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Symmetric Composition Algebras JOURNAL OF ALGEBRA 196, 283]300Ž. 1997 ARTICLE NO. JA977071 Symmetric Composition Algebras Alberto Elduque* Departamento de Matematicas y Computacion, Uni¨ersidad de La Rioja, CORE 26004, Logrono,Ä Spain Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Communicated by Efim Zelmano¨ Received December 18, 1996 A construction of all the Okubo algebras over fields of characteristic 3 is provided and the complete classification of the composition algebras with associa- tive norm over fields is finished. Q 1997 Academic Press 1. INTRODUCTION An algebra A over a field F is said to be a composition algebra if there exists a strictly nondegenerate quadratic formŽ. called the norm n: A ª F such that nxyŽ.snxny Ž.Ž. Ž.1 for any x, y g A. The form being strictly nondegenerate means that its polarization, defined by nxŽ.,ysnx Žqy .ynx Ž.Ž.yny, is a nondegenerate bilinear form. Linearization ofŽ. 1 gives nxyŽ.Ž.Ž.Ž.,xz s nyx,zx s nxny,z, Ž.2 nxyŽ.Ž.Ž.Ž.,zt q nzy,xt s nx,zny,t, for any x, y, z, t g A. *Supported by the Spanish DGICYTŽ. PB 94-1311-C03-03 and by the Universidad de La RiojaŽ. Convenio Caja Rioja, 95PYB06MBC and 96PYB07PBC . Present address: Departa- mento de Matematicas, Universidad de Zaragoza, 50009 Zaragoza, Spain. E-mail: [email protected]. 283 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. 284 ALBERTO ELDUQUE Unital composition algebras, also called Hurwitz algebras, constitute a well-known class of algebras of dimension 1, 2, 4, or 8, and are classified by the ``Generalized Hurwitz Theorem''w ZSSS82, Chap. II; KMRT,Ž. 35.16x . Through the work of Peterssonwx P69 , Okubo w O78, O95 x , Okubo and Myungwx O-M80 , Okubo and Osborn w O-O181, O-O281 x , Faulkner wx F88 , Elduque and Myungwx E-M91, E-M93 , and Elduque and Perez wx E-P96 , those composition algebras in which the norm is associative, that is, it satisfies nxyŽ.Ž.,zsnx,yz Ž.3 for any x, y, z, have been considered. Because of their nice and symmetric properties, these composition algebras of dimension G 2 with associative norm have been called sym- metric composition algebras inwx KMRT and this notation will be used here. It turns outwx M86, p. 86 that for a strictly nondegenerate quadratic form ndefined on an algebra, conditionsŽ. 1 and Ž. 3 are equivalent to the validity of Ž.xy x s xyx Ž.snxy Ž. Ž.4 for any x, y. In particular, by taking an element 0 / x g A with nxŽ./0, Ž.4 shows that the left and right multiplications by x are bijective, whence it followsŽ seewx K53. that any symmetric composition algebra is finite dimensional, its dimension being restricted to 2, 4, or 8. The linearization ofŽ. 4 ; Ž.xy z q Ž.zy x s xyz Ž.qzyx Ž.snx Ž,zy .,5 Ž. will be used throughout the paper. Two classes of symmetric composition algebras have played a key role: the para-Hurwitz and Okubo algebras. Given a Hurwitz algebra C, with multiplication xy, the para-Hurwitz algebra associated to C is the algebra defined on the same vector space C, but with new multiplication x ? y s xy Žseewx O-M80. , where x ¬ x is the standard conjugation of C. By definition, the Okubo algebras are the forms of the pseudo-octonion algebra}which will be defined shortly}that is, those algebras which become isomorphic to the pseudo-octonion algebra after extending scalars. The original definition inwx O78 of the pseudo-octonion algebra required some restrictions on the ground field. In order to avoid this, a new definition, which extended the one given by Okubo, valid over any field was proposed inwx E-P96 , inspired in a previous construction of Petersson wxP69 . Given a Hurwitz algebra C and an automorphism t of C such that 3 t s 1, the Petersson algebra Ct is the algebra defined on C with the new SYMMETRIC COMPOSITION ALGEBRAS 285 multiplication t ty1 x) y s xy Žseew E-P96,Ž. 1.2x. The name Petersson algebra appeared for the first time inwx KMRT, Sect. 36B . Now, in case C s CFŽ.is Zorn's vector matrix algebra au CFŽ.s :a,bgF,u,¨gF=F=F ½5ž/¨b with multiplication a u a9 u9 aa9 q u?¨9 au9 q b9uy¨ =¨9 s , ž/ž/¨b¨9b9 ž/a9¨qb¨9qu=u9bb9 q u ? u9 where u ? ¨ and u = ¨ denote the usual dot and vector product in V s Ž. 3 F=F=F, let us take any w g End F V with minimum polynomial X y 1 and define t au a uw s Žw*.1 , ž/¨b ž/¨y b where w* is the adjoint relative to the dot product. Then t is an automorphism of CFŽ.of order 3 and the Petersson algebra CF Ž.t Ž.unique up to isomorphism is called the pseudo-octonion algebra and denoted by PF8Ž.. The main results inwx E-P96 , extending those inw O-O181, O-O281x , assert that the symmetric composition algebras with nonzero idempotents are exactly the Petersson algebras, and that these are either para-Hurwitz or some Okubo algebras constructed in a specific way. Since it is knownw O-O181; O-O281; KMRT,Ž. 36.11x that given a symmetric composition algebra, either it already contains a nonzero idem- potent or one finds these idempotents after a scalar extension of degree threeŽ. see also the paragraph before Corollary 3.4 here , it follows that the symmetric composition algebras are the forms of the para-Hurwitz alge- bras and the Okubo algebras. On the other hand, any form of a para- Hurwitz algebra of dimension / 2 is again a para-Hurwitz algebraw E-P96, Lemma 3.3x , so only the forms of the two-dimensional para-Hurwitz algebras and the Okubo algebras need to be determined. Along the same lines of the definition by Okubo of the pseudo-octonion algebra, but independently, Faulknerwx F88 obtained composition algebras from separable alternative algebras of generic degree 3 over fields of 286 ALBERTO ELDUQUE characteristic /3. Following this idea, the symmetric composition alge- bras over fields of characteristic / 3 are classified inwx E-M93 as follows. Let F be a field of characteristic / 3 and assume first that F contains a primitive cube root v of 1. Let A be a separable alternative algebra of degree 3 over F with generic minimum polynomial 3 2 pxŽ.l s l y Tx Ž.lqSx Ž.lyNx Ž.1, for a linear form T, a quadratic form S, and a cubic one N. Consider the Ä Ž. 4 subspace A0 s x g A: Txs0 with multiplication 2v 1 2 q x y s v xy y v yx y TxyŽ.1. ) 3 Ž. Then A0 , ) is a composition algebra relative to S. If, on the contrary, F does not contain the primitive cube roots of 1, consider the quadratic field extension K s Fwxv obtained by adjoining one of them, v,to F. Let A be a separable alternative algebra of degree 3 over K with generic minimum polynomial pxŽ.l as above and equipped J 2 with an involution J of the second kind Žv s v .. Consider the F-sub- Ä J Ž. 4 Ž. space AÄ0 s x g A: x syx and Txs0 , with the well defined multiplication 2v 1 2 q x y s v xy y v yx y TxyŽ.1. ) 3 Ž.Ä <Ä Then A0 , ) is a composition algebra relative to S A 0 . Moreover, we have THEOREM 1.1wx E-M93 . Let C be an algebra o¨er a field F of characteris- tic / 3 with dimF C ) 1. Ž.i If F contains the cube roots of 1 Ž.type I,then C is a composition algebra with associati¨e bilinear form if and only if there exists a separable Ž. alternati¨e algebra A of degree 3 o¨er F such that C ( A0 , ) . Two such algebras C's are isomorphic if and only if so are their correspond- ing alternati¨e algebras. Ž.ii If F does not contain the cube roots of 1 Ž.type II and K s Fwxv 2 with v / 1 s v , then C is a composition algebras with associati¨e bilinear form if and only if there exists a separable alternati¨e algebra A o¨er K with an Ž.Ä in¨olution of the second kind such that C ( A0 , ) . Two such algebras C's are isomorphic if and only if so are their correspond- ing alternati¨e algebras as algebras with in¨olution. Actually, inwx E-M93 the ground field is assumed to have characteristic /2, 3, but everything is valid also in characteristic 2. The only necessary SYMMETRIC COMPOSITION ALGEBRAS 287 changes affect the argument in Section 6 of the latter reference for types IIc and Ic. For type IIc and following the notation in that paper, A s K = C JJ w with K s k and C s C, the algebra D s Äx g C: x s x4is Hurwitz with the restrictions of the trace t and norm n of C, and, although the Ž. decomposition D s Fe [ D0 wxE-M93, end of p. 2496 and 6.1 inwx E-M93 are not valid for characteristic 2, the mapŽ. with x ? y s xy Ž.Ž .
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