(??) ’ (??) 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

∼