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A0 — L by

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

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

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