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By Mal'tsev N., Kuz'min E. N.

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BALLESTEROS ET AL. Here a is the permutation map u(a 0 b) = b 0 a and , if P == (Pkl) is a (m x m) matrix with entries in B , the k-th component of (P0Xh is (P0Xh := 2::;:1 pk/ 0 XI . 4. QUE Cayley-Klein Algebras We restrict in this section to th e study of the algebras generating the two and three dimensional CK systems. The former is a family of 3-d Lie algebra depending on two parameters which contains as particular cases so(3),so(2, 1), the 2-d euclidean e(2) and the (1 + 1) Galilei and Poincare algebras .

T2] J . Tits, Algebres alternatives, algebres de Jordan, et algebres de Lie exceptionelle, Nederl. Akad. Weten. Proc. Ser. A , so (1966) ,223-237. [Z] E. Zelmanov, Lie algebras graded by finite root sys tems, preprint , (1992) . 1991 Mathematics Subject Classifications : Primary 17B20, 17B70, 17B25 BERNSTEIN REPRESENTATIONS J. BERNAD , A. ILTYAIWV and C. MARTINEZ· Departamento de Matemati cas, Universidad de Oviedo 33007 Oviedo, Spain Abstract. In this paper is defined the notion of Bernstein representation and it is proved that every irreducible module over a nuclear Bernstein algebra is one-dimensional.

2. l :B ..... End(M) a linear mapping. A. P. CB-6/91. 39 S. ), Non-Associative Algebra and Its Applications, 39-45 . © 1994 Kluwer Academic Publishers. 40 J . BERNAD ET AL . We say that p : B -+ M is a Berns tein representation if the explit algebra B=B-+M with the homomorphism w:-+ End(M ) defined by w(b + m) = w(b) is a Bernstein algebra. The I<-module M is called B-B ernstein modul e. 1 Let (B,w) be a Bernstein algebra and p: B representation . Then -+ E nd (M) a Bernst ein (2) Proof. (b2). Since J1.

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A basis for the identities of the algebra of second-order matrices over a finite field by Mal'tsev N., Kuz'min E. N.


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