Birrepresentations in a locally nilpotent variety

Abstract

It is known that commutative algebras satisfying the identity of degree four ((yx)x)x +γy((xx)x) = 0, with γ in the field and γ 6= −1 are locally nilpotent. In this paper we study the birrepresentations of an algebra A that belongs to a variety V of locally nilpotent algebras. We prove that if the split null extension of a birrepresentation of an algebra A ∈ V 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 birrepresentation 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.