Birrepresentations in a locally nilpotent variety


  • Manuel Arenas Universidad de Chile.
  • Alicia Labra Universidad de Chile.



Locally nilpotent algebra, vector space, birrepresentation, álgebra localmente nilpotente, espacio vectorial, birrepresentación.


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.

Author Biographies

Manuel Arenas, Universidad de Chile.

Departamento de Matematicas, Facultad de Ciencias.

Alicia Labra, Universidad de Chile.

Departamento de Matemáticas, Facultad de Ciencias.


[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).

[CHL] I. Correa, I. R. Hentzel, A. Labra, Nilpotency of Commutative Finitely Generated Algebras Satisfying L3x+γLx3 = 0, γ = 1, 0 Journal of Algebra 330, pp. 48-59, (2011).

[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.



How to Cite

M. Arenas and A. Labra, “Birrepresentations in a locally nilpotent variety”, Proyecciones (Antofagasta, On line), vol. 33, no. 1, pp. 123-132, Mar. 2017.