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Local Triality and Some Related Algebras
Journal of Algebra 244, 828᎐844 Ž2001. doi:10.1006rjabr.2001.8889, available online at http:rrwww.idealibrary.com on
Local Triality and Some Related Algebras Alberto Elduque1 Departamento de Matematicas, Facultad de Ciencias, Uni¨ ersidad de Zaragoza, ´ 50009 Zaragoza, Spain E-mail: [email protected]
and Susumu Okubo 2 Department of Physics and Astronomy, Uni¨ ersity of Rochester, Rochester, New York 14627 Communicated by Efim Zelmano¨ Received May 1, 2001
Eight dimensional symmetric composition algebras give very nice formulae for triality and local triality. These latter formulae will be extended to any algebra in a suitable variety. The simple algebras in this variety will be shown to be precisely the symmetric composition algebras over fields, while the prime algebras are central orders in the simple ones. 䊚 2001 Academic Press
1. INTRODUCTION It is well known Žsee wvBS, KMRTx and the references therein . that given any Cayley᎐Dickson algebra Ž C, n., where n denotes its norm, and any element in the reduced orthogonal group O⬘Ž C, n., there are new elements ⬘, ⬙ g O⬘Ž C, n., determined up to a common sign, such that
Ž xy . s ⬘ Ž x . ⬙ Ž y . ,
Ž 1.1.
1 Supported by the Spanish DGES ŽPb 97-1291-C03-03. and by a grant from the Spanish Direccion Superior e Investigacion ´ General de Ensenanza ˜ ´ Cientifica ŽPrograma de Estancias de Investigadores Espanoles en Centros de Investigacion ˜ ´ Extranjeros., while the first author was visiting the University of Wisconsin at Madison. He thanks its Department of Mathematics and, especially, Professor Georgia Benkart. 2 Supported in part by U.S. Department of Energy Grant DE-F402-ER 40685.
828 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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for any x, y g C ŽPrinciple of Triality.. This provides the spin group SpinŽ C, n. with an automorphism of order 3 whose fixed subgroup is the group of automorphisms of C, an algebraic group of type G 2 . The local version or Principle of Local Triality Žsee wSc, Chap. IIIx. asserts that, assuming characteristic / 2, for any d in the orthogonal Lie algebra oŽ C, n., there are d⬘, d⬙ g oŽ C, n., uniquely determined, such that d Ž xy . s d⬘ Ž x . y q xd⬙ Ž y . ,
Ž 1.2.
for any x, y g C. Again this gives an automorphism of order 3 of oŽ C, n. such that its fixed subalgebra is the Lie algebra of derivations Der C, which is a classical simple Lie algebra of type G 2 if the characteristic is / 2, 3 Žbut it is not simple in characteristic 3ᎏsee wAEMNx.. It has been noted, however, in wKMRT, Chap. VIIIx that the situation above becomes more symmetrical if the so-called symmetric composition algebras Žof dimension 8. are used instead of the Cayley᎐Dickson algebras. An algebra A over a field F is said to be a symmetric composition algebra if it is endowed with a regular quadratic form q, the norm, satisfying q Ž xy . s q Ž x . q Ž y . q Ž xy, z . s q Ž x, yz .
Ž 1.3.a. Ž 1.3.b.
for any x, y, z g A, where q Ž x, y . s q Ž x q y . y q Ž x . y q Ž y .. It is known from wO1x that Eqs. Ž1.3. are equivalent to the validity of
Ž xy . x s x Ž yx . s q Ž x . y
Ž 1.4.
for any x, y g A. There are two types of symmetric composition algebras, the para-Hurwitz algebras Žand some forms of them in dimension 2. and the pseudo-octonion algebras Žsee wOO, EP, KMRT, E1x.. The para-Hurwitz algebras are defined as follows: given any Hurwitz algebra C with norm q and multiplication x ⭈ y, its canonical involution is given by x ¬ x s q Ž1, x .1 y x. Then the new algebra C, defined over C but with the new multiplication xy s x ⭈ y,
Ž 1.5.
is said to be the para-Hurwitz algebra related to C. It satisfies Ž1.3. Žor Ž1.4.. with the same norm q. We will restrict ourselves in what follows to considerations of the Principle of Local Triality only. Given an eight-dimensional symmetric composition algebra A with norm q over a field of characteristic / 2, it is shown in wKMRT, Chap. VIIIx that for any d 0 g oŽ A, q . there are uniquely determined d1 , d 2 g oŽ A, q . such that d 0 Ž xy . s d1 Ž x . y q xd 2 Ž y .
Ž 1.6.
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ELDUQUE AND OKUBO
for any x, y g A. Since Ž1.6. is equivalent, in view of Ž1.3.b., to q Ž d 0 Ž x . , yz . q q Ž d1 Ž y . , zx . q q Ž d 2 Ž z . , xy . s 0
Ž 1.7.
for any x, y, z g A, the maps d 0 , d1 , d 2 can be permuted cyclically in Ž1.6.. such triples Ž d 0 , d1 , d 2 . satisfying Ž1.6. Žor Ž1.7.. are called related triples. The map d 0 ¬ d1 is an automorphism of order 3 of oŽ A, q . whose fixed subalgebra is the derivation algebra of A, which for characteristic / 2, 3 is a simple Lie algebra of type G 2 or A 2 , depending on A being a para-Hurwitz or a pseudo-octonion algebra. In wE2x, some explicit formulae were computed for this automorphism. Actually, the results in wE2, Section 5x show that the related triples for A are spanned by the triples
ž
x , y , R x L y y
1 2
q Ž x, y . I, L x R y y
1 2
/
q Ž x, y . I ,
where x, y Ž z . s q Ž y, z . x y q Ž x, z . y. But, because of Ž1.4., q Ž x, y . z s Ž xz . y q Ž yz . x s x Ž zy . q y Ž zx . and hence Rx Ly y Lx R y y
1 2 1 2
q Ž x, y . I s q Ž x, y . I s
1 2 1 2
Ž R x Ly y R y Lx . , Ž Lx R y y Ly R x . .
Also,
x , y Ž z . s Ž zx . y q Ž yx . z y Ž zy . x y Ž xy . z s z Ž xy . q y Ž xz . y z Ž yx . y x Ž yz . , so
x , y s R y , R x y Lw x , y x s L y , L x q Rw x , y x . Hence the above spanning set of related triples is the set of those Ž d 0 , d1 , d 2 . g oŽ A, q . 3 with d 0 s d 0 Ž x, y . s Rw x , y x y L x , L y s y Ž Lw x , y x q R x , R y d1 s d1 Ž x, y . s d 2 s d 2 Ž x, y . s
1 2 1 2
Ž R x Ly y R y Lx . Ž Lx R y y Ly R x .
. Ž 1.8.
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for some x, y g A. In this way, the related triples are described in terms of the left and right multiplications in A. In this paper, two varieties of Žnonassociative. algebras will be introduced. Given a unital commutative and associative ground ring ⌽, let S be the variety of ⌽-algebras satisfying the identities
Ž xy . x s x Ž yx .
Ž flexible law .
Ux y , z s Ux , y z ,
Ž 1.9.a. Ž 1.9.b.
where Ux s L x R x s R x L x Žbecause of Ž1.9a.. and Ux, y s Uxqy y Ux y Uy . These are natural identities in view of Ž1.3. and Ž1.4.. Moreover, after studying some consequences of these identities and giving some examples, it will be shown that, assuming ⌽ acts without 2-torsion Ž2 x s 0 implies x s 0., the simple algebras in S are precisely the symmetric composition algebras over a field extension of ⌽ Ža field K with a unital ring homomorphism ⌽ ª K .. Thus the symmetric composition algebras over a field F are characterized as the central simple algebras in the variety S over F. Prime algebras in S will also be determined in terms of symmetric composition algebras. It must be remarked here that the first appearance of symmetric composition algebras in the literature Žalthough not with this name., to the authors’ knowledge, happened in the paper wPx, where the simple algebras with an idempotent over an algebraically closed field of characteristic / 2, 3, in the variety defined by the flexible law and by the identity of degree 5 given by Ux y s Ux Uy ŽU as above., were shown to be symmetric composition algebras Žeither para-Hurwitz algebras or the so-called Petersson algebrasᎏsee wKMRT, Chap. VIIIx.. Under some restrictions, the identity Ux Uy s Ux y follows from Ž1.9. Žsee Proposition 2.3.. Also under some restrictions on the algebra A in the variety S , the operators Ux are in the centroid of the algebra A, in particular for semiprime algebras in S Žassuming no 2-torsion.. This suggests the consideration of the subvariety S C of S defined by Ž1.9.a., Ž1.9.b., and the extra identity Ux Ž yz . s Ž Ux y . z.
Ž 1.9.c.
The study of the varieties S and S C will be the objective of the next section. Another purpose of this paper is to show that, assuming 12 g ⌽, the same formulae for d 0 Ž x, y ., d1Ž x, y ., and d 2 Ž x, y . in Ž1.8. remain valid for any algebra in S C , although the proof of this fact is necessarily very different from the proof in wE2x, which was only valid for eight-dimensional symmetric composition algebras. That is, for any algebra A in S C , it will be shown in Section 4 that d i Ž x, y . Ž u¨ . s Ž d iq1 Ž x, y . u . ¨ q u Ž d iq2 Ž x, y . ¨ .
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ELDUQUE AND OKUBO
Žindices modulo 3. for any x, y, u, ¨ g A.
2. THE VARIETIES S AND S C Let ⌽ be a unital commutative and associative ring and let S be the variety of algebras over ⌽ defined by Ž1.9.a, b., that is,
LEMMA 2.1.
Ž xy . x s x Ž yx . ,
Ž 2.1.a.
Ux y , z s Ux , y z .
Ž 2.1.b.
Let A be an algebra in S . Then for any x, y, z g A,
Ž Ux y . z s y Ž Ux z . ,
Ž 2.2.
UU x y , z s Uy , U x z .
Ž 2.3.
Proof. For Ž2.2., compute Ž zx .Ž xy . in two ways using Ž2.1.a.,
Ž zx . Ž xy . s Uz , x y x y Ž Ž xy . x . z s Uz , x y x y Ž Ux y . z, Ž zx . Ž xy . s Uz x , y x y y Ž x Ž zx . . s Uz x , y x y y Ž Ux z . , and Ž2.2. follows since Uz, x y s Uz x, y by Ž2.1.b.. For Ž2.3., just use Ž2.1. to get UU x y , z s UŽ x y . x , z s Ux y , x z s Uy , Ž x z . x s Uy , U x z .
LEMMA 2.2. Let A be an algebra in S . Then I s x g A : Ux s 0 s Ux, A 4 is an ideal of A. Proof. Let x g I and y, z g A, then Ux y, z s Ux, y z g Ux, A s 0 for any z g A, so Ux y, A s 0. Also, Ux y z s Ž Ž xy . z . Ž xy . s y Ž Ž zy . x . Ž xy . s Ž Ž xy . x . Ž zy .
since Ux , z s 0, since Ux y , A s 0,
s Ž Ux y . Ž zy . s 0. Therefore xy g I, and similarly one proves that yx g I, as required. Note that Ž2.1.b. immediately implies that J s x g A : Ux, A s 04 is an ideal of A. If ⌽ acts without 2-torsion on A and x g J, then for any
LOCAL TRIALITY AND RELATED ALGEBRAS
833
y g A, 0 s Ux, x y s 2Ux y, so Ux y s 0 and x g I. Therefore I s J if A has no 2-torsion. PROPOSITION 2.3.
Let A be an algebra in the ¨ ariety S .
Ži. If x g A : Ux, A s 04 s 0, then, for any x g A, Ux belongs to the centroid ⌫ Ž A. of A Ž that is, Ux Ž yz . s ŽUx y . z s y ŽUx z . for any x, y, z g A.. Žii. If Ux g ⌫ Ž A. for any x g A and ⌽ acts without 2-torsion on A, then Ux y s Ux Uy for any x, y g A. Žiii. If x g A : xA s Ax s 04 s 0 and Ux y s Ux Uy for any x, y g A, then Ux g ⌫ Ž A. for any x g A. Proof. For Ži., let J s x g A : Ux, A s 04 . Given t, x, y, z g A, Ut , U x Ž y z . s UU x t , y z
by Ž 2.3. ,
s UŽU x t . y , z
by Ž 2.1.b . ,
s UtŽU x y ., z
by Ž 2.2. ,
s Ut , ŽU x y . z
by Ž 2.1.b . .
Hence Ux Ž yz . y ŽUx y . z g J s 0 and Ux Ž yz . s ŽUx y . z, which together with Ž2.2. show that Ux g ⌫ Ž A.. For Žii., given any x, y g A, Ux L y s L y Ux s LU x y and Ux R y s R y Ux s RU x y because Ux g ⌫ Ž A.. Hence Ux Uy s Ux L y R y s LU x y R y
and
Ux Uy s Ux L y R y s L y Ux R y s L y RU x y ,
so 2Ux Uy s LU x y R y q L y RU x y s UU x y , y s UŽ x y . x , y s Ux y , x y s 2Ux y , and, because of our assumption on the action of ⌽ on A, Ux y s Ux Uy Žs Uy Ux since Ux g ⌫ Ž A... Finally, for Žiii., given any t, x, y, z g A, t Ž Ux Ž yz . . s Ž Ux t . Ž yz .
Ž by Ž 2.2. .
s UU x t , z y y z Ž y Ž Ux t . . s UxŽ t x ., z y y z Ž Ž Ux y . t .
Ž by Ž 2.2. .
s Ut x , z x y y z Ž Ž Ux y . t .
Ž by Ž 2.1.b. .
s Ut , z Ž Ux y . y z Ž Ž Ux y . t . Ž since Uab s UaUb for any a, b . s t Ž Ž Ux y . z . .
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Hence AŽUx Ž yz . y ŽUx y . z . s 0. Similarly one proves that
Ž Ux Ž yz . . t s Ž yz . Ž Ux t . s Uy , U t z y Ž Ž Ux t . z . y x
s Uy x , t x z y Ž t Ž Ux z . . y s Uy , t Ž Ux z . y Ž t Ž Ux z . . y s Ž y Ž Ux z . . t s Ž Ž Ux y . z . t , so Ux Ž yz . y ŽUx y . z s Ux Ž yz . y y ŽUx z . g x g A : Ax s xA s 04 s 0, as required. In view of Proposition 2.3, it is natural to consider the subvariety S C of S defined by Ž2.1.a., Ž2.1.b., and Ux Ž yz . s Ž Ux y . z s x Ž Uy z . .
Ž 2.1.c.
Before proceeding, let us pause to show an example of an algebra in S but not in S C : EXAMPLE 2.4. Let A be the free module over ⌽ with a basis a, b, c, d, e, f 4 and a commutative product determined by ab s c, ac s d, ae s yf , bc s e, bd s yf , c 2 s f , and all the other products among basic elements equal to 0. Note that A is flexible Ž2.1.a. since it is commutative. Also, A2 s span² c, d, e, f :, A3 s span² d, e, f :, A4 s span² f :, and A5 s 0. Let us check that UA, A 2 s 0, which immediately implies Ž2.1.b .. Since A 5 s 0, UA , A 2 A s span²Ua, c a, Ua, c b, Ub, c a, Ub, c b :. But Ua, c a s a2 c q Ž ca . a s da s 0, Ua, c b s Ž ab . c q Ž cb . a s c 2 q ae s f y f s 0, Ub , c a s Ž ba. c q Ž ca . b s c 2 q bd s f y f s 0, Ub , c b s b 2 c q Ž bc . b s be s 0. Therefore, A belongs to the variety S . Now, note that UaŽ b 2 . s Ua 0 s 0, but
Ž Ua b . b s Ž Ž ab . a. b s Ž ca. b s db s yf / 0, so Ž2.1.c. is not satisfied and A is not in S C .
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LOCAL TRIALITY AND RELATED ALGEBRAS
The variety S C is contained in a wider variety, that of the flexible Malcev-admissible algebras Žsee wMx and references therein for results on these algebras.. THEOREM 2.5.
Any algebra in the ¨ ariety S C is Malce¨ -admissible.
Proof. Following wO2x, we note first that for any x, y in an algebra A in S C,
w x, y x , x s Ž xy y yx . x y x Ž xy y yx . s Ž xy . x q x Ž yx . y Ž yx . x y x Ž xy . s 2Ux y y 2Ux , y x q x 2 ( y,
Ž 2.4.
where a( b s ab q ba. Then, for any x, y, z g A,
w y, z x , x , x s y x, w y, z x , x s y2Ux w y, z x q 2Ux , w y , z x x y x 2 ( w y, z x
Ž 2.5.
and
w z, x x , x , y s y w x, z x , x , y s y2 w Ux z, y x q 2 w Ux , z x, y x y x 2 ( z, y s y2Ux w z, y x q 2Ux , z w x, y x y x 2 ( z, y
Ž 2.6.
since Ux and Ux, z are in the centroid. We next set w s w x, y x for simplicity and evaluate
w x, y x , z , x y w x, y x , w x, z x s w w, z x , x y w, w x, z x s w w, z x , x q w x, z x , w s 2Ux , w z y 2Ux , z w y 2Uw , z x q Ž x ( w . ( z, where we have used the linearization of Ž2.4.. But Ux , w s Ux , w x , y x s Uw x , x x, y s 0 by Ž2.1.b. and x ( w s x ( w x, y x s x Ž xy y yx . q Ž xy y yx . x s x Ž xy . y Ž yx . x
by Ž 2.1.a . ,
s Ux , y x y yx 2 y Uy , x x q x 2 y s x2, y .
Ž 2.7.
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Therefore Ž2.7. becomes
w x, y x , z , x y w x, y x , w x, z x s y2Ux , z w x, y x y 2Uw x , y x, z x q x 2 , y ( z. Adding this to Ž2.5. and Ž2.6. gives the Malcev identity
w y, z x , x , x q w z, x x , x , y q w x, y x , z , x y w x, y x , w x, z x s y2Ux w y, z x q 2Ux , w y , z x x y x 2 ( w y, z x y 2Ux w z, y x q 2Ux , z w x, y x y x 2 ( z, y y 2Ux , z w x, y x y 2Uw x , y x, z x q x 2 , y ( zs0 since Uw x, y x, z s Ux, w y, z x by Ž2.1.b. and w x 2 ( z, y x s w x 2 , y x( z y x 2 (w y, z x by flexibility Ž2.1.a. Žad y is contained in DerŽ A, (... Given a Malcev algebra M with multiplication w x, y x, its nucleus is N Ž M . s x g M : J Ž x, M, M . s 0 4 , where J Ž x, y, z . s ww x, y x, z x q ww y, z x, x x q ww z, x x, y x is the Jacobian of x, y, z. For any x g N Ž M ., ad x : y ¬ w x, y x is a derivation of M. Also, in the absence of 2-torsion, for any x, y g M the operator d Ž x, y . s ad w x , y x q ad x , ad y is a derivation of M Žsee wSax.. Hence, if the ground ring acts without 2-torsion, given any flexible Malcev-admissible algebra A, both ad x , for x g N Ž A., and d Ž x, y ., for x, y g A, are derivations of both Ž A, w , x. and Ž A, (. and hence of A. We finish this section with some examples of algebras in S C , which are not symmetric composition algebras. EXAMPLE 2.6. Any commutative and associative algebra A is a S Calgebra. EXAMPLE 2.7. Let F be a field of characteristic / 3 containing the cubic roots 1, , 2 of 1 and let A be the algebra with a basis e, f 4 and multiplication given by e 2 s e, f 2 s 0, ef s y f , fe s y 2 f . A is easily seen to be flexible and, besides, Ue s id, Uf s Ue, f s 0, from which Ž2.1.b. and Ž2.1.c. follow at once.
LOCAL TRIALITY AND RELATED ALGEBRAS
837
EXAMPLE 2.8. Given a quaternion algebra Q with standard involution x ¬ x over a field F and a nonzero scalar g F, the Cayley᎐Dickson algebra C Ž Q, . is the algebra defined on Q [ Q with multiplication given by
Ž x 0 , x1 . Ž y 0 , y1 . s Ž x 0 y 0 q y1 x1 , y1 x 0 q x1 y 0 . and involution given by
Ž x 0 , x 1 . s Ž x 0 , yx 1 . . Therefore, the product in the associated para-Cayley algebra C Ž Q, . is
Ž x 0 , x 1 . ⭈ Ž y 0 , y 1 . s Ž x 0 , x 1 . Ž y 0 , y 1 . s Ž x 0 , yx 1 . Ž y 0 , yy1 . s Ž x 0 y 0 q y 1 x 1 , yy 1 x 0 y x 1 y 0 . , and this is a symmetric composition algebra, so it is an algebra in S C . By an argument of continuity in the Zariski topology, the same is true of the algebra C Ž Q, 0 . with multiplication
Ž x 0 , x 1 . ⭈ Ž y 0 , y 1 . s Ž x 0 y 0 , yy1 x 0 y x 1 y 0 . , which is no longer a symmetric composition algebra.
3. SIMPLE AND PRIME ALGEBRAS IN S In order to deal with the simple and prime algebras in the variety S , we first check that the ideal J s x g A : Ux, A s 04 , considered in the previous section, is trivial for these algebras. More generally, we have PROPOSITION 3.1. 0 s Ux, A 4 s 0.
Let A be a semiprime algebra in S , then x g A : Ux s
Proof. By Lemma 2.2, I s x g A : Ux s 0 s Ux, A 4 is an ideal of A. Assume I / 0, then we can take an element 0 / x g I with x 2 s 0. Otherwise, if y s x 2 / 0, then y 2 s x 2 x 2 s Ux, x 2 x y Ž x 2 x . x s Ux, x 2 x y ŽUx x . x s 0 and x can be replaced with y. Note that given any ideal K of A contained in I, K 2 is again an ideal of A Žsince Ž u¨ . a s Uu, a¨ y Ž a¨ . u s yŽ a¨ . u g K 2 for any u, ¨ g K and a g A.. Let us denote by K the ideal generated by x g I above Ž x 2 s 0.. Let us first prove that K 2 s xK q Kx. It is clear that xK q Kx : K 2 . For the converse, note that given a, b g A and z g I, L a L b z s a Ž bz . s yz Ž ba. s yR b a z, R a R b z s Ž zb . a s y Ž ab . z s yL ab z.
Ž 3.1.
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ELDUQUE AND OKUBO
Therefore, K is the span of the elements
Ž La R a 1
2
L a 3 ⭈⭈⭈ . Ž x .
and
Ž R a La 1
2
R a 3 ⭈⭈⭈ . Ž x . .
If u s Ž R a1 L a 2 R a 3 ⭈⭈⭈ .Ž x . and ¨ g K, by Ž3.1., u¨ s Ž R ¨ R a1 L a 2 R a 3 ⭈⭈⭈ . Ž x . s y Ž L¨ a1 L a 2 R a 3 ⭈⭈⭈ . Ž x . s Ž R a 2 Ž ¨ a1 . R a 3 ⭈⭈⭈ . Ž x . s ⭈⭈⭈ s Ta Ž x . , where T s L or R and a g K. Hence u¨ g Kx or u¨ g xK. Similarly, if ¨ s Ž L a1 R a 2 L a 3 ⭈⭈⭈ .Ž x . and u g K, either u¨ g Kx or u¨ g xK. Finally, if u s Ž L a1 R a 2 L a 3 ⭈⭈⭈ .Ž x . and ¨ s Ž R b 1 L b 2 R b 3 ⭈⭈⭈ .Ž x ., u s a1 u1 and ¨ s ¨ 1 b1 with u1 s Ž R a 2 L a 3 ⭈⭈⭈ .Ž x . and ¨ 1 s Ž L b 2 R b 3 ⭈⭈⭈ .Ž x .. Then, u¨ s Ž a1 u1 . Ž ¨ 1 b1 . s y Ž Ž ¨ 1 b1 . u1 . a1 s Ž Ž u1 b1 . ¨ 1 . a1 s y Ž a1¨ 1 . Ž u1 b1 . and, by the arguments above, a1¨ 1 s Ta x and u1 b1 s Tb x for some a, b g A. Now note that
Ž ax . Ž bx . s y Ž Ž bx . x . a s Ž x 2 b . a s 0, Ž xa. Ž xb . s yb Ž x Ž xa. . s b Ž ax 2 . s 0, Ž ax . Ž xb . s y Ž Ž xb . x . a s Ž Ux b . a s 0,
Ž 3.2.
Ž xa. Ž bx . s y Ž Ž bx . a. x g Kx s yx Ž b Ž xa. . g xK , and the proof of the fact that K 2 s xK q Kx is complete. To finish the proof of the proposition, note that because of Ž3.2., Ž xK q Kx . 2 s Ž xK .Ž Kx . : Ž xK l Kx . and Ž xK l Kx . 2 : Ž Kx .Ž xK . s 0. Hence ŽŽ K 2 . 2 . 2 s 0 and K s 0 by semiprimeness, a contradiction. Hence, in view of Proposition 2.3.i, if ⌽ acts without 2-torsion, the semiprime algebras Žand hence the simple or prime ones. in S and S C coincide. THEOREM 3.2. Let A be a simple algebra in S such that ⌽ acts without 2-torsion on it. Then there is a field extension F of ⌽ such that A is a symmetric composition algebra o¨ er F.
LOCAL TRIALITY AND RELATED ALGEBRAS
839
Con¨ ersely, any symmetric composition algebra o¨ er an extension field of ⌽ is a simple algebra in the ¨ ariety S . Proof. Any symmetric composition algebra A over a field F, with quadratic form q: A ª F, satisfies Ž2.1.a. and Ž2.1.b., where Ux is just the multiplication by the scalar q Ž x .. Besides, since Ž xy . z q Ž zy . x s q Ž x, z . y for any x, y, z g A and q is nondegenerate, any element y g A is in the ideal generated by any other element x g A, so A is simple. Let A now be a simple algebra in S and assume that ⌽ acts without 2-torsion. Then, by Proposition 3.1, I s J s 0, where J s x g A : Ux, A s 04 . Proposition 2.3.i shows that Ux g ⌫ Ž A., which is a field F, an extension of ⌽, by simplicity. Now, U: A ª F; x ¬ Ux is a regular quadratic form, since J s 0; and A satisfies Ž xy . x s x Ž yx . s Ux y, which is Ž1.4.. Therefore, A is a symmetric composition algebra. As for prime algebras in S , we have THEOREM 3.3. Let A be a prime algebra in S on which ⌽ acts without 2-torsion. Then A is a central order in a symmetric composition algebra o¨ er a field extension F of ⌽. Con¨ ersely, any ⌽-algebra which is such a central order is a prime algebra in the ¨ ariety S . Proof. By primeness, the centroid S s ⌫ Ž A. of the prime algebra A is an integral domain, extension of ⌽, which acts without 2-torsion on A by our assumptions. Again by Proposition 3.1, J s x g A : Ux, A s 04 s 0 and, by Proposition 2.3.i, Ux g S for any x g S. Let F be the quotient field of S. Since Ux g S for any x g A, the algebra of fractions Sy1A satisfies Ž xy . x s x Ž yx . s q Ž x . y, where q Ž a . s Uars 2 g F. The q thus defined is a s quadratic form and it is nondegenerate because J s 0. Hence Sy1A is a symmetric composition algebra, as required. The converse is clear. COROLLARY 3.4. Assume that the ground ring ⌽ is a field F of characteristic / 2 and let A be a finite-dimensional semiprime algebra in S . Then A is a direct sum of symmetric composition algebras o¨ er finite extension fields of F. Proof. Since A is semiprime, A is a subdirect sum of prime algebras. By finite dimensionality, there are prime ideals P1 , . . . , Pr of A such that P1 l ⭈⭈⭈ l Pr s 0. Take r minimal. Now ArP1 is prime but its centroid is a finite-dimensional integral domain over F, so it is a field K 1 extension of F. By the proof of Theorem 3.3, ArP1 is a symmetric composition algebra over K 1 and hence it is simple. But then P1 is a maximal ideal of A and, therefore, A s P1 [ Ž P2 l ⭈⭈⭈ l Pr . ( P1 [ ArP1. Now P1 ( ArP2 l ⭈⭈⭈ l Pr is semiprime, and an induction on r completes the proof.
840
ELDUQUE AND OKUBO
4. LOCAL TRIALITY IN S C Throughout this section, all the algebras considered will be over a ground ring ⌽ containing 12 . Given any elements x, y in an algebra A in the variety S C , consider the endomorphisms of A given by the same formulae Ž1.8. that work for eight-dimensional symmetric-composition algebras. That is, d 0 Ž x, y . s Uy , y x y Ux , y y s R w x , y x y L x , L y s y Ž Lw x , y x q R x , R y d1 Ž x, y . s d 2 Ž x, y . s
1 2 1 2
.,
Ž 4.1.a.
Ž R x Ly y R y Lx . ,
Ž 4.1.b.
Ž Lx R y y Ly R x . .
Ž 4.1.c.
Then, we have PROPOSITION 4.1. With the same assumptions as abo¨ e, for any a, b, x, y, u, ¨ g A and j s 0, 1, 2, Ži. Ud Ž a, b. x, y q Ux, d Ž a, b. y s 0, j j Žii. Ux d j Ž a, b . s d j ŽUx a, b . s d j Ž a, Ux b . s d j Ž a, b .Ux , and Žiii. w d j Ž a, b ., d j Ž u, ¨ .x s d j Ž a, Ub, u¨ . q d j Ž b, Ua, ¨ u. y d j Ž a, Ub, ¨ u. yd j Ž b, Ua, u¨ .. Proof. Item Ži. follows from Ž2.1.b. and Žii. follows from Ž2.1.c.. Let us check first Žiii. for j s 0. Since Ux, y is in the centroid for any x, y g A, d 0 Ž a, b . , d 0 Ž u, ¨ . Ž z . s Ub , U¨ , z uyUu , z ¨ a y Ua, U¨ , z uyUu , z ¨ b y U¨ , Ub , z ayUa , z b u q Uu , Ub , z ayUa , z b¨ s Ub , uU¨ , z a y Ub , ¨ Uu , z a y Ua, uU¨ , z b q Ua, ¨ Uu , z b y U¨ , aUb , z u q U¨ , b Ua, z u q Uu , aUb , z¨ y Uu , b Ua, z¨ s Ž UUb , u ¨ , z a y Ua, z Ub , u¨ . y Ž UUb , ¨ u , z a y Ua, z Ub , ¨ u . y Ž UUa , u ¨ , z b y Ub , z Ua, u¨ . q Ž UUa , ¨ u , z b y Ub , z Ua, ¨ u . s Ž d 0 Ž a, Ub , u¨ . y d 0 Ž a, Ub , ¨ u . y d 0 Ž b, Ua, u¨ . q d 0 Ž b, Ua, ¨ u . . Ž z . .
841
LOCAL TRIALITY AND RELATED ALGEBRAS
Now, for j s 1 Žthe case j s 2 is similar., R a L b R u L¨ s R a Ž Ub , u y L u R b . L¨ s Ub , u R a L¨ y R a L u R b L¨ s Ub , u R a L¨ y Ž Ua, u y R u L a . Ž Ub , ¨ y R ¨ L b . s Ub , u R a L¨ y Ua, uUb , ¨ q Ub , ¨ R u L a q Ua, u R ¨ L b y R u L a R ¨ L b s Ub , u R a L¨ y Ua, u Ž Ub , ¨ y R ¨ L b . q Ub , ¨ R u L a y R u Ž Ua, ¨ y L¨ R a . L b s Ub , u R a L¨ q Ub , ¨ R u L a y Ua, u R b L¨ y Ua, ¨ R u L b q R u L¨ R a L b . Thus
w R a L b , R u L¨ x s R a LUb , u ¨ q RUb , u L a y R b LUa , u ¨ y RUa , u L b . ¨
¨
Letting a l b and u l ¨ leads to 4 d1 Ž a, b . , d1 Ž u, ¨ . s w R a L b y R b L a , R u L¨ y R ¨ L u x s R a LUb , u ¨ q RUb , ¨ u L a y R b LUa , u ¨ y RUa , ¨ u L b y R a LUb , ¨ u y RUb , u ¨ L a q R b LUa , ¨ u q RUa , u ¨ L b y R b LUa , u ¨ y RUa , ¨ u L b q R a LUb , u ¨ q RUb , ¨ u L a q R b LUa , ¨ u q RUa , u ¨ L b y R a LUb , ¨ u y RUb , u ¨ L a s 2 Ž Ž R a LUb , u ¨ y RUb , u ¨ L a . y Ž R b LUa , u ¨ y RUa , u ¨ L b . y Ž R a LUb , ¨ u y RUb , ¨ u L a . q Ž R b LUa , ¨ u y RUa , ¨ u L b . . s 4 Ž d1 Ž a, Ub , u¨ . q d1 Ž b, Ua, ¨ u . y d1 Ž a, Ub , ¨ u . y d1 Ž b, Ua, u¨ . . .
Proposition 4.1.iii shows that, for any j s 0, 1, 2, the linear span of the d j Ž a, b .’s forms a Lie subalgebra of End ⌽ Ž A.. Actually, it is a subalgebra of the Lie multiplication algebra. Given a quaternion algebra Q over a field F with norm q and the corresponding para-quaternion algebra Q, the d 0 Ž x, y .’s are just the usual generators of the orthogonal Lie algebra oŽ Q, q ., d 0 Ž x, y . s q Ž y, y . x y q Ž x, y . y, so the Lie algebra spanned by the d 0 Ž x, y .’s is oŽ Q, q ..
842
ELDUQUE AND OKUBO
On the other hand, if x ⭈ y denotes the product in Q, then xy s x ⭈ y is the product in Q and, for any x, y g Q, R x L y Ž z . s Ž yz . x s Ž y ⭈ z . x s y ⭈ z ⭈ x s z ⭈ y ⭈ x and hence d1Ž x, y . is one-half of the right multiplication in A by y ⭈ x y x ⭈ y. Thus the Lie algebra spanned by the d1Ž x, y .’s is the Lie algebra of right multiplications in Q by elements of zero trace; R Qⴢ 0 . Similarly, the Lie algebra spanned by the d 2 Ž x, y .’s is the Lie algebra of left multiplications by elements of zero trace, LⴢQ 0 . It is well-known that oŽ Q, n. s LⴢQ 0 [ R Qⴢ 0 . THEOREM 4.2. a, b, x, y g A,
Let A be an algebra in the ¨ ariety S C . Then for any
d 0 Ž a, b . Ž xy . s Ž d1 Ž a, b . x . y q x Ž d 2 Ž a, b . y . ,
Ž 4.2.a.
d1 Ž a, b . Ž xy . s Ž d 2 Ž a, b . x . y q x Ž d 0 Ž a, b . y . ,
Ž 4.2.b.
d 2 Ž a, b . Ž xy . s Ž d 0 Ž a, b . x . y q x Ž d1 Ž a, b . y . .
Ž 4.2.c.
Proof. Because of Ž2.1.b.,
Ž d1Ž a, b . x . y q x Ž d 2 Ž a, b . y . s 12 Ž Ž Ž bx . a y Ž ax . b . y q x Ž a Ž yb . y b Ž ya . . . s
1 2
Ž Ub x , y a y Ž ya. Ž bx . y Ua x , y b q Ž yb . Ž ax . qUx , y b a y Ž yb . Ž ax . y Ux , y a b q Ž ya . Ž bx . .
s Ub , x y a y Ua, x y b s d 0 Ž a, b . Ž xy . , thus obtaining Ž4.2.a.. Now, Eq. Ž4.2.b. is equivalent to d1 Ž a, b . R y s R y d 2 Ž a, b . q R d 0 Ž a, b. y .
Ž 4.3.
However, thanks to Ž2.1.c., R d 0 Ž a, b. y s RUb , y ayUa , y b s Ub , y R a y Ua, y R b . Hence Ž4.3. becomes
Ž R a L b y R b L a . R y y R y Ž L a R b y L b R a . s 2 Ž Ub , y R a y Ua, y R b . . Ž 4.4. Since R a L b R y s R a Ž Ub , y y L y R b . s R aUb , y y R a L y R b s Ub , y R a y Ž Ua, y y R y L a . R b s Ub , y R a y Ua, y R b q R y L a R b ,
LOCAL TRIALITY AND RELATED ALGEBRAS
843
R a L b R y y R y L a R b s Ub , y R a y Ua, y R b .
Ž 4.5.
then
Interchanging a and b, one gets R b L a R y y R y L b R a s Ua, y R b y Ub , y R a ,
Ž 4.6.
and by subtracting Ž4.6. from Ž4.5. one gets Ž4.4. and thus Ž4.2.b.. Equation Ž4.2.c. is proved along the same lines. As a consequence of Theorem 4.2, given an algebra A in S C , for any x, y g A, the linear transformation d Ž x, y . s 2 Ž d 0 Ž x, y . q d1 Ž x, y . q d 2 Ž x, y . .
Ž 4.7.
is a derivation of A. Besides, since the d i Ž x, y .’s are obtained from left and right multiplications by the elements x and y, it is clear that for any a,b, x, y g A and i s 0, 1, 2, d Ž a, b . , d i Ž x, y . s d i Ž d Ž a, b . x, y . q d i Ž x, d Ž a, b . y . ,
Ž 4.8.
d Ž a, b . , d Ž x, y . s d Ž d Ž a, b . x, y . q d Ž x, d Ž a, b . y . ,
Ž 4.9.
so the span d Ž A, A. of the d Ž x, y .’s is a subalgebra of the inner derivation algebra of A. From Ž4.1., d Ž x, y . s Rw x , y x y L x , L y y Lw x , y x y R x , R y q R x L y y R y Lx q Lx R y y Ly R x s y Ž Lw x , y x q L x , L y y R w x , y x q R x , R y y R x , Ly y Lx , R y s y Ž ad w x , y x q ad x , ad y
.
.,
where ad x s L x y R x . As mentioned in Section 2, the linear maps d Ž x, y . s ad w x, y x q wad x , ad y x are derivations in any flexible Malcev-admissible algebra. Now, if C is a Hurwitz algebra over a field F with multiplication x ⭈ y and C denotes the associated para-Hurwitz algebra, with multiplication xy s x ⭈ y, then ⴢ
w x, y x s x ⭈ y y y ⭈ x s y Ž x ⭈ y y y ⭈ x. s x ⭈ y y y ⭈ x s w x, y x ,
844
ELDUQUE AND OKUBO
since the standard involution in C is minus the identity on commutators. Hence for any x g C s C, ad x s ad ⴢx Žthe adjoint map in C . and d Ž x, y . s y Ž ad wⴢ x , y x q ad ⴢx , ad ⴢy
..
Using the alternative law in C, it is not difficult to check that in this case d Ž x, y . s y2 Ž ad wⴢ x , y x y 3 Lⴢx , R ⴢy
.,
thus recovering the usual expression for the derivations in these algebras Žsee wSc, Ž3.70.x..
REFERENCES wAEMNx wvBSx wE1x wE2x wEPx wKMRTx wMx wO1x wO2x wOOx wPx wSax wScx
P. Alberca Bjerregaard, A. Elduque, C. Martın and F. J. Navarro ´ Gonzalez, ´ Marquez, On the Cartan᎐Jacobson theorem, preprint. ´ F. van der Blij and T. A. Springer, Octaves and triality, Nieuw Arch. Wisk. Ž3. 8 Ž1960., 158᎐169. A. Elduque, Symmetric composition algebras, J. Algebras 196 Ž1997., 283᎐300. A. Elduque, Triality and automorphisms and derivations of compositions algebras, Linear Algebra Appl. 314 Ž2000., 49᎐74. A. Elduque and J. M. Perez, Composition algebras with associative bilinear form, ´ Commun. Algebra 24 Ž1996., 1091᎐1116. M. A. Knus, A. S. Merkurjev, M. Rost, and J. P. Tignol, ‘‘The Book of Involutions,’’ American Mathematical Society Colloquium Publications, Vol. 44, American Mathematical Society, Providence, RI, 1998. H. C. Myung, ‘‘Malcev-Admissible Algebras,’’ Birkhauser, Boston, 1986. ¨ S. Okubo, Deformation of the Lie-admissible pseudo-composition algebras into the octonion algebras, Hadronic J. 1 Ž1978., 1383᎐1431. S. Okubo, Classification of flexible composition algebras I, Hadronic J. 5 Ž1982., 1564᎐1612. S. Okubo and J. M. Osborn, Algebras with non-degenerate associative symmetric bilinear form permitting composition, I and II, Commun. Algebra 9 Ž1981., 1233᎐1261, 2015᎐2073. H. P. Petersson, Eine Identitat Grades, der gewisse Isotope von Kompo¨ funften ¨ sitions-Algebren genugen, Math. Z. 109 Ž1969., 217᎐238. ¨ A. A. Sagle, Malcev algebras, Trans. Amer. Math. Soc. 101 Ž1961., 426᎐458. R. D. Schafer, ‘‘An Introduction to Nonassociative Algebras,’’ Academic Press, New York, 1966.
Local Triality and Some Related Algebras Alberto Elduque1 Departamento de Matematicas, Facultad de Ciencias, Uni¨ ersidad de Zaragoza, ´ 50009 Zaragoza, Spain E-mail: [email protected]
and Susumu Okubo 2 Department of Physics and Astronomy, Uni¨ ersity of Rochester, Rochester, New York 14627 Communicated by Efim Zelmano¨ Received May 1, 2001
Eight dimensional symmetric composition algebras give very nice formulae for triality and local triality. These latter formulae will be extended to any algebra in a suitable variety. The simple algebras in this variety will be shown to be precisely the symmetric composition algebras over fields, while the prime algebras are central orders in the simple ones. 䊚 2001 Academic Press
1. INTRODUCTION It is well known Žsee wvBS, KMRTx and the references therein . that given any Cayley᎐Dickson algebra Ž C, n., where n denotes its norm, and any element in the reduced orthogonal group O⬘Ž C, n., there are new elements ⬘, ⬙ g O⬘Ž C, n., determined up to a common sign, such that
Ž xy . s ⬘ Ž x . ⬙ Ž y . ,
Ž 1.1.
1 Supported by the Spanish DGES ŽPb 97-1291-C03-03. and by a grant from the Spanish Direccion Superior e Investigacion ´ General de Ensenanza ˜ ´ Cientifica ŽPrograma de Estancias de Investigadores Espanoles en Centros de Investigacion ˜ ´ Extranjeros., while the first author was visiting the University of Wisconsin at Madison. He thanks its Department of Mathematics and, especially, Professor Georgia Benkart. 2 Supported in part by U.S. Department of Energy Grant DE-F402-ER 40685.
828 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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for any x, y g C ŽPrinciple of Triality.. This provides the spin group SpinŽ C, n. with an automorphism of order 3 whose fixed subgroup is the group of automorphisms of C, an algebraic group of type G 2 . The local version or Principle of Local Triality Žsee wSc, Chap. IIIx. asserts that, assuming characteristic / 2, for any d in the orthogonal Lie algebra oŽ C, n., there are d⬘, d⬙ g oŽ C, n., uniquely determined, such that d Ž xy . s d⬘ Ž x . y q xd⬙ Ž y . ,
Ž 1.2.
for any x, y g C. Again this gives an automorphism of order 3 of oŽ C, n. such that its fixed subalgebra is the Lie algebra of derivations Der C, which is a classical simple Lie algebra of type G 2 if the characteristic is / 2, 3 Žbut it is not simple in characteristic 3ᎏsee wAEMNx.. It has been noted, however, in wKMRT, Chap. VIIIx that the situation above becomes more symmetrical if the so-called symmetric composition algebras Žof dimension 8. are used instead of the Cayley᎐Dickson algebras. An algebra A over a field F is said to be a symmetric composition algebra if it is endowed with a regular quadratic form q, the norm, satisfying q Ž xy . s q Ž x . q Ž y . q Ž xy, z . s q Ž x, yz .
Ž 1.3.a. Ž 1.3.b.
for any x, y, z g A, where q Ž x, y . s q Ž x q y . y q Ž x . y q Ž y .. It is known from wO1x that Eqs. Ž1.3. are equivalent to the validity of
Ž xy . x s x Ž yx . s q Ž x . y
Ž 1.4.
for any x, y g A. There are two types of symmetric composition algebras, the para-Hurwitz algebras Žand some forms of them in dimension 2. and the pseudo-octonion algebras Žsee wOO, EP, KMRT, E1x.. The para-Hurwitz algebras are defined as follows: given any Hurwitz algebra C with norm q and multiplication x ⭈ y, its canonical involution is given by x ¬ x s q Ž1, x .1 y x. Then the new algebra C, defined over C but with the new multiplication xy s x ⭈ y,
Ž 1.5.
is said to be the para-Hurwitz algebra related to C. It satisfies Ž1.3. Žor Ž1.4.. with the same norm q. We will restrict ourselves in what follows to considerations of the Principle of Local Triality only. Given an eight-dimensional symmetric composition algebra A with norm q over a field of characteristic / 2, it is shown in wKMRT, Chap. VIIIx that for any d 0 g oŽ A, q . there are uniquely determined d1 , d 2 g oŽ A, q . such that d 0 Ž xy . s d1 Ž x . y q xd 2 Ž y .
Ž 1.6.
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ELDUQUE AND OKUBO
for any x, y g A. Since Ž1.6. is equivalent, in view of Ž1.3.b., to q Ž d 0 Ž x . , yz . q q Ž d1 Ž y . , zx . q q Ž d 2 Ž z . , xy . s 0
Ž 1.7.
for any x, y, z g A, the maps d 0 , d1 , d 2 can be permuted cyclically in Ž1.6.. such triples Ž d 0 , d1 , d 2 . satisfying Ž1.6. Žor Ž1.7.. are called related triples. The map d 0 ¬ d1 is an automorphism of order 3 of oŽ A, q . whose fixed subalgebra is the derivation algebra of A, which for characteristic / 2, 3 is a simple Lie algebra of type G 2 or A 2 , depending on A being a para-Hurwitz or a pseudo-octonion algebra. In wE2x, some explicit formulae were computed for this automorphism. Actually, the results in wE2, Section 5x show that the related triples for A are spanned by the triples
ž
x , y , R x L y y
1 2
q Ž x, y . I, L x R y y
1 2
/
q Ž x, y . I ,
where x, y Ž z . s q Ž y, z . x y q Ž x, z . y. But, because of Ž1.4., q Ž x, y . z s Ž xz . y q Ž yz . x s x Ž zy . q y Ž zx . and hence Rx Ly y Lx R y y
1 2 1 2
q Ž x, y . I s q Ž x, y . I s
1 2 1 2
Ž R x Ly y R y Lx . , Ž Lx R y y Ly R x . .
Also,
x , y Ž z . s Ž zx . y q Ž yx . z y Ž zy . x y Ž xy . z s z Ž xy . q y Ž xz . y z Ž yx . y x Ž yz . , so
x , y s R y , R x y Lw x , y x s L y , L x q Rw x , y x . Hence the above spanning set of related triples is the set of those Ž d 0 , d1 , d 2 . g oŽ A, q . 3 with d 0 s d 0 Ž x, y . s Rw x , y x y L x , L y s y Ž Lw x , y x q R x , R y d1 s d1 Ž x, y . s d 2 s d 2 Ž x, y . s
1 2 1 2
Ž R x Ly y R y Lx . Ž Lx R y y Ly R x .
. Ž 1.8.
LOCAL TRIALITY AND RELATED ALGEBRAS
831
for some x, y g A. In this way, the related triples are described in terms of the left and right multiplications in A. In this paper, two varieties of Žnonassociative. algebras will be introduced. Given a unital commutative and associative ground ring ⌽, let S be the variety of ⌽-algebras satisfying the identities
Ž xy . x s x Ž yx .
Ž flexible law .
Ux y , z s Ux , y z ,
Ž 1.9.a. Ž 1.9.b.
where Ux s L x R x s R x L x Žbecause of Ž1.9a.. and Ux, y s Uxqy y Ux y Uy . These are natural identities in view of Ž1.3. and Ž1.4.. Moreover, after studying some consequences of these identities and giving some examples, it will be shown that, assuming ⌽ acts without 2-torsion Ž2 x s 0 implies x s 0., the simple algebras in S are precisely the symmetric composition algebras over a field extension of ⌽ Ža field K with a unital ring homomorphism ⌽ ª K .. Thus the symmetric composition algebras over a field F are characterized as the central simple algebras in the variety S over F. Prime algebras in S will also be determined in terms of symmetric composition algebras. It must be remarked here that the first appearance of symmetric composition algebras in the literature Žalthough not with this name., to the authors’ knowledge, happened in the paper wPx, where the simple algebras with an idempotent over an algebraically closed field of characteristic / 2, 3, in the variety defined by the flexible law and by the identity of degree 5 given by Ux y s Ux Uy ŽU as above., were shown to be symmetric composition algebras Žeither para-Hurwitz algebras or the so-called Petersson algebrasᎏsee wKMRT, Chap. VIIIx.. Under some restrictions, the identity Ux Uy s Ux y follows from Ž1.9. Žsee Proposition 2.3.. Also under some restrictions on the algebra A in the variety S , the operators Ux are in the centroid of the algebra A, in particular for semiprime algebras in S Žassuming no 2-torsion.. This suggests the consideration of the subvariety S C of S defined by Ž1.9.a., Ž1.9.b., and the extra identity Ux Ž yz . s Ž Ux y . z.
Ž 1.9.c.
The study of the varieties S and S C will be the objective of the next section. Another purpose of this paper is to show that, assuming 12 g ⌽, the same formulae for d 0 Ž x, y ., d1Ž x, y ., and d 2 Ž x, y . in Ž1.8. remain valid for any algebra in S C , although the proof of this fact is necessarily very different from the proof in wE2x, which was only valid for eight-dimensional symmetric composition algebras. That is, for any algebra A in S C , it will be shown in Section 4 that d i Ž x, y . Ž u¨ . s Ž d iq1 Ž x, y . u . ¨ q u Ž d iq2 Ž x, y . ¨ .
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ELDUQUE AND OKUBO
Žindices modulo 3. for any x, y, u, ¨ g A.
2. THE VARIETIES S AND S C Let ⌽ be a unital commutative and associative ring and let S be the variety of algebras over ⌽ defined by Ž1.9.a, b., that is,
LEMMA 2.1.
Ž xy . x s x Ž yx . ,
Ž 2.1.a.
Ux y , z s Ux , y z .
Ž 2.1.b.
Let A be an algebra in S . Then for any x, y, z g A,
Ž Ux y . z s y Ž Ux z . ,
Ž 2.2.
UU x y , z s Uy , U x z .
Ž 2.3.
Proof. For Ž2.2., compute Ž zx .Ž xy . in two ways using Ž2.1.a.,
Ž zx . Ž xy . s Uz , x y x y Ž Ž xy . x . z s Uz , x y x y Ž Ux y . z, Ž zx . Ž xy . s Uz x , y x y y Ž x Ž zx . . s Uz x , y x y y Ž Ux z . , and Ž2.2. follows since Uz, x y s Uz x, y by Ž2.1.b.. For Ž2.3., just use Ž2.1. to get UU x y , z s UŽ x y . x , z s Ux y , x z s Uy , Ž x z . x s Uy , U x z .
LEMMA 2.2. Let A be an algebra in S . Then I s x g A : Ux s 0 s Ux, A 4 is an ideal of A. Proof. Let x g I and y, z g A, then Ux y, z s Ux, y z g Ux, A s 0 for any z g A, so Ux y, A s 0. Also, Ux y z s Ž Ž xy . z . Ž xy . s y Ž Ž zy . x . Ž xy . s Ž Ž xy . x . Ž zy .
since Ux , z s 0, since Ux y , A s 0,
s Ž Ux y . Ž zy . s 0. Therefore xy g I, and similarly one proves that yx g I, as required. Note that Ž2.1.b. immediately implies that J s x g A : Ux, A s 04 is an ideal of A. If ⌽ acts without 2-torsion on A and x g J, then for any
LOCAL TRIALITY AND RELATED ALGEBRAS
833
y g A, 0 s Ux, x y s 2Ux y, so Ux y s 0 and x g I. Therefore I s J if A has no 2-torsion. PROPOSITION 2.3.
Let A be an algebra in the ¨ ariety S .
Ži. If x g A : Ux, A s 04 s 0, then, for any x g A, Ux belongs to the centroid ⌫ Ž A. of A Ž that is, Ux Ž yz . s ŽUx y . z s y ŽUx z . for any x, y, z g A.. Žii. If Ux g ⌫ Ž A. for any x g A and ⌽ acts without 2-torsion on A, then Ux y s Ux Uy for any x, y g A. Žiii. If x g A : xA s Ax s 04 s 0 and Ux y s Ux Uy for any x, y g A, then Ux g ⌫ Ž A. for any x g A. Proof. For Ži., let J s x g A : Ux, A s 04 . Given t, x, y, z g A, Ut , U x Ž y z . s UU x t , y z
by Ž 2.3. ,
s UŽU x t . y , z
by Ž 2.1.b . ,
s UtŽU x y ., z
by Ž 2.2. ,
s Ut , ŽU x y . z
by Ž 2.1.b . .
Hence Ux Ž yz . y ŽUx y . z g J s 0 and Ux Ž yz . s ŽUx y . z, which together with Ž2.2. show that Ux g ⌫ Ž A.. For Žii., given any x, y g A, Ux L y s L y Ux s LU x y and Ux R y s R y Ux s RU x y because Ux g ⌫ Ž A.. Hence Ux Uy s Ux L y R y s LU x y R y
and
Ux Uy s Ux L y R y s L y Ux R y s L y RU x y ,
so 2Ux Uy s LU x y R y q L y RU x y s UU x y , y s UŽ x y . x , y s Ux y , x y s 2Ux y , and, because of our assumption on the action of ⌽ on A, Ux y s Ux Uy Žs Uy Ux since Ux g ⌫ Ž A... Finally, for Žiii., given any t, x, y, z g A, t Ž Ux Ž yz . . s Ž Ux t . Ž yz .
Ž by Ž 2.2. .
s UU x t , z y y z Ž y Ž Ux t . . s UxŽ t x ., z y y z Ž Ž Ux y . t .
Ž by Ž 2.2. .
s Ut x , z x y y z Ž Ž Ux y . t .
Ž by Ž 2.1.b. .
s Ut , z Ž Ux y . y z Ž Ž Ux y . t . Ž since Uab s UaUb for any a, b . s t Ž Ž Ux y . z . .
834
ELDUQUE AND OKUBO
Hence AŽUx Ž yz . y ŽUx y . z . s 0. Similarly one proves that
Ž Ux Ž yz . . t s Ž yz . Ž Ux t . s Uy , U t z y Ž Ž Ux t . z . y x
s Uy x , t x z y Ž t Ž Ux z . . y s Uy , t Ž Ux z . y Ž t Ž Ux z . . y s Ž y Ž Ux z . . t s Ž Ž Ux y . z . t , so Ux Ž yz . y ŽUx y . z s Ux Ž yz . y y ŽUx z . g x g A : Ax s xA s 04 s 0, as required. In view of Proposition 2.3, it is natural to consider the subvariety S C of S defined by Ž2.1.a., Ž2.1.b., and Ux Ž yz . s Ž Ux y . z s x Ž Uy z . .
Ž 2.1.c.
Before proceeding, let us pause to show an example of an algebra in S but not in S C : EXAMPLE 2.4. Let A be the free module over ⌽ with a basis a, b, c, d, e, f 4 and a commutative product determined by ab s c, ac s d, ae s yf , bc s e, bd s yf , c 2 s f , and all the other products among basic elements equal to 0. Note that A is flexible Ž2.1.a. since it is commutative. Also, A2 s span² c, d, e, f :, A3 s span² d, e, f :, A4 s span² f :, and A5 s 0. Let us check that UA, A 2 s 0, which immediately implies Ž2.1.b .. Since A 5 s 0, UA , A 2 A s span²Ua, c a, Ua, c b, Ub, c a, Ub, c b :. But Ua, c a s a2 c q Ž ca . a s da s 0, Ua, c b s Ž ab . c q Ž cb . a s c 2 q ae s f y f s 0, Ub , c a s Ž ba. c q Ž ca . b s c 2 q bd s f y f s 0, Ub , c b s b 2 c q Ž bc . b s be s 0. Therefore, A belongs to the variety S . Now, note that UaŽ b 2 . s Ua 0 s 0, but
Ž Ua b . b s Ž Ž ab . a. b s Ž ca. b s db s yf / 0, so Ž2.1.c. is not satisfied and A is not in S C .
835
LOCAL TRIALITY AND RELATED ALGEBRAS
The variety S C is contained in a wider variety, that of the flexible Malcev-admissible algebras Žsee wMx and references therein for results on these algebras.. THEOREM 2.5.
Any algebra in the ¨ ariety S C is Malce¨ -admissible.
Proof. Following wO2x, we note first that for any x, y in an algebra A in S C,
w x, y x , x s Ž xy y yx . x y x Ž xy y yx . s Ž xy . x q x Ž yx . y Ž yx . x y x Ž xy . s 2Ux y y 2Ux , y x q x 2 ( y,
Ž 2.4.
where a( b s ab q ba. Then, for any x, y, z g A,
w y, z x , x , x s y x, w y, z x , x s y2Ux w y, z x q 2Ux , w y , z x x y x 2 ( w y, z x
Ž 2.5.
and
w z, x x , x , y s y w x, z x , x , y s y2 w Ux z, y x q 2 w Ux , z x, y x y x 2 ( z, y s y2Ux w z, y x q 2Ux , z w x, y x y x 2 ( z, y
Ž 2.6.
since Ux and Ux, z are in the centroid. We next set w s w x, y x for simplicity and evaluate
w x, y x , z , x y w x, y x , w x, z x s w w, z x , x y w, w x, z x s w w, z x , x q w x, z x , w s 2Ux , w z y 2Ux , z w y 2Uw , z x q Ž x ( w . ( z, where we have used the linearization of Ž2.4.. But Ux , w s Ux , w x , y x s Uw x , x x, y s 0 by Ž2.1.b. and x ( w s x ( w x, y x s x Ž xy y yx . q Ž xy y yx . x s x Ž xy . y Ž yx . x
by Ž 2.1.a . ,
s Ux , y x y yx 2 y Uy , x x q x 2 y s x2, y .
Ž 2.7.
836
ELDUQUE AND OKUBO
Therefore Ž2.7. becomes
w x, y x , z , x y w x, y x , w x, z x s y2Ux , z w x, y x y 2Uw x , y x, z x q x 2 , y ( z. Adding this to Ž2.5. and Ž2.6. gives the Malcev identity
w y, z x , x , x q w z, x x , x , y q w x, y x , z , x y w x, y x , w x, z x s y2Ux w y, z x q 2Ux , w y , z x x y x 2 ( w y, z x y 2Ux w z, y x q 2Ux , z w x, y x y x 2 ( z, y y 2Ux , z w x, y x y 2Uw x , y x, z x q x 2 , y ( zs0 since Uw x, y x, z s Ux, w y, z x by Ž2.1.b. and w x 2 ( z, y x s w x 2 , y x( z y x 2 (w y, z x by flexibility Ž2.1.a. Žad y is contained in DerŽ A, (... Given a Malcev algebra M with multiplication w x, y x, its nucleus is N Ž M . s x g M : J Ž x, M, M . s 0 4 , where J Ž x, y, z . s ww x, y x, z x q ww y, z x, x x q ww z, x x, y x is the Jacobian of x, y, z. For any x g N Ž M ., ad x : y ¬ w x, y x is a derivation of M. Also, in the absence of 2-torsion, for any x, y g M the operator d Ž x, y . s ad w x , y x q ad x , ad y is a derivation of M Žsee wSax.. Hence, if the ground ring acts without 2-torsion, given any flexible Malcev-admissible algebra A, both ad x , for x g N Ž A., and d Ž x, y ., for x, y g A, are derivations of both Ž A, w , x. and Ž A, (. and hence of A. We finish this section with some examples of algebras in S C , which are not symmetric composition algebras. EXAMPLE 2.6. Any commutative and associative algebra A is a S Calgebra. EXAMPLE 2.7. Let F be a field of characteristic / 3 containing the cubic roots 1, , 2 of 1 and let A be the algebra with a basis e, f 4 and multiplication given by e 2 s e, f 2 s 0, ef s y f , fe s y 2 f . A is easily seen to be flexible and, besides, Ue s id, Uf s Ue, f s 0, from which Ž2.1.b. and Ž2.1.c. follow at once.
LOCAL TRIALITY AND RELATED ALGEBRAS
837
EXAMPLE 2.8. Given a quaternion algebra Q with standard involution x ¬ x over a field F and a nonzero scalar g F, the Cayley᎐Dickson algebra C Ž Q, . is the algebra defined on Q [ Q with multiplication given by
Ž x 0 , x1 . Ž y 0 , y1 . s Ž x 0 y 0 q y1 x1 , y1 x 0 q x1 y 0 . and involution given by
Ž x 0 , x 1 . s Ž x 0 , yx 1 . . Therefore, the product in the associated para-Cayley algebra C Ž Q, . is
Ž x 0 , x 1 . ⭈ Ž y 0 , y 1 . s Ž x 0 , x 1 . Ž y 0 , y 1 . s Ž x 0 , yx 1 . Ž y 0 , yy1 . s Ž x 0 y 0 q y 1 x 1 , yy 1 x 0 y x 1 y 0 . , and this is a symmetric composition algebra, so it is an algebra in S C . By an argument of continuity in the Zariski topology, the same is true of the algebra C Ž Q, 0 . with multiplication
Ž x 0 , x 1 . ⭈ Ž y 0 , y 1 . s Ž x 0 y 0 , yy1 x 0 y x 1 y 0 . , which is no longer a symmetric composition algebra.
3. SIMPLE AND PRIME ALGEBRAS IN S In order to deal with the simple and prime algebras in the variety S , we first check that the ideal J s x g A : Ux, A s 04 , considered in the previous section, is trivial for these algebras. More generally, we have PROPOSITION 3.1. 0 s Ux, A 4 s 0.
Let A be a semiprime algebra in S , then x g A : Ux s
Proof. By Lemma 2.2, I s x g A : Ux s 0 s Ux, A 4 is an ideal of A. Assume I / 0, then we can take an element 0 / x g I with x 2 s 0. Otherwise, if y s x 2 / 0, then y 2 s x 2 x 2 s Ux, x 2 x y Ž x 2 x . x s Ux, x 2 x y ŽUx x . x s 0 and x can be replaced with y. Note that given any ideal K of A contained in I, K 2 is again an ideal of A Žsince Ž u¨ . a s Uu, a¨ y Ž a¨ . u s yŽ a¨ . u g K 2 for any u, ¨ g K and a g A.. Let us denote by K the ideal generated by x g I above Ž x 2 s 0.. Let us first prove that K 2 s xK q Kx. It is clear that xK q Kx : K 2 . For the converse, note that given a, b g A and z g I, L a L b z s a Ž bz . s yz Ž ba. s yR b a z, R a R b z s Ž zb . a s y Ž ab . z s yL ab z.
Ž 3.1.
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ELDUQUE AND OKUBO
Therefore, K is the span of the elements
Ž La R a 1
2
L a 3 ⭈⭈⭈ . Ž x .
and
Ž R a La 1
2
R a 3 ⭈⭈⭈ . Ž x . .
If u s Ž R a1 L a 2 R a 3 ⭈⭈⭈ .Ž x . and ¨ g K, by Ž3.1., u¨ s Ž R ¨ R a1 L a 2 R a 3 ⭈⭈⭈ . Ž x . s y Ž L¨ a1 L a 2 R a 3 ⭈⭈⭈ . Ž x . s Ž R a 2 Ž ¨ a1 . R a 3 ⭈⭈⭈ . Ž x . s ⭈⭈⭈ s Ta Ž x . , where T s L or R and a g K. Hence u¨ g Kx or u¨ g xK. Similarly, if ¨ s Ž L a1 R a 2 L a 3 ⭈⭈⭈ .Ž x . and u g K, either u¨ g Kx or u¨ g xK. Finally, if u s Ž L a1 R a 2 L a 3 ⭈⭈⭈ .Ž x . and ¨ s Ž R b 1 L b 2 R b 3 ⭈⭈⭈ .Ž x ., u s a1 u1 and ¨ s ¨ 1 b1 with u1 s Ž R a 2 L a 3 ⭈⭈⭈ .Ž x . and ¨ 1 s Ž L b 2 R b 3 ⭈⭈⭈ .Ž x .. Then, u¨ s Ž a1 u1 . Ž ¨ 1 b1 . s y Ž Ž ¨ 1 b1 . u1 . a1 s Ž Ž u1 b1 . ¨ 1 . a1 s y Ž a1¨ 1 . Ž u1 b1 . and, by the arguments above, a1¨ 1 s Ta x and u1 b1 s Tb x for some a, b g A. Now note that
Ž ax . Ž bx . s y Ž Ž bx . x . a s Ž x 2 b . a s 0, Ž xa. Ž xb . s yb Ž x Ž xa. . s b Ž ax 2 . s 0, Ž ax . Ž xb . s y Ž Ž xb . x . a s Ž Ux b . a s 0,
Ž 3.2.
Ž xa. Ž bx . s y Ž Ž bx . a. x g Kx s yx Ž b Ž xa. . g xK , and the proof of the fact that K 2 s xK q Kx is complete. To finish the proof of the proposition, note that because of Ž3.2., Ž xK q Kx . 2 s Ž xK .Ž Kx . : Ž xK l Kx . and Ž xK l Kx . 2 : Ž Kx .Ž xK . s 0. Hence ŽŽ K 2 . 2 . 2 s 0 and K s 0 by semiprimeness, a contradiction. Hence, in view of Proposition 2.3.i, if ⌽ acts without 2-torsion, the semiprime algebras Žand hence the simple or prime ones. in S and S C coincide. THEOREM 3.2. Let A be a simple algebra in S such that ⌽ acts without 2-torsion on it. Then there is a field extension F of ⌽ such that A is a symmetric composition algebra o¨ er F.
LOCAL TRIALITY AND RELATED ALGEBRAS
839
Con¨ ersely, any symmetric composition algebra o¨ er an extension field of ⌽ is a simple algebra in the ¨ ariety S . Proof. Any symmetric composition algebra A over a field F, with quadratic form q: A ª F, satisfies Ž2.1.a. and Ž2.1.b., where Ux is just the multiplication by the scalar q Ž x .. Besides, since Ž xy . z q Ž zy . x s q Ž x, z . y for any x, y, z g A and q is nondegenerate, any element y g A is in the ideal generated by any other element x g A, so A is simple. Let A now be a simple algebra in S and assume that ⌽ acts without 2-torsion. Then, by Proposition 3.1, I s J s 0, where J s x g A : Ux, A s 04 . Proposition 2.3.i shows that Ux g ⌫ Ž A., which is a field F, an extension of ⌽, by simplicity. Now, U: A ª F; x ¬ Ux is a regular quadratic form, since J s 0; and A satisfies Ž xy . x s x Ž yx . s Ux y, which is Ž1.4.. Therefore, A is a symmetric composition algebra. As for prime algebras in S , we have THEOREM 3.3. Let A be a prime algebra in S on which ⌽ acts without 2-torsion. Then A is a central order in a symmetric composition algebra o¨ er a field extension F of ⌽. Con¨ ersely, any ⌽-algebra which is such a central order is a prime algebra in the ¨ ariety S . Proof. By primeness, the centroid S s ⌫ Ž A. of the prime algebra A is an integral domain, extension of ⌽, which acts without 2-torsion on A by our assumptions. Again by Proposition 3.1, J s x g A : Ux, A s 04 s 0 and, by Proposition 2.3.i, Ux g S for any x g S. Let F be the quotient field of S. Since Ux g S for any x g A, the algebra of fractions Sy1A satisfies Ž xy . x s x Ž yx . s q Ž x . y, where q Ž a . s Uars 2 g F. The q thus defined is a s quadratic form and it is nondegenerate because J s 0. Hence Sy1A is a symmetric composition algebra, as required. The converse is clear. COROLLARY 3.4. Assume that the ground ring ⌽ is a field F of characteristic / 2 and let A be a finite-dimensional semiprime algebra in S . Then A is a direct sum of symmetric composition algebras o¨ er finite extension fields of F. Proof. Since A is semiprime, A is a subdirect sum of prime algebras. By finite dimensionality, there are prime ideals P1 , . . . , Pr of A such that P1 l ⭈⭈⭈ l Pr s 0. Take r minimal. Now ArP1 is prime but its centroid is a finite-dimensional integral domain over F, so it is a field K 1 extension of F. By the proof of Theorem 3.3, ArP1 is a symmetric composition algebra over K 1 and hence it is simple. But then P1 is a maximal ideal of A and, therefore, A s P1 [ Ž P2 l ⭈⭈⭈ l Pr . ( P1 [ ArP1. Now P1 ( ArP2 l ⭈⭈⭈ l Pr is semiprime, and an induction on r completes the proof.
840
ELDUQUE AND OKUBO
4. LOCAL TRIALITY IN S C Throughout this section, all the algebras considered will be over a ground ring ⌽ containing 12 . Given any elements x, y in an algebra A in the variety S C , consider the endomorphisms of A given by the same formulae Ž1.8. that work for eight-dimensional symmetric-composition algebras. That is, d 0 Ž x, y . s Uy , y x y Ux , y y s R w x , y x y L x , L y s y Ž Lw x , y x q R x , R y d1 Ž x, y . s d 2 Ž x, y . s
1 2 1 2
.,
Ž 4.1.a.
Ž R x Ly y R y Lx . ,
Ž 4.1.b.
Ž Lx R y y Ly R x . .
Ž 4.1.c.
Then, we have PROPOSITION 4.1. With the same assumptions as abo¨ e, for any a, b, x, y, u, ¨ g A and j s 0, 1, 2, Ži. Ud Ž a, b. x, y q Ux, d Ž a, b. y s 0, j j Žii. Ux d j Ž a, b . s d j ŽUx a, b . s d j Ž a, Ux b . s d j Ž a, b .Ux , and Žiii. w d j Ž a, b ., d j Ž u, ¨ .x s d j Ž a, Ub, u¨ . q d j Ž b, Ua, ¨ u. y d j Ž a, Ub, ¨ u. yd j Ž b, Ua, u¨ .. Proof. Item Ži. follows from Ž2.1.b. and Žii. follows from Ž2.1.c.. Let us check first Žiii. for j s 0. Since Ux, y is in the centroid for any x, y g A, d 0 Ž a, b . , d 0 Ž u, ¨ . Ž z . s Ub , U¨ , z uyUu , z ¨ a y Ua, U¨ , z uyUu , z ¨ b y U¨ , Ub , z ayUa , z b u q Uu , Ub , z ayUa , z b¨ s Ub , uU¨ , z a y Ub , ¨ Uu , z a y Ua, uU¨ , z b q Ua, ¨ Uu , z b y U¨ , aUb , z u q U¨ , b Ua, z u q Uu , aUb , z¨ y Uu , b Ua, z¨ s Ž UUb , u ¨ , z a y Ua, z Ub , u¨ . y Ž UUb , ¨ u , z a y Ua, z Ub , ¨ u . y Ž UUa , u ¨ , z b y Ub , z Ua, u¨ . q Ž UUa , ¨ u , z b y Ub , z Ua, ¨ u . s Ž d 0 Ž a, Ub , u¨ . y d 0 Ž a, Ub , ¨ u . y d 0 Ž b, Ua, u¨ . q d 0 Ž b, Ua, ¨ u . . Ž z . .
841
LOCAL TRIALITY AND RELATED ALGEBRAS
Now, for j s 1 Žthe case j s 2 is similar., R a L b R u L¨ s R a Ž Ub , u y L u R b . L¨ s Ub , u R a L¨ y R a L u R b L¨ s Ub , u R a L¨ y Ž Ua, u y R u L a . Ž Ub , ¨ y R ¨ L b . s Ub , u R a L¨ y Ua, uUb , ¨ q Ub , ¨ R u L a q Ua, u R ¨ L b y R u L a R ¨ L b s Ub , u R a L¨ y Ua, u Ž Ub , ¨ y R ¨ L b . q Ub , ¨ R u L a y R u Ž Ua, ¨ y L¨ R a . L b s Ub , u R a L¨ q Ub , ¨ R u L a y Ua, u R b L¨ y Ua, ¨ R u L b q R u L¨ R a L b . Thus
w R a L b , R u L¨ x s R a LUb , u ¨ q RUb , u L a y R b LUa , u ¨ y RUa , u L b . ¨
¨
Letting a l b and u l ¨ leads to 4 d1 Ž a, b . , d1 Ž u, ¨ . s w R a L b y R b L a , R u L¨ y R ¨ L u x s R a LUb , u ¨ q RUb , ¨ u L a y R b LUa , u ¨ y RUa , ¨ u L b y R a LUb , ¨ u y RUb , u ¨ L a q R b LUa , ¨ u q RUa , u ¨ L b y R b LUa , u ¨ y RUa , ¨ u L b q R a LUb , u ¨ q RUb , ¨ u L a q R b LUa , ¨ u q RUa , u ¨ L b y R a LUb , ¨ u y RUb , u ¨ L a s 2 Ž Ž R a LUb , u ¨ y RUb , u ¨ L a . y Ž R b LUa , u ¨ y RUa , u ¨ L b . y Ž R a LUb , ¨ u y RUb , ¨ u L a . q Ž R b LUa , ¨ u y RUa , ¨ u L b . . s 4 Ž d1 Ž a, Ub , u¨ . q d1 Ž b, Ua, ¨ u . y d1 Ž a, Ub , ¨ u . y d1 Ž b, Ua, u¨ . . .
Proposition 4.1.iii shows that, for any j s 0, 1, 2, the linear span of the d j Ž a, b .’s forms a Lie subalgebra of End ⌽ Ž A.. Actually, it is a subalgebra of the Lie multiplication algebra. Given a quaternion algebra Q over a field F with norm q and the corresponding para-quaternion algebra Q, the d 0 Ž x, y .’s are just the usual generators of the orthogonal Lie algebra oŽ Q, q ., d 0 Ž x, y . s q Ž y, y . x y q Ž x, y . y, so the Lie algebra spanned by the d 0 Ž x, y .’s is oŽ Q, q ..
842
ELDUQUE AND OKUBO
On the other hand, if x ⭈ y denotes the product in Q, then xy s x ⭈ y is the product in Q and, for any x, y g Q, R x L y Ž z . s Ž yz . x s Ž y ⭈ z . x s y ⭈ z ⭈ x s z ⭈ y ⭈ x and hence d1Ž x, y . is one-half of the right multiplication in A by y ⭈ x y x ⭈ y. Thus the Lie algebra spanned by the d1Ž x, y .’s is the Lie algebra of right multiplications in Q by elements of zero trace; R Qⴢ 0 . Similarly, the Lie algebra spanned by the d 2 Ž x, y .’s is the Lie algebra of left multiplications by elements of zero trace, LⴢQ 0 . It is well-known that oŽ Q, n. s LⴢQ 0 [ R Qⴢ 0 . THEOREM 4.2. a, b, x, y g A,
Let A be an algebra in the ¨ ariety S C . Then for any
d 0 Ž a, b . Ž xy . s Ž d1 Ž a, b . x . y q x Ž d 2 Ž a, b . y . ,
Ž 4.2.a.
d1 Ž a, b . Ž xy . s Ž d 2 Ž a, b . x . y q x Ž d 0 Ž a, b . y . ,
Ž 4.2.b.
d 2 Ž a, b . Ž xy . s Ž d 0 Ž a, b . x . y q x Ž d1 Ž a, b . y . .
Ž 4.2.c.
Proof. Because of Ž2.1.b.,
Ž d1Ž a, b . x . y q x Ž d 2 Ž a, b . y . s 12 Ž Ž Ž bx . a y Ž ax . b . y q x Ž a Ž yb . y b Ž ya . . . s
1 2
Ž Ub x , y a y Ž ya. Ž bx . y Ua x , y b q Ž yb . Ž ax . qUx , y b a y Ž yb . Ž ax . y Ux , y a b q Ž ya . Ž bx . .
s Ub , x y a y Ua, x y b s d 0 Ž a, b . Ž xy . , thus obtaining Ž4.2.a.. Now, Eq. Ž4.2.b. is equivalent to d1 Ž a, b . R y s R y d 2 Ž a, b . q R d 0 Ž a, b. y .
Ž 4.3.
However, thanks to Ž2.1.c., R d 0 Ž a, b. y s RUb , y ayUa , y b s Ub , y R a y Ua, y R b . Hence Ž4.3. becomes
Ž R a L b y R b L a . R y y R y Ž L a R b y L b R a . s 2 Ž Ub , y R a y Ua, y R b . . Ž 4.4. Since R a L b R y s R a Ž Ub , y y L y R b . s R aUb , y y R a L y R b s Ub , y R a y Ž Ua, y y R y L a . R b s Ub , y R a y Ua, y R b q R y L a R b ,
LOCAL TRIALITY AND RELATED ALGEBRAS
843
R a L b R y y R y L a R b s Ub , y R a y Ua, y R b .
Ž 4.5.
then
Interchanging a and b, one gets R b L a R y y R y L b R a s Ua, y R b y Ub , y R a ,
Ž 4.6.
and by subtracting Ž4.6. from Ž4.5. one gets Ž4.4. and thus Ž4.2.b.. Equation Ž4.2.c. is proved along the same lines. As a consequence of Theorem 4.2, given an algebra A in S C , for any x, y g A, the linear transformation d Ž x, y . s 2 Ž d 0 Ž x, y . q d1 Ž x, y . q d 2 Ž x, y . .
Ž 4.7.
is a derivation of A. Besides, since the d i Ž x, y .’s are obtained from left and right multiplications by the elements x and y, it is clear that for any a,b, x, y g A and i s 0, 1, 2, d Ž a, b . , d i Ž x, y . s d i Ž d Ž a, b . x, y . q d i Ž x, d Ž a, b . y . ,
Ž 4.8.
d Ž a, b . , d Ž x, y . s d Ž d Ž a, b . x, y . q d Ž x, d Ž a, b . y . ,
Ž 4.9.
so the span d Ž A, A. of the d Ž x, y .’s is a subalgebra of the inner derivation algebra of A. From Ž4.1., d Ž x, y . s Rw x , y x y L x , L y y Lw x , y x y R x , R y q R x L y y R y Lx q Lx R y y Ly R x s y Ž Lw x , y x q L x , L y y R w x , y x q R x , R y y R x , Ly y Lx , R y s y Ž ad w x , y x q ad x , ad y
.
.,
where ad x s L x y R x . As mentioned in Section 2, the linear maps d Ž x, y . s ad w x, y x q wad x , ad y x are derivations in any flexible Malcev-admissible algebra. Now, if C is a Hurwitz algebra over a field F with multiplication x ⭈ y and C denotes the associated para-Hurwitz algebra, with multiplication xy s x ⭈ y, then ⴢ
w x, y x s x ⭈ y y y ⭈ x s y Ž x ⭈ y y y ⭈ x. s x ⭈ y y y ⭈ x s w x, y x ,
844
ELDUQUE AND OKUBO
since the standard involution in C is minus the identity on commutators. Hence for any x g C s C, ad x s ad ⴢx Žthe adjoint map in C . and d Ž x, y . s y Ž ad wⴢ x , y x q ad ⴢx , ad ⴢy
..
Using the alternative law in C, it is not difficult to check that in this case d Ž x, y . s y2 Ž ad wⴢ x , y x y 3 Lⴢx , R ⴢy
.,
thus recovering the usual expression for the derivations in these algebras Žsee wSc, Ž3.70.x..
REFERENCES wAEMNx wvBSx wE1x wE2x wEPx wKMRTx wMx wO1x wO2x wOOx wPx wSax wScx
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