(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

and the proof is reduced to checking the assertion for the products v v, v x, x v

and x x, where it is easy.

For symmetric compositions of dimension 2, we do not have to assume in Propo-

sition (??) that F contains a contains a primitive cube root of unity since a sepa-

rable alternative algebra A of degree 3 and dimension 3 is ´tale commutative and

e

1

the multiplication reduces to a b = ab ’ 3 TA (ab). Thus (??) and (??) imply:

(34.28) Proposition. Let F be a ¬eld of characteristic di¬erent from 3 and let

(S, , n) be a symmetric composition algebra over F of dimension 2. Then S

(L0 , ) for a unique cubic ´tale F -algebra L. The algebra S is para-quadratic if and

e

only if L is not a ¬eld.

To obtain a complete description of symmetric compositions in dimension 4

and 8 we may take the Structure Theorem (??) for cubic alternative algebras into

account:

(34.29) Theorem (Elduque-Myung). Let F be ¬eld with char = 3 which contains

a primitive cube root of unity. There exist equivalences of groupoids

Scomp4 ≡ Hurw 4 ≡ Hurw 4 ≡ A1

and

1 1

Scomp8 ≡ Hurw 8 Oku ≡ G2 A2 .

However we did not prove (??) and we shall give an alternate proof of these

equivalences. Observe that this will yield, in turn, a proof of Theorem (??)!

We postpone the proof of (??) and begin with an example:

(34.30) Example. Let V be a 2-dimensional vector space over F . We view ele-

ments of EndF (V • F ) of trace zero as block matrices

φ v End(V ) V

∈

V—

f ’ tr(φ) F

The product of such two blocks is given by

φ v φ v φ—¦φ +v—¦f φ(v ) ’ tr(φ )v

· =

f ’ tr(φ) f ’ tr(φ ) f —¦ φ ’ tr(φ)f f (v ) + tr(φ) tr(φ )

where (v —¦ f )(x) = vf (x). With the multiplication (x, y) ’ x y de¬ned in (??)

1

and the quadratic form n = ’ 3 SEndF (V •F ) ,

SV = 1 C EndF (V • F ) = EndF (V • F )0 ,

is a symmetric composition algebra and

’1V 0 End(V ) V

e= ∈

V—

0 2 F

is a nontrivial idempotent. By Proposition (??), (??), the map

•(x) = e (e x) = bn (e, x)e ’ x e

is an automorphism of SV , of order 3, such that SV , reduces to the Petersson

algebra SV • . The corresponding Z/3Z-grading is

S2 = V — .

S0 = EndF (V ), S1 = V,

In particular we have dimF S0 = 4. The converse also holds (Elduque-P´rez [?,

e

Theorem 3.5]) in view of the following:

§34. SYMMETRIC COMPOSITIONS 475

(34.31) Proposition. Let F be a ¬eld of characteristic not 3 which contains a

primitive cube root of unity ω. Let (S, , n) be a symmetric composition algebra

of dimension 8 with a nontrivial idempotent e. Let •(x) = e (e x) and let

S = S0 • S1 • S2 be the Z/3Z-grading of (S, , n) de¬ned by • (see (??)). If

dimF S0 = 4, there exists a 2-dimensional vector space V such that (S, ) S V • =

1

C End(V • F ) .

Proof : Since dimF S0 = 4 we must have dimF S1 = dimF S2 = 2, and S1 , S2 are

maximal isotropic direct summands of S by Lemma (??). Let (x1 , x2 ) be a basis

of S1 and let (f1 , f2 ) be a basis of S2 such that bn (xi , fj ) = δij . Since

(xi + fi ) u (xi + fi ) = (xi + fi ) u (xi + fi ) = u

holds for all u we have

(1) xi (u xi ) = 0 = (xi u) xi

(2) fj (u fj ) = 0 = (fj u) fj

and

(3) xi (u fi ) + fi (u xi ) = u

for all u ∈ S0 . Thus, by choosing u = e and using that

x = ’ωe x = ’ω 2 x e, e = ’ω 2 e f

f = ’ωf

for x ∈ S1 , resp. f ∈ S2 , we see that xi xi = 0 = fj fj . Moreover x1 x2 =

0 = x2 x1 and f1 f2 = 0 = f2 f1 , so S1 S1 = 0 = S2 S2 . Since u xi ∈ Si ,

u fj ∈ S2 for u ∈ S0 , (??) implies that

(f1 x1 , f1 x2 , x1 f1 , x1 f2 )

generates S0 , hence is a basis of S0 and S1 S2 = S0 = S2 S1 . We now de¬ne an

F -linear map ψ : S ’ M3 (F ) on basis elements by

« « « «

001 000 0 00 000

0 0 0, x2 ’ 0 0 1, f1 ’ 0 0 0, f2 ’ 0 0 0

x1 ’

000 000 3 00 030

and

1

xi fj ’ µxi fj + (1 ’ µ)fj xi ’ tr(xi fj ),

3

1

fj xi ’ µfj xi + (1 ’ µ)xi fj ’ tr(xi fj )

3

where multiplication on the right is in M3 (F ) and µ = 1’ω . We leave it as a

3

∼

(lengthy) exercise to check that ψ is an isomorphism of composition algebras S ’

’

C M3 (F ) = SV with V = F 2 .

1

Proof of Theorem (??): Let (S, , n) be a symmetric composition algebra. Since

twisted forms of Okubo algebras are Okubo and twisted forms of para-Hurwitz

algebras of dimension ≥ 4 are para-Hurwitz, we may assume by Lemma (??) that

S contains a nontrivial idempotent. Let S = S0 • S1 • S2 be the grading given

by Lemma (??). Assume ¬rst that dim S = 4. If dim S0 = 4, then S = S0

is para-Hurwitz; and if dim S0 = 2, S is para-Hurwitz by Proposition (??). If

dim S = 8 and dim S0 = 2, S is para-Hurwitz by Proposition (??); if dim S = 8

and dim S0 = 4, S is Okubo by Proposition (??).

476 VIII. COMPOSITION AND TRIALITY

34.D. Alternative algebras with unitary involutions. To overcome the

condition on the existence of a primitive cube root of unity in F , one considers

separable cubic alternative algebras B over the quadratic extension K = F (ω) =

F [X]/(X 2 + X + 1) which admit a unitary involution „ .

(34.32) Proposition. Let K be a quadratic separable ¬eld extension of F with

conjugation ι. Cubic separable unital alternative K-algebras with a unitary involu-

tion „ are of the following types:

(1) L = K — L0 for L0 cubic ´tale over F and „ = ι — 1.

e

(2) Central simple associative algebras of degree 3 over K with a unitary involution.

(3) Products K — (K — C) where C is a Hurwitz algebra of dimension 4 or 8 over F

and „ = (ι, ι — π) where π is the conjugation of the Hurwitz algebra C.

Proof : A cubic separable unital K-algebra is of the types described in Theorem

(??). (??) then follows by Galois descent and (??) follows from Proposition (??)

(the case of a Cayley algebra being proved as the quaternion case).

Assume that F has characteristic di¬erent from 3 and that K = F (ω) is a

¬eld. Let 2Sepalt m (3) be the groupoid of alternative F -algebras (B, „ ) which are

separable cubic of dimension m over K and have unitary involutions. The generic

polynomial of degree 3 on B with coe¬cients in K restricts to a polynomial function

on Sym(B, „ ) with coe¬cients in F . Let

Sym(B, „ )0 = { x ∈ Sym(B, „ ) | TB (x) = 0 }.

on Sym(B, „ )0 as in (??):

We de¬ne a multiplication

x y = µxy + (1 ’ µ)yx ’ 1 TB (yx)1.

(34.33) 3

1

The element x y lies in Sym(B, „ )0 since ι(µ) = 1 ’ µ. Let n(x) = ’ 3 TB (x2 ).

A description of Sym(B, „ ) (and of ) in cases (??) and (??) is obvious, however

less obvious in case (??).

(34.34) Lemma. Let C be a Hurwitz algebra over F , let K be a quadratic ´tale

e

F -algebra, and let B be the alternative K-algebra B = K — (K — C) with unitary

involution „ = (ι, ι — π). Then:

(1)

Sym(B, „ )0 = TC (z), ξ — z + ι(ξ) — π(z) z∈C

where ξ is a (¬xed ) generator of K such that TK (ξ) = ’1.

(2) If K = F (ω) where ω is a primitive cube root of unity, the map z ’ ω — z +

∼

ι(ω) — π(z) is an isomorphism (C, ) ’ Sym(B, „ )0 , .

’

Proof : (??) We obviously have

Sym(B, „ )0 ⊃ TC (z), ξ — z + ι(ξ) — π(z) z∈C ,

hence the claim by dimension count, since the map

Sym(B, „ )0 ’ TC (z), ξ — z + ι(ξ) — π(z) z∈C

given by z ’ ξ — z + ι(ξ) — π(z) is an isomorphism of vector spaces.

(??) follows from (??) and Proposition (??).

§34. SYMMETRIC COMPOSITIONS 477

(34.35) Theorem. Let F be a ¬eld of characteristic not 3 which does not contain

a primitive cube root of unity ω and let K = F (ω). For any cubic separable alterna-

tive K-algebra with a unitary involution „ , the F -vector space Sym(B, „ )0 , is a

symmetric composition algebra. Conversely, for any symmetric composition algebra

(S, ), the unital alternative K-algebra B = K · 1 • (K — S) with the multiplication

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

for x, y ∈ B 0 , admits the unitary involution (ι, ι — 1S ) and the functors

C : (B, „ ) ’ Sym(B, „ )0 ,

2

,

2

A : (S, ) ’ B = K · 1 • (K — S), (ι, ι — 1)

de¬ne an equivalence of groupoids

2

Sepalt m+1 (3) ≡ Scomp m

for m = 2, 4 and 8.

Proof : To check that Sym(B, „ )0 , is a symmetric composition algebra over F ,

we may assume that K = F — F . Then B is of the form (A, Aop ) and „ (a, bop ) =

(b, aop ) for A separable alternative of degree 3 as in the associative case (see (??)).

By projecting on the ¬rst factor, we obtain an isomorphism

Sym(B, „ )0 A0 = { x ∈ A | TA (x) = 0 },

and the product on A0 is as in (??). Thus the composition Sym(B, „ )0 , is

0

isomorphic to (A , ) and hence is a symmetric composition.

Conversely, if (S, ) is a symmetric composition algebra over F , then (S, ) — K

is a symmetric composition algebra over K and (K · 1 • K — S, ·) is a K-alternative

algebra by Theorem (??). The fact that (ι, ι — 1) is a unitary involution on B

follows from the de¬nition of the multiplication of B. That 2 C and 2A de¬ne an

equivalence of groupoids then follows as in (??).

(34.36) Corollary. Let (S, ) be a symmetric composition algebra with norm n.

The following conditions are equivalent:

(1) S contains a nontrivial idempotent.

(2) the cubic form N (x) = bn (x x, x) is isotropic.

(3) the alternative algebra 1A(S) (if F contains a primitive cube root of unity,

otherwise 2A(S)) has zero divisors.

Proof : (??) ’ (??) By Proposition (??) the map •(x) = e (e x) is an auto-

morphism of (S, ) of order ¤ 3 and the corresponding subalgebra S0 (see Lemma

(??)) has dimension at least 2. For every nonzero x ∈ S0 with bn (x, e) = 0 we have

x e = e x = ’x, so that bn (x x, x) = ’bn (x x, e x) = n(x)bn (x, e) = 0.

The implication (??) ’ (??) is Lemma (??).

We check that (??) ” (??): If S is Okubo, then 2A(S) is central simple with

zero divisors and NrdB (x) = bn (x x, x) for x ∈ Sym(B, „ )0 , hence the claim in

this case. Since K — K — S always has zero divisors we are left with showing that

N is always isotropic on a para-Hurwitz algebra. Take x = 0 with bn (x, 1) = 0;

then bn (x x, x) = bn ’n(x)1, x = 0.

In the following classi¬cation of Elduque-Myung [?, p. 2487] “unique” always

means up to isomorphism:

478 VIII. COMPOSITION AND TRIALITY

(34.37) Theorem (Classi¬cation of symmetric compositions). Let F be a ¬eld of

characteristic = 3 and let (S, ) be a symmetric composition algebra over F .

(L0 , ) for a unique cubic ´tale F -algebra L. The

(1) If dimF S = 2, then S e

algebra S is para-quadratic if and only if L is not a ¬eld.

(2) If dimF S = 4, then S is isomorphic to a para-quaternion algebra (Q, ) for a

unique quaternion algebra Q.

(3) If dimF S = 8, then S is either isomorphic to

(a) a para-Cayley algebra (C, ) for a unique Cayley algebra C,

(b) an algebra of the form (A0 , ) for a unique central simple F -algebra A of

degree 3 if F contains a primitive cube root of unity, or

(c) S is of the form Sym(B, „ )0 , for a unique central simple F (ω)-algebra

B of degree 3 with an involution of the second kind if F does not contain a

primitive cube root of unity ω.

Proof : (??) was already proved in Proposition (??). Let K = F (ω). By Theo-

Sym(B, „ )0 , for some alternative K-algebra B with

rem (??) we have (S, )

a unitary involution „ . By Proposition (??) cases (??.??) and (??.??) occur if B

is central simple over K, and cases (??) and (??.??) occur when B K • K — C

for some Hurwitz algebra C of dimension 4 or 8. In view of Lemma (??), we must

have in the two last cases (S, ) (C , ) so that (S, ) is a para-Hurwitz algebra,

as asserted.

We call symmetric composition algebras as in (??.??) or (??.??) Okubo algebras

(the case where ω lies in F is not new!), we denote the corresponding groupoids

by 1Oku (when K = F — F ), 2Oku (when ω ∈ F ) and we set Oku = 1Oku

2

Oku. Assume that F does not contain a primitive cube root of unity ω and

let 2A2 be the full subgroupoid of A2 whose objects are algebras of degree 3 over

F (ω) with involution of the second kind. Since we have equivalences of groupoids

1

Oku ≡ 1A2 and 2Oku ≡ 2A2 , we also call symmetric composition in iOku symmetric

compositions of type iA2 .

(34.38) Corollary. Let F be ¬eld of characteristic di¬erent from 3. There exist

equivalences of groupoids

Scomp4 ≡ Hurw 4 ≡ Hurw 4 ≡ A1

and

1 2

Scomp 8 ≡ Hurw 8 Oku ≡ G2 A2 .

A2

34.E. Cohomological invariants of symmetric compositions. Three co-

homological invariants classify central simple algebras (B, „ ) of degree 3 with uni-

tary involutions „ (see Theorem (??)): f1 ∈ H 1 (F, µ2 ) (which determines the cen-

ter K), g2 ∈ H 2 F, (µ3[K] (which determines the K-algebra B) and f3 ∈ H 3 (F, µ2 )

(which determines the involution). We have a corresponding classi¬cation for sym-

metric compositions:

(34.39) Proposition. Let F be a ¬eld of characteristic not 2, 3. Dimension 8

symmetric compositions of

(1) type G2 are classi¬ed by one cohomological invariant f3 ∈ H 3 (F, µ2 ).

(2) type 1A2 (if ω ∈ F ) by one invariant g2 ∈ H 2 (F, µ3 ).

§35. CLIFFORD ALGEBRAS AND TRIALITY 479

(3) type 2A2 by two cohomological invariants g2 ∈ H 2 (F, Z/3Z) and f3 ∈ H 3 (F, µ2 ).

Proof : The claims follow from Theorem (??) and the corresponding classi¬cations

of Cayley algebras, resp. central simple algebras. For (??), one also has to observe

that if K = F (ω), then the action on µ3 twisted by a cocycle γ de¬ning K is the

usual action of the Galois group. Thus (µ3 )γ = Z/3Z.

Observe that a symmetric composition algebra S of type 1A2 with g2 = 0

comes from a division algebra A over F . However, its norm is always hyperbolic

(see Example (??)), hence f3 (S) = 0. On the other hand, in view of the following

example, there exist composition algebras of type 2A2 with invariants g2 (S) = 0

and f3 (S) = 0.

(34.40) Example. Over R there are no compositions of type 1A2 but there exist

compositions of type 2A2 with K = C and B = M3 (C). There are two classes of

involutions of the second type on M3 (C), the standard involution x ’ „ (x) where

„ (x) is the hermitian conjugate, and the involution Int(d) —¦ „ : x ’ d„ (x)d’1 ,

with d = diag(’1, 1, 1), which is distinguished. Since tr(x2 ) > 0 for any nonzero

hermitian 3 — 3 matrix x, the norm on Sym(B, „ )0 is anisotropic. The restriction

of the norm to Sym B, Int(d) —¦ „ 0 is hyperbolic. Observe that f3 Sym(B, „ )0 = 0

and f3 Sym B, Int(d) —¦ „ 0 = 0.

§35. Cli¬ord Algebras and Triality

35.A. The Cli¬ord algebra. Let (S, ) be a symmetric composition algebra

of dimension 8 over F with norm n. Let C(n) be the Cli¬ord algebra and C0 (n)

the even Cli¬ord algebra of (S, n). Let „ be the involution of C(n) which is the

identity on V . Let rx (y) = y x and x (y) = x y.

(35.1) Proposition. For any » ∈ F — , the map S ’ EndF (S • S) given by

0 »x

x’

rx 0

induces isomorphisms

∼

±S : C(»n), „ ’ EndF (S • S), σn⊥n

’

and

∼

±S : C0 (»n), „ ’ EndF (S), σn — EndF (S), σn ,

’

of algebras with involution.

Proof : We have rx —¦ x (y) = x —¦ rx (y) = »n(x) · y by Lemma (??). Thus the

existence of the map ±S follows from the universal property of the Cli¬ord algebra.

The fact that ±S is compatible with involutions is equivalent to

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

for all x, y, z, u in S. This formula follows from the associativity of n, since

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

The map ±S is an isomorphism by dimension count, since C(n) is central simple.

480 VIII. COMPOSITION AND TRIALITY

Let (V, q) be a quadratic space of even dimension. We call the class of the

Cli¬ord algebra [C(V, q)] ∈ Br(F ) the Cli¬ord invariant of (V, q). It follows from

Proposition (??) that, for any symmetric composition (S, , n) of dimension 8, the

discriminant and the Cli¬ord invariant of (S, »n) are trivial. Conversely, a quadratic

form of dimension 8 with trivial discriminant and trivial Cli¬ord invariant is, by the

following Proposition, similar to the norm n of a Cayley (or para-Cayley) algebra C:

(35.2) Proposition. Let (V, q) be a quadratic space of dimension 8. The following

condition are equivalent:

(1) (V, q) has trivial discriminant and trivial Cli¬ord invariant,