Birrepresentations in a locally nilpotent variety

• Arenas, Manuel ^[1] ; Labra, Alicia ^[1]
  1. [1] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
• 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
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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.

• Referencias bibliográficas
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