Lie algebras with an algebraic adjoint representation revisited

Abstract

A well-known theorem due to E. Zelmanov proves that PI-Lie algebras with an algebraic adjoint representation over a field of characteristic zero are locally finite-dimensional. In particular, a Lie algebra (over a field of characteristic zero) whose adjoint representation is algebraic of bounded degree is locally finite-dimensional. In this paper it is proved that a prime nondegenerate PI-Lie algebra with an algebraic adjoint representation over a field of characteristic zero is simple and finite-dimensional over its centroid, which is an algebraic field extension of the base field. We also give a new and shorter proof of the local finiteness of Lie algebras with an algebraic adjoint representation of bounded degree.