ 0 mod 3.

In particular (S0 , , n) ‚ C• is a para-Hurwitz algebra of even dimension and S1

(resp. S2 ) is a maximal isotropic subspace of S1 • S2 .

(2) Conversely, any Z/3Z-grading of C de¬nes an automorphism • of order 3 of C,

hence a Petersson algebra C• .

Proof : Claim (??) follows easily from the fact that ω i , i = 0, 1, 2, are the eigen-

values of the automorphism •. For (??) we take the identity on degree 0 elements,

multiplication by ω on degree 1 elements and multiplication by ω 2 on degree 2

elements.

If • = 1, S0 in Lemma (??) must have dimension 2 or 4 (being a para-Hurwitz

algebra). We show next that C• is para-Hurwitz if dim S0 = 2. The case dim S0 = 4

and dim C• = 8 corresponds to a di¬erent type of symmetric composition, discussed

in the next subsection.

(34.13) Proposition (Elduque-P´rez). Let F be a ¬eld of characteristic not 3, let

e

C be a Hurwitz algebra over F , let • be an F -automorphism of C of order 3. Then

S0 = { x ∈ C | •(x) = x }

is a para-Hurwitz algebra of dimension 2 or 4. The Petersson algebra C • is iso-

morphic to a para-Hurwitz algebra if and only if dim S0 = 2.

§34. SYMMETRIC COMPOSITIONS 467

Proof : The ¬rst claim is clear. For the second claim we use an argument in Elduque-

P´rez [?, proof of Proposition 3.4]. If dim C = 2 there is nothing to prove. Thus

e

by Remark (??) we may assume that F contains a primitive cube root of unity. To

simplify notations we denote the multiplication in C by (x, y) ’ xy and we put

n = NC for the norm of C. Let xi ∈ Si , i = 1, 2; we have x2 = n(xi ) = 0 by

i

Lemma (??), so that

bn (x1 x2 , x2 x1 ) = bn (x1 , x2 )2

by (??). Furthermore (x1 x2 )(x2 x1 ) = x1 (x2 x2 )x1 = 0 (by Artin™s theorem, see (??))

and

(x1 x2 )2 ’ bn (x1 x2 , 1)x1 x2 + n(x1 x2 ) · 1 = 0

implies that (x1 x2 )2 = ’bn (x1 , x2 )x1 x2 . Choosing x1 , x2 such that bn (x1 , x2 ) =

’1, we see that e1 = x1 x2 is an idempotent of C and it is easily seen that e2 =

1 ’ e1 = x2 x1 . We claim that if dim S0 = 2, then e1 = y1 y2 for any pair (y1 , y2 ) ∈

S1 — S2 such that bn (y1 , y2 ) = ’1. We have S0 = F · e1 • F · e2 if dim S0 = 2, so

that the claim will follow if we can show that bn (e1 , y1 y2 ) = 0. Let y1 = »x1 + x1

with bn (x1 , x2 ) = 0. By using (??) and the fact that n(Si ) = 0 for i = 1, 2, we

deduce

bn (e1 , y1 y2 ) = bn x1 x2 , (»x1 + x1 )y2

= n(x1 )bn (x2 , »y2 ) + bn (x1 x2 , x1 y2 )

= ’bn (x1 y2 , x1 x2 ).

However x1 x2 satis¬es

(x1 x2 )2 ’ bn (x1 x2 , 1)x1 x2 + n(x1 x2 ) = 0

hence (x1 x2 )2 = 0, since bn (x1 x2 , 1) = ’bn (x1 , x2 ) = 0. Since the algebra S0 is

´tale, we must have x1 x2 = 0 and, as claimed, bn (e1 , y1 y2 ) = 0. Similarly we have

e

e2 = y2 y1 for (y1 , y2 ) ∈ S1 — S2 such that bn (y1 , y2 ) = ’1. It follows that

e1 y1 = (1 ’ e2 )y1 = (1 ’ y2 y1 )y1 = y1 ,

y1 e1 = y1 (y1 y2 ) = 0 = e2 y1 ,

y1 e2 = y 1 , e2 y2 = y 2 = y 2 e1 and e1 y2 = 0 = y2 e2 ,

so that

S1 = { x ∈ C | e1 x = x = xe2 }

and

S2 = { x ∈ C | e2 x = x = xe1 }.

The element

e = ω 2 e1 + ωe2

is a para-unit of C• , since

e x = (ω 2 e1 + ωe2 )(ω 2i x) = ’(ωe1 + ω 2 e2 )ω 2i x = ’x

for x ∈ Si and since

e (ωe1 ’ ω 2 e2 ) = (ω 2 e1 + ωe2 )(’ω 2 e1 + ω 2 e2 ) = (’ωe1 + ω 2 e2 )

for ωe1 ’ ω 2 e2 ∈ e⊥ ‚ S0 . The claim then follows from Lemma (??).

468 VIII. COMPOSITION AND TRIALITY

34.C. Cubic separable alternative algebras. Following Faulkner [?] we

now give another approach to symmetric composition algebras over ¬elds of char-

acteristic not 3. We ¬rst recall some useful identities holding in cubic alternative

algebras. Let A be a ¬nite dimensional unital separable alternative F -algebra of

degree 3 and let

PA,a (X) = X 3 ’ TA (a)X 2 + SA (a)X ’ NA (a)1

be its generic minimal polynomial. The trace TA is linear, the form SA is quadratic

and the norm NA is cubic. As was observed in the introduction to this chapter we

have

NA (X ’ a · 1) = PA,a (X), NA (xy) = NA (x)NA (y), and TA (xy) = TA (yx).

Let

bSA (x, y) = SA (x + y) ’ SA (x) ’ SA (y),

x# = x2 ’ TA (x)x + SA (x) · 1

and

x — y = (x + y)# ’ x# ’ y # .

Note that

NA (x) = xx# = x# x.

Observe that the #-operation and the —-product are de¬ned for any cubic

algebra. They will be systematically used in Chapter IX for cubic Jordan algebras.

(34.14) Lemma. (1) NA (1) = 1, SA (1) = TA (1) = 3, 1# = 1, 1 — 1 = 2,

(2) (xy)# = y # x# ,

(3) SA (x) = TA (x# ), SA (x# ) = TA (x)NA (x), NA (x# ) = NA (x)2 ,

(4) bSA (x, y) = TA (x — y).

(5) NA (x + »y) = »3 NA (y) + »2 TA (x · y # ) + »TA (x# · y) + NA (x) for x, y ∈ A,

and » ∈ F .

(6) The coe¬cient of ±βγ in NA (±x + βy + γz) is TA x(y — z) and TA x(y — z)

is symmetric in x, y, and z.

(7) bSA (x, 1) = 2TA (x),

(8) x — 1 = TA (x) · 1 ’ x,

(9) TA (xy) = TA (x)TA (y) ’ bSA (x, y),

(10) TA (xz)y = TA x(zy) .

Proof : We may assume that F is in¬nite and identify polynomials through their

coe¬cients. NA (1) = 1 follows from the multiplicativity of NA , so that 1# = 1

and 1 — 1 = 2. Putting a = 1 in NA (X ’ a · 1) = PA,a (X) gives PA,1 (X) =

(X ’ 1)3 NA (1) = (X ’ 1)3 , hence SA (1) = TA (1) = 3.

By density it su¬ces to prove (??) for x, y such that NA (x) = 0 = NA (y).

Then (xy)# = (xy)’1 NA (xy) = y ’1 NA (y)x’1 NA (x) = y # x# .

Again by density, it su¬ces to prove (??) for x such that NA (x) = 0. We then

have NA (x ’ ») = NA (1 ’ »x’1 )NA (x). Comparing the coe¬cients of » gives (??),

and (??) follows by linearizing (??).

(??) follows by computing NA (x + »y) = NA (xy ’1 + »)NA (y).

The ¬rst claim of (??) follows by computing the coe¬cient of ±βγ in NA ±x +

(βy + γz) (and using (??)). The last claim of (??) then is clear by symmetry.

§34. SYMMETRIC COMPOSITIONS 469

(??) follows from (??), since

bSA (x, 1) = TA (x — 1) = TA x(1 — 1) = 2TA (x)

and (??) implies (??).

For (??) we have

bSA (x, y) = TA (x — y)1 = TA (x — 1)y

= TA TA (x) · 1 ’ x y = TA (x)TA (y) ’ TA (xy).

Finally, by linearizing

TA x(yx) = TA (xy)x = TA (yx2 ) = TA y x# + xTA (x) ’ SA (x)1 ,

we obtain

TA x(yz) + TA z(yx) = TA (xy)z + TA (zy)x

= TA y(x — z) + TA (yx)TA (z) + TA (yz)TA (x)

’ TA (y)bSA (x, z)

= TA y(x — z) + TA (yx)TA (z) + TA (yz)TA (x)

+ TA (xz)TA (y) ’ TA (x)TA (y)TA (z),

so that by (??) TA x(yz) + TA z(yx) = TA (xy)z + TA (zy)x is symmetric in

x, y and z. It follows that

TA x(yz) + TA z(yx) = TA y(xz) + TA z(xy)

and TA (xz)y = TA x(zy) , as claimed.

(34.15) Proposition. A cubic alternative algebra is separable if and only if the

bilinear trace form T (x, y) = TA (xy) is nonsingular.

Proof : By (??) T is associative, hence the claim follows from Dieudonn´™s Theo-

e

rem (??).

We recall:

∼

(34.16) Proposition. For any isomorphism ± : A ’ A of cubic unital alterna-

’

tive algebras we have

TA ±(x) = TA (x), SA ±(x) = SA (x), NA ±(x) = NA (x).

Proof : The polynomial pA ,±(a) (X) is a minimal generic polynomial for A, hence

the claim by uniqueness.

(34.17) Theorem. Let A be a cubic separable unital alternative algebra over F of

dimension > 1. Then either :

(1) A L, for some unique (up to isomorphism) cubic ´tale algebra L over F ,

e

(2) A F —Q where Q is a unique (up to isomorphism) quaternion algebra over F ,

(3) A F — C where C is a unique (up to isomorphism) Cayley algebra over F ,

(4) A is isomorphic to a unique (up to isomorphism) central simple associative

algebra of degree 3.

In particular such an algebra has dimension 3, 5 or 9. In case (??) the generic

minimal polynomial is the characteristic polynomial, in case (??) and (??) the

product of the generic minimal polynomial pF,a (X) = X ’ a of F with the generic

minimal polynomial pC,c (X) = X 2 ’ TC (c)X + NC (c) · 1 of C = Q or C = C and

in case (??) the reduced characteristic polynomial.

470 VIII. COMPOSITION AND TRIALITY

Proof : The claim is a special case of Theorem (??).

Let 1An denote the category of central simple algebras of degree n + 1 over

F . Let I : Sepalt m’1 (2) ’ Sepalt m (3) be the functor C ’ F — C. Theorem

(??) gives equivalences of groupoids Sepalt 3 (3) ≡ S3 , Sepalt 5 (3) ≡ I(1A1 ), and

Sepalt 9 (3) I(G2 ) 1A2 .

We assume from now on (and till the end of the section) that F is a ¬eld of

characteristic di¬erent from 3. Let A be cubic alternative separable over F and let

A0 = { x ∈ A | TA (x) = 0 }.

1 1 1

Since x = 3 TA (x) · 1 + x ’ 3 TA (x) · 1 and TA x ’ 3 TA (x) = 0 we have A =

F · 1 • A0 and the bilinear trace form T : (x, y) ’ TA (xy) is nonsingular on A0 . By

Lemma (??) the polar of the quadratic form SA on A0 is ’T . Thus the restriction

of SA to A0 is a nonsingular quadratic form.

We further assume that F contains a primitive cube root of unity ω and set

µ = 1’ω . We de¬ne a multiplication on A0 by

3

1

(34.18) x y = µxy + (1 ’ µ)yx ’ 3 TA (yx)1.

This type of multiplication was ¬rst considered by Okubo [?] for matrix algebras

and by Faulkner [?] for cubic alternative algebras.

(34.19) Proposition. The algebra (A0 , ) is a symmetric composition algebra with

1

norm n(x) = ’ 3 SA (x).

Proof : The form n is nonsingular, since SA is nonsingular. We check that

1

(x y) x = x (y x) = ’ 3 SA (x)y = n(x)y.

Lemma (??) will then imply that (A0 , ) is a symmetric composition algebra. We

have 3µ(1 ’ µ) = 1. It follows that

(34.20)

(x y) x = x (y x) = µ2 xyx + (1 ’ µ)2 xyx + µ(1 ’ µ)(yx2 + x2 y)

’ 3 TA (xy)x ’ 3 µTA (xyx)1 ’ 1 (1 ’ µ)TA (xyx)1

1 1

3

= [1 ’ 2µ(1 ’ µ)]xyx + µ(1 ’ µ)(yx2 + x2 y)

’ 1 TA (xy)x ’ 3 TA (xyx)1

1

3

= 1 (xyx + yx2 + x2 y) ’ 3 TA (xy)x ’ 3 TA (xyx)1.

1 1

3

By evaluating TA on the generic polynomial, we obtain 3NA (x) = TA (x3 ) for ele-

ments in A0 . Thus

x3 + SA (x)x ’ 1 TA (x3 )1 = 0

(34.21) 3

holds for all x ∈ A0 . Since it su¬ces to prove (??) over a ¬eld extension, we may

assume that F is in¬nite. Replacing x by x + »y in (??), the coe¬cient of » must

then be zero. Hence we are lead to the identity

xyx + yx2 + x2 y ’ TA (xy)x + SA (x)y ’ TA (xyx)1 = 0

for all x, y ∈ A0 , taking into account that bSA (x, y) = ’TA (xy) on A0 . Combining

this with equation (??) shows that

(x y) x = x (y x) = ’ 1 SA (x)y = n(x)y

3

as claimed.

§34. SYMMETRIC COMPOSITIONS 471

Hence we have a functor 1 C : Sepalt m+1 (3) ’ Scomp m for m = 2, 4 and 8

given by A ’ (A0 , n). We now construct a functor 1A in opposite direction; a

straightforward computation shows that (??) is equivalent to

(34.22) xy = (1 + ω)x y ’ ωy x + bn (x, y) · 1

for the multiplication in A0 ‚ A. Thus, given a symmetric composition (S, ), it is

natural to de¬ne a multiplication (x, y) ’ x · y = xy on A = F • S by (??) for x,

y ∈ S, and by 1 · x = x = x · 1. Let 1A be the functor (S, ) ’ (F • S, ·).

(34.23) Theorem (Elduque-Myung). The functors 1 C and 1A de¬ne an equiva-

lence of groupoids

Sepalt m+1 (3) ≡ Scomp m

for m = 2, 4 and 8.

Proof : We ¬rst show that A = 1A(S) = F • S is a separable alternative algebra of

degree 3: Let x = ±1 + a, ± ∈ F and a ∈ S. We have

x2 = ±2 + bn (a, a) 1 + 2±a + a a

and

xx2 = x2 x = [±3 + 3±bn (a, a) + bn (a a, a)]1 + [3±2 + 3n(a)]a + 3±(a a).

It follows that

x3 ’ 3±x2 + 3±2 ’ 3n(a) x = [±3 ’ 3n(a)± + bn (a a, a)]1

so that elements of A satisfy a polynomial condition of degree 3

pA,x (X) = X 3 ’ TA (x)X 2 + SA (x)X ’ NA (x)1 = 0

with

SA (x) = 3±2 ’ 3n(a)

TA (x) = 3±,

and

NA (x) = ±3 ’ 3±n(a) + bn (a a, a)

for x = ±1 + a. To show that A is of degree 3 we may assume that the ground

¬eld F is in¬nite and we need an element x ∈ A such that the set {1, x, x2 } is

linearly independent. Because x2 = x x + (x, x)1 for x ∈ S, it su¬ces to have

x ∈ S such that {1, x, x x} is linearly independent. Since TA (1) = 3, while

TA (x) = 0 = TA (x x), the only possible linear dependence is between x and x x.

If {x, x x} is linearly dependent for all x ∈ S, there is a map f : S ’ F such

that x x = f (x)x for x ∈ S. By the following Lemma (??) f is linear. Since

x x x = n(x)x we get n(x) = f (x)2 . This is only possible if dimF S = 1. We

next check that A is alternative. It su¬ces to verify that

a2 b = a(ab) and ba2 = (ba)a for a, b ∈ S.

We have

a2 b = [a a + (a, a)]b = (1 + ω)(a a) b ’ ωb (a a) + bn (a a, b)

472 VIII. COMPOSITION AND TRIALITY

and

a(ab) = a[(1 + ω)a b ’ ωb a + bn (a, b)]

= (1 + ω)[a (a b) ’ ω(a b) a + bn (a, a b)]

’ ω[(1 + ω)a (b a) ’ ω(b a) a + bn (a, b a)] + bn (a, b)a.

By (??) we have

(a a) b + (b a) a = bn (a, b)a = b (a a) + a (a b).

This, together with the identities

bn (a, a b) = bn (a a, b) = bn (b, a a) = bn (b a, a) = bn (a, b a)

which follow from the associativity of n, implies that a2 b = a(ab). The proof of

ba2 = (ba)a is similar. Thus A is alternative of degree 3. We next check that A is

separable. We have for x = ± + a, y = β + b,

xy = ±β + βa + ±b + (1 + ω)a b ’ ωb a + bn (a, b)

so that

T (x, y) = TA (xy) = 3±β + 3bn (a, b)

is a nonsingular bilinear form. Since the trace form of a cubic alternative algebra is

associative (Lemma (??)), A is separable by Dieudonn´™s Theorem (??). We ¬nally

e

have an equivalence of groupoids since

(F • S)0 = S and F • A0 = A

and since formulas (??) and (??) are equivalent.

(34.24) Lemma. Let F be an in¬nite ¬eld and let (S, ) be an F -algebra. If there

exists a map f : S ’ F such that x x = f (x)x for all x ∈ S, then f is linear.

Proof : (Elduque) If S is 1-dimensional, the claim is clear. So let (e1 , . . . , en ) be a

xi ei and ei ej = k ak ek , we have i,j ak xi xj = f (x)xk

basis of S. For x = ij ij

’1

for any k. Thus f (x) = gk (x)xk for some quadratic homogeneous polynomial

gk (x) and k = 1, . . . , n in the Zariski open set

D(xk ) = { x ∈ S | xk = 0 }.

For any pair i, j we have gi (x)xj = gj (x)xi in D(xi ) © D(xj ), so by density

gi (x)xj = gj (x)xi holds for any x ∈ S. Unique factorization over the polyno-

mial ring F [x1 , . . . xn ] shows that there exists a linear map φ : S ’ F such that

gi (x) = xi φ(x). It is clear that f = φ.

(34.25) Remark. Let A be central simple of degree 3 over F and assume that F

has characteristic di¬erent from 3 and that F contains a primitive cube root of

unity. The form n from (??) is then hyperbolic on A0 : by Springer™s Theorem

(see [?, p. 119]) we may assume that A is split, and in that case the claim is easy

to check directly. Hence, if A and A are of degree 3 and are not isomorphic, the

0

compositions (A0 , ) and (A , ) are nonisomorphic (by (??)) but have isometric

norms. This is in contrast with Cayley (or para-Cayley) composition algebras.

§34. SYMMETRIC COMPOSITIONS 473

(34.26) Remark. The polar of a cubic form N is

N (x, y, z) = N (x + y + z) ’ N (x + y) ’ N (x + z) ’ N (y + z)

+ N (x) + N (y) + N (z)

and N is nonsingular if its polar is nonsingular, i.e., if N (x, y, z) = 0 for all x, y im-

plies that z = 0. Let A be an F -algebra. If char F = 2, 3 a necessary and su¬cient

condition for A to admit a nonsingular cubic form N which admits composition (i.e.,

such that N (xy) = N (x)N (y)) is that A is cubic separable alternative and N NA

(see Schafer [?, Theorem 3]). Thus, putting x = ± · 1 + a ∈ F · 1 • A0 , the multi-

plicativity of NA (x) = ±3 ’ 3±n(a) + bn (a a, a) for the multiplication (x, y) ’ xy

of A is equivalent by Proposition (??) to the multiplicativity of n = ’ 1 SA for the

3

multiplication (a, b) ’ a b of A0 . It would be nice to have a direct proof!

A symmetric composition algebra isomorphic to a composition (A0 , ) for A

central simple of degree 3 is called an Okubo composition algebra or a composition

algebra of type 1A2 since its automorphism group is a simple adjoint algebraic group

of type 1A2 . Twisted forms of Okubo algebras are again Okubo algebras. The

groupoid of Okubo composition algebras over a ¬eld F containing a primitive cube

root of unity is denoted 1Oku. We have an equivalence of groupoids 1Oku ≡ 1A2 (if

F contains a primitive cube root of unity).

For para-Hurwitz compositions of dimension 4 or 8 we have the following situ-

ation:

(34.27) Proposition. Let I : Hurw m ’ Sepalt m+1 (3) be the functor C ’ F — C,

P : Hurw m ’ Hurw m the para-Hurwitz functor and J : Hurw m ’ Scomp m the

inclusion. Then the map

·C : C ’ (F — C)0 z ’ TC (z), ωz + ω 2 z

given by

is a natural transformation between the functors 1 C —¦ I and J —¦ P, i.e., the diagram

1

C

Sepalt m+1 (3) ’ ’ ’ Scomp m

’’

¦ ¦

I¦ ¦J

P

’ ’ ’ Hurw m

’’

Hurw m

commutes up to ·C .

Proof : It su¬ces to check that ·C is an isomorphism of the para-Hurwitz algebra

(C, ) with the symmetric composition algebra (F — C)0 , . We shall use that

TA (x) = ξ + TC (c), SA (x) = NC (c) + ξTC (c) and NA (x) = ξNC (c) for A = F — C,

ξ ∈ F , c ∈ C and TC the trace and NC the norm of C. If char F = 2, we decompose

C = F · 1 • C 0 and set u = (2, ’1) ∈ A0 . We then have

·C (βe + x) = βu + (1 + 2ω)x.

The element u satis¬es u u = u and (0, x) u = u (0, x) = (0, ’x) for x ∈ C 0 .

Thus it su¬ces to check the multiplicativity of ·C on products of elements in C 0 ,

in which case the claim follows by a tedious but straightforward computation. If

char F = 2, we choose v ∈ C with TC (v) = 1, to have C = F · v • C 0 . We then

have

·C (βv + x) = (β, βv + x + ω 2 β)

474 VIII. COMPOSITION AND TRIALITY