Birrepresentations in a locally nilpotent variety
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Arenas, Manuel [1] ; Labra, Alicia [1]
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[1] Universidad de Chile
Universidad de Chile
Santiago, Chile
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- Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 33, Nº. 1, 2014, págs. 123-132
- Idioma: inglés
- DOI: 10.4067/S0716-09172014000100009
- Enlaces
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Resumen
It is known that commutative algebras satisfying the identity of degree four ((yx)x)x + γ((xx)x) = 0, with γ in the field and γ ≠ —1 are locally nilpotent. In this paper we study the birrepresentations of an algebra A that belongs to a variety ν of locally nilpotent algebras. We prove that if the split null extension of a birrepresentation of an algebra A ∈ ν by a vector space M is locally nilpotent, then it is trivial or reducible. As corollaries we get that if A is finitely generated, then every birrepresentation is trivial or reducible and that every finite-dimensional birrepresentation is equivalent to a birrepre-sentation consisting of strictly upper triangular matrices. We also prove that the multiplicative universal envelope of a finitely generated algebra in V is nilpotent, therefore it is finite-dimensional.
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Referencias bibliográficas
- Citas [BEL] A. Behn, A. Elduque, A. Labra, A class of Locally Nilpotent Commutative Algebras, International Journal of Algebra and Computation,...
- [CHL] I. Correa, I. R. Hentzel, A. Labra, Nilpotency of Commutative Finitely Generated Algebras Satisfying L3 x + γLx3 = 0, γ =...
- [Eil] S. Eilenberg, Extensions of general algebras. Ann. Soc. Polon. Math. 21, pp. 125-134, (1948).
- [Um] U. Umirbaev, Universal enveloping algebras and derivations of Poisson algebras. Arxiv. 1102 0366v 2 feb. 2011.
