**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.

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**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 + yL_{x}3 =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.*