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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.33 no.1 Antofagasta mar. 2014

http://dx.doi.org/10.4067/S0716-09172014000100009 

 

Birrepresentations in a locally nilpotent variety

 

Manuel Arenas * and Alicia Labra
Universidad de Chile, Chile


ABSTRACT

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.

*Supported by Fondecyt 1120844.
Supported by Fondecyt 1120844.


 

REFERENCES

[BEL] A. Behn, A. Elduque, A. Labra, A class of Locally Nilpotent Commutative Algebras, International Journal of Algebra and Computation, 21, No. 5, pp. 763 - 774, (2011).         [ Links ]

[CHL] I. Correa, I. R. Hentzel, A. Labra, Nilpotency of Commutative Finitely Generated Algebras Satisfying LX + yLx3 =0,ã = 1, 0 Journal of Algebra 330, pp. 48-59, (2011).         [ Links ]

[Eil] S. Eilenberg, Extensions of general algebras. Ann. Soc. Polon. Math. 21, pp. 125-134, (1948).         [ Links ]

[Um] U. Umirbaev, Universal enveloping algebras and derivations of Pois-son algebras. Arxiv. 1102 0366v 2 feb. 2011.         [ Links ]

 

Manuel Arenas
Departamento de Matematicas, Facultad de Ciencias
Universidad de Chile Casilla 653, Santiago. Chile
e-mail : mcarenas@yahoo.com

Alicia Labra
Departamento de Matemáticas, Facultad de Ciencias
Universidad de Chile Casilla 653, Santiago. Chile
e-mail : alimat@uchile.cl

Chile Received : June 2013. Accepted : November 2013.

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