x — y = [(1 — ρ)(x)] [(1 — ρ2 )(y)]

√

and Q(x) = (n — 1)(x). Let v = 3 ». Taking (1, v, v ’1 ) as a basis of L over F , we

can write any element of A0 — L as a sum x = a + bv + cv ’1 with a, b, c ∈ A0 , and

we have, (using that µω 2 + ω(1 ’ µ) = 0 and µω + ω 2 (1 ’ µ) = ’1),

β(x) = x — x = (a + bω 2 v + cωv ’1 ) —1(a + bωv + cω 2 v ’1 )

= a2 ’ 3 TA (a2 ) ’ bc + 3 TA (bc)

1 1

+ v[»’1 c2 ’ 3 TA (c2 ) ’ ab + 1 TA (ab)]

1

3

+ v ’1 [» b2 ’ 1 TA (b2 ) ’ ca + 1 TA (ca)]

3 3

= a a ’ (bc)0 + v»’1 [c c ’ (ba)0 ] + v ’1 »[b b ’ (ca)0 ]

where x0 = x ’ 1 TA (x) for x ∈ A. The form Q is given by

3

Q(a + bv + cv ’1 ) = ’ 3 SA—L (a + bv + cv ’1 )

1

1

= ’ 3 [SA (a) + SA (c)»’1 v + SA (b2 )»v ’1 ]

1

+ 3 [TA (bc) + TA (ab)v + TA (ca)v ’1 ]

504 VIII. COMPOSITION AND TRIALITY

and the norm N by

N (a + bv + cv ’1 ) = bQ a + bv + cv ’1 , β(a + bv + cv ’1 )

= bn (a, a a) + »bn (b, b b) + »’1 bn (c, c c)

+ 1 [bSA a, (bc)0 + bSA b, (ca)0 + bSA c, (ab)0 ]

3

= NA (a) + »NA (b) + »’1 NA (c) ’ TA (abc)

since bn (a, a a) = NA (a) and BSA a, (bc)0 = ’T a, (bc)0 = ’TA (abc).

Assume now that F does not necessarily contain a primitive cube root of unity

ω. Replacing F by F (ω) = F [X]/(X 2 + X + 1), we may de¬ne — on A0 — F (ω) — L.

However, since ω does not explicitly appear in the above formulas for β and Q

restricted to A0 —L, we obtain for any algebra A of degree 3 over F of characteristic 3

and for any » ∈ F — a twisted composition “(A, ») = (A0 — L, L, Q, β) over L =

√

F ( 3 »). A twisted composition “ similar to a composition “(A, ») for A associative

central simple and » ∈ F — is said to be a composition of type 1A2 .

Any pair (φ, ψ) ∈ AutF (A) — AutF (L) induces an automorphism of “(A, L).

Thus we have a morphism of group schemes PGL3 —µ3 ’ Spin8 S3 and:

(36.33) Proposition. Twisted compositions “(A, ») of type 1A2 are classi¬ed by

the image of H 1 (F, PGL3 —µ3 ) in H 1 (F, Spin8 S3 ).

(36.34) Remark. If F contains a primitive cube root of unity, µ3 = A3 and

the image under the morphism PGL3 —µ3 ’ Spin8 S3 of the group PGL3 =

Spin8 A3 is contained in Spin8 .

Let now (B, „ ) be central simple of degree 3 over a quadratic ´tale F -algebra K,

e

—

with a unitary involution „ . For ν ∈ K we have a twisted K-composition “(B, ν)

√

over K( 3 ν) which we would like (under certain conditions) to descent to a twisted

F -composition.

(36.35) Proposition. If NK (ν) = 1, then:

√

(1) There is an ι-semilinear automorphism ι of K( 3 ν) of order 2 which maps ν

to ν ’1 ; its set of ¬xed elements is a cubic ´tale F -algebra L with disc(L) K —F (ω)

e

(where ω is a primitive cube root of unity).

(2) There is an ι-semilinear automorphism of order 2 of the twisted K-composi-

tion “(B, „ ) such that its set of ¬xed elements is a twisted F -composition “(B, „, ν)

Sym(B, „ )0 • B 0 ; under this isomorphism we have, for

over L with “(B, „, ν) √ √

z = (x, y) ∈ Sym(B, „ )0 • B 0 and v = 3 ν ∈ K( 3 ν) = L — K,

SB (x) + TL—K/L SB (y)»v ’1 + TB (xy)v + TB y„ (y) ,

1

Q(z) = 3

β(z) = x2 ’ 3 TB (x2 ) ’ y„ (y) + 3 TB y„ (y)

1 1

+ ν[„ (y)2 ’ 3 TB „ (y)2 ] ’ xy + 1 TB (xy),

1

3

N (z) = NB (x) + νNB (y) + ν ’1 NB „ (y) ’ TB xy„ (y) .

Proof : (??) This is Proposition (??). √

(??) We have “(B, ν) = B 0 —K K( 3 ν) and take as our „ -semilinear automor-

√

phism the map „ = „ — „ . We write B 0 —K K( 3 ν) = B 0 • B 0 v • B 0 v ’1 . The

isomorphism Sym(B, „ )0 •B 0 “(B, „, ν) is then given by (x, y) ’ x+yv+„ (y)v ’1

and it is easy to check that its image lies in the descended object “(B, „, ν). The

formulas for Q, β, and N follow from the corresponding formulas for type 1A2 .

§36. TWISTED COMPOSITIONS 505

(36.36) Example. If K = F (ω), ω a primitive cubic root of 1, then the L given

by Proposition (??) is cyclic and the twisted composition “(B, „, ν) is the twisted

composition associated to the cyclic composition Sym(B, „ )0 — L.

A twisted composition isomorphic to a composition “(B, „, ν) is said to be of

type 2A2 . We have a homomorphism

GU3 (K) — (µ3 )γ ’ Spin8 S3

where γ is a cocycle de¬ning K and the analogue of Proposition (??) is:

(36.37) Proposition. Twisted compositions “(B, „, ν) of type 2A2 are classi¬ed

by the image of H 1 F, PGU3 (K) — (µ3 )γ in H 1 (F, Spin8 S3 ).

36.E. The dimension 2 case. If (V, L, Q, β) is a twisted composition with

rankL V = 2, then V admits, in fact, more structure:

(36.38) Proposition. Let (V, L, Q, β) be a twisted composition with dim L V = 2.

There exists a quadratic ´tale F -algebra K which operates on V and a nonsingular

e

L — K-hermitian form h : V — V ’ L — K of rank 1 such that Q(x) = h(x, x),

NK — » where » can be chosen such that NL/F (») ∈ F —2 .

x ∈ V . Hence Q

Furthermore the algebra K is split if Q is isotropic.

Proof : For generic v ∈ V , we may assume that Q(v) = » ∈ L— , bQ v, β(v) =

a ∈ F — , and that {v, β(v)} are linearly independent over L (see also the following

Remark). Then v1 = v, v2 = »β(v) is an L-basis of V , and

Q(x1 v1 + x2 v2 ) = x2 + ax1 x2 + nL/F (»)x2 · »

1 2

Thus 4NL/F (») ’ a2 = detL Q is nonzero and the quadratic F -algebra

K = F [x]/ x2 + ax + NL/F (»)

is ´tale. Let ιK be the conjugation map of K. Let ξ = x+ x2 +ax+NL/F (») ∈ K.

e

We de¬ne a K-module structure on V by putting

ξv1 = v2 and ξv2 = ’av1 ’ NL/F (»)v2 .

Thus v = v1 is a basis element for the L — K-module V . We then de¬ne

h(·1 v, ·2 v) = ·1 »¯2

·

for ·1 , ·2 ∈ L — K, and ·1 = (1 — ιK )(·1 ). In particular we have » = h(v, v)

¯

for the chosen element v. The fact that Q(x) = h(x, x), x ∈ V , follows from the

formula Q(x1 v1 + x2 v2 ) = x2 + ax1 x2 + NL/F (»)x2 · ». The last claim, i.e., that

1 2

—2

NL/F (») ∈ F , follows by choosing v of the form v = β(u). If Q is isotropic, Q

is hyperbolic by Proposition (??) and Corollary (??), hence Q NK — 1 and the

claim follows from Springer™s theorem.

(36.39) Remark. If bQ v, β(v) = 0 for v = 0 or if {v, β(v)} is linearly dependent

over L, the twisted composition is induced from a Hurwitz algebra (see Theo-

rem (??)).

506 VIII. COMPOSITION AND TRIALITY

Exercises

1. Let C be a separable alternative algebra of degree 2 over F . Show that π : x ’ x

is the unique F -linear automorphism of C such that x + π(x) ∈ F · 1 for all

x ∈ C.

2. Let (C, N ), (C , N ) be Hurwitz algebras of dimension ¤ 4. Show that an

∼

isometry N ’ N which maps 1 to 1 is either an isomorphism or an anti-

’

isomorphism. Give an example where this is not the case for Cayley algebras.

3. A symmetric composition algebra with identity is 1-dimensional.

4. (Petersson [?]) Let K be quadratic ´tale with norm N = NK and conjugation

e

x ’ x. Composition algebras (K, ) are either K (as a Hurwitz algebra) or”up

to isomorphism”of the form

(a) x y = xy,

(b) x y = xy, or

(c) x y = uxy for some u ∈ K such that N (u) = 1.

Compositions of type (??) are symmetric.

5. The split Cayley algebra over F can be regarded as the set of all matrices (Zorn

matrices) ± β with ±, β ∈ F and a, b ∈ F 3 , with multiplication

a

b

±a γ c ±γ + a · d ±c + δa ’ (b § d)

=

bβ d δ γb + βd + (a § c) βδ + b · c

where a · d is the standard scalar product in F 3 and b § d the standard vector

product (cross product). The conjugation is given by

±a β ’a

π =

bβ ’b ±

and the norm by

±a

= ±β ’ a · b.

n

bβ

6. Let K be a quadratic ´tale F -algebra and let (V, h) be a ternary hermitian space

e

over K with trivial (hermitian) discriminant, i.e., there exists an isomorphism

∼

φ : §3 (V, h) ’ 1 . For any v, w ∈ V , let v — w ∈ V be determined by the

’

condition h(u, v — w) = φ(u § v § w).

(a) Show that the vector space C(K, V ) = K • V is a Cayley algebra under

the multiplication

(a, v) (b, w) = ab ’ h(v, w), aw + bv + v — w

and the norm n (a, v) = NK/F (a) + h(v, v).

(b) Conversely, if C is a Cayley algebra and K is a quadratic ´tale subalgebra,

e

⊥

then V = K admits the structure of a hermitian space over K and

C C(K, V ).

(c) AutF (C, K) = SU3 (K).

There exists a monomorphism SL3 Z/2Z ’ G where G is split simple of

(d)

type G2 (i.e., “ A2 ‚ G2 ” ) such that H 1 (F, SL3 Z/2Z) ’ H 1 (F, G) is

surjective.

7. (a) Let Q be a quaternion algebra and let C = C(Q, a) be the Cayley algebra

Q • vQ with v 2 = a. Let AutF (C, Q) be the subgroup of automorphisms

of AutF (C) which map Q to Q. Show that there is an exact sequence

φ

1 ’ SL1 (Q) ’ AutF (C, Q) ’ AutF (Q) ’ 1

’

EXERCISES 507

where φ(y)(a + vb) = a + (vy)b for y ∈ SL1 (Q).

(b) The map SL1 (Q) — SL1 (Q) ’ AutF (C) induced by

(u, x) ’ (a + vb) ’ uau + (vx)(ubu)

is a group homomorphism (i.e., “A2 — A2 ‚ G2 ”).

(Elduque) Let S = (F4 , ) be the unique para-quadratic F2 -algebra. Show

8.

that 1-dimensional algebras and S are the only examples of power-associative

symmetric composition algebras.

9. Let F be a ¬eld of characteristic not 3. Let A be a central simple F -algebra

of degree 3. Compute the quadratic forms TA (x2 ) and SA (x) on A and on A0 ,

and determine their discriminants and their Cli¬ord invariants.

Let » ∈ F — and let (Q, n) be a quaternion algebra. Construct an isomorphism

10.

C(»Q, n), σ M2 (Q), σn⊥n .

Hint: Argue as in the proof of (??).

11. Let (C, , n) be a Cayley algebra and let (C, ) be the associated para-Cayley

algebra, with multiplication x y = x y. Show that

(x a) (a y) = a a (x y) .

(By using the Theorem of Cartan-Chevalley this gives another approach to

triality.)

12. (Elduque) Let C be a Cayley algebra, let (C, ) be the associated para-Cayley

algebra, and let (C• , ) be a Petersson algebra. Let t be a proper similitude

of (C, n), with multiplier µ(t).

(a) If t+ , t’ are such that µ(t)’1 t(x y) = t’ (x) t+ (y), show that

µ(t)’1 t(x y) = •’1 t’ •(x) •t+ •’1 (y).

(b) If θ+ is the automorphism of Spin(C, n) as de¬ned in Proposition (??)

¯

and if θ+ is the corresponding automorphism with respect to ¯, show that

—

¯+ = C(•)θ+ = θ+ C(•).

θ

13. Compute Spin(C, n) for (C, n) a symmetric composition algebra of dimension 2,

resp. 4.

14. Let C be a twisted Hurwitz composition over F — F — F .

(a) If C is a quaternion algebra, show that

AutF (C) = (C — — C — — C — )Det /F — S3

where

(C — — C — — C — )Det = { (a, b, c) ∈ C — | NC (a) = NC (b) = NC (c) }

and S3 acts by permuting the factors.

(b) If C is quadratic,

AutF (C) = SU1 (C) — SU1 (C) (Z/2Z — S3 )

where Z/2Z operates on SU1 (C) — SU1 (C) through (a, b) ’ (a, b) and S3

operates on SU1 (C) — SU1 (C) as in Lemma (??).

15. Describe the action of S3 (triality) on the Weyl group (Z/2Z)3 S4 of a split

simple group of type D4 .

508 VIII. COMPOSITION AND TRIALITY

Notes

§??. The notion of a generic polynomial, which is classical for associative alge-

bras, was extended to strictly power-associative algebras by Jacobson. A systematic

treatment is given in Chap. IV of [?], see also McCrimmon [?].

§??. Octonions (or the algebra of octaves) were discovered by Graves in 1843

and described in letters to Hamilton (see Hamilton [?, Vol. 3, Editor™s Appendix 3,

p. 648]); however Graves did not publish his result and octonions were rediscovered

by Cayley in 1845 [?, I, p. 127, XI, p. 368“371]. Their description as pairs of

quaternions (the “Cayley-Dickson process”) can be found in Dickson [?, p. 15].

Dickson was also the ¬rst to notice that octonions with positive de¬nite norm

function form a division algebra [?, p. 72].

The observation that x(xa) = (xx)a = (ax)x holds in an octonion algebra dates

back to Kirmse [?, p. 76]. The fact that Cayley algebras satisfy the alternative law

was conjectured by E. Artin and proved by Artin™s student Max Zorn in [?]. Artin™s

theorem (that a subalgebra of an alternative algebra generated by two elements is

associative) and the structure theorem (??) ¬rst appeared in [?]. The description of

split octonions as “vector matrices”, as well as the abstract Cayley-Dickson process,

are given in a later paper [?] of Zorn. The fact that the Lie algebra of derivations

of a Cayley algebra is of type G2 and the fact that the group of automorphisms

of the Lie algebra of derivations of a Cayley algebra is isomorphic to the group of

automorphisms of the Cayley algebra if F is a ¬eld of characteristic zero, is given

in Jacobson [?]. In this connection we observe that the Lie algebra of derivations of

the split Cayley algebra over a ¬eld of characteristic 3 has an ideal of dimension 7,

hence is not simple. The fact that the group of automorphisms of a Cayley algebra

is of type G2 is already mentioned without proof by E. Cartan [?, p. 298] [?, p. 433].

Other proofs are found in Freudenthal [?], done by computing the root system, or

in Springer [?], done by computing the dimension of the group and applying the

classi¬cation of simple algebraic groups. In [?] no assumption on the characteristic

of the base ¬eld is made.

Interesting historical information on octonions can be found in the papers of van

der Blij [?] and Veldkamp [?], see also the book of van der Waerden [?, Chap. 10].

The problem of determining all composition algebras has been treated by many

authors (see Jacobson [?] for references). Hurwitz [?] showed that the equation

(x2 + · · · + x2 )(y1 + · · · + yn ) = z1 + · · · + zn

2 2 2 2

1 n

has a solution given by bilinear forms z1 , . . . , zn in the variables x = (x1 , . . . , xn ),

y = (y1 , . . . , yn ) exactly for n = 1, 2, 4, and 8. The determination of all composition

algebras with identity over a ¬eld of characteristic not 2 is due to Jacobson [?]. We

used here the proof of van der Blij-Springer [?], which is also valid in characteristic 2.

A complete classi¬cation of composition algebras (even those without an identity)

is known in dimensions 2 (Petersson [?]) and 4 (Stamp¬‚i-Rollier [?]).

§??. Compositions algebras with associative norms were considered indepen-

dently by Petersson [?], Okubo [?], and Faulkner [?]. We suggest calling them

symmetric composition algebras in view of their very nice (and symmetric) proper-

ties. Applications of these algebras in physics can be found in a recent book [?] by

S. Okubo.

Petersson showed that over an algebraically closed ¬eld symmetric composi-

tions are either para-Hurwitz or, as we call them, Petersson compositions. Okubo

NOTES 509

described para-Cayley Algebras and “split Okubo algebras” as examples of sym-

metric composition algebras. In the paper [?] of Okubo-Osborn it is shown that

over an algebraically closed ¬eld these two types are the only examples of symmetric

composition algebras.

The fact that the trace zero elements in a cubic separable alternative algebra

carry the structure of a symmetric algebra was noticed by Faulkner [?]. The clas-

si¬cation of symmetric compositions, as given in Theorem (??), is due to Elduque-

Myung [?]. However they applied the Zorn Structure Theorem for separable alter-

native algebras, instead of invoking (as we do) the eigenspace decomposition of the

operator e for e an idempotent. The idea to consider such eigenspaces goes back

to Petersson [?]. A similar decomposition for the operator ade is used by Elduque-

Myung in [?]. Connections between the di¬erent constructions of symmetric alge-

bras are clearly described in Elduque-P´rez [?]. We take the opportunity to thank

e

A. Elduque, who detected an error in our ¬rst draft and who communicated [?] to

us before its publication.

Let (A0 , ) be a composition of type 1A2 . It follows from Theorem (??) that

AutF (A0 , ) AutF (A). This can also be viewed in terms of Lie algebras: Since

0

∼

x y’y x = µ(xy’yx), any isomorphism of compositions ± : (A0 , ) ’ (A , ) also

’

0

∼

induces a Lie algebra isomorphism L(A0 ) ’ L(A ). Conversely, (and assuming

’

0

∼

that F has characteristic 0) any isomorphism of Lie algebras L(A0 ) ’ L(A ) ’

∼

extends to an algebra isomorphism A ’ A or the negative of an anti-isomorphism

’

∼

of algebras A ’ A (Jacobson [?, Chap. X, Theorem 10]). However the negative of

’

an anti-isomorphism of algebras cannot restrict to an isomorphism of composition

algebras. In particular we see that AutF (A0 , ) is isomorphic to the connected

component AutF L(A0 ) 0 of AutF L(A0 ) .

§??. We introduce triality using symmetric composition algebras of dimen-

sion 8 and their Cli¬ord algebras. Most of the results for compositions of type G2

can already be found in van der Blij-Springer [?], Springer [?], Wonenburger [?], or

Jacobson [?, p. 78], [?]. However the presentation through Cli¬ord algebras given

here, which goes back to [?], is di¬erent. The use of symmetric compositions also

has the advantage of giving very symmetric formulas for triality. The isomorphism

∼

of algebras C(S, n) ’ EndF (S • S) for symmetric compositions of dimension 8

’

can already be found in the paper [?] of Okubo and Myung. A di¬erent approach

to triality can be found in the book of Chevalley [?].

Triality in relation to Lie groups is discussed brie¬‚y by E. Cartan [?, Vol. II,

1

§139] as an operation permuting the vector and the 2 -spinor representations of D4 .

The ¬rst systematic treatment is given in Freudenthal [?], where local triality (for

Lie algebras) and global triality is discussed.

There is also an (older) geometric notion of triality between points and spaces of

two kinds on a (complex) 6-dimensional quadric in P7 . These spaces correspond to

maximal isotropic spaces of the quadric given by the norm of octonions. Geometric

triality goes back to Study [?] and E. Cartan [?, pp. 369-370], see also [?, I, pp. 563“

565]; A systematic study of geometric triality is given in Vaney [?], Weiss [?], see

also Kuiper [?]. Geometric applications can be found in the book on “Punktreihen-

geometrie” of Weiss [?].

The connection between triality and octonions, already noticed by Cartan,

is used systematically by Vaney and Weiss. The existence of triality is, in fact,

“responsible” for the existence of Cayley algebras (see Tits [?]). A systematic

510 VIII. COMPOSITION AND TRIALITY

description of triality in projective geometry in relation to the theory of groups is

given in Tits [?].

The paper of van der Blij-Springer [?] gives a very nice introduction to triality

in algebra and geometry. There is also another survey article, by Adams [?].

§??. The notion of a twisted composition (due to Rost) was suggested by the

construction of cyclic compositions, due to Springer [?]. Many results of this section,

for example Theorem (??), were inspired by the notes [?].

CHAPTER IX

Cubic Jordan Algebras

The set of symmetric elements in an associative algebra with involution admits

the structure of a Jordan algebra. One aim of this chapter is to give some insight

into the relationship between involutions on central simple algebras and Jordan

algebras. After a short survey on central simple Jordan algebras in §??, we spe-

cialize to Jordan algebras of degree 3 in §??; in particular, we discuss extensively