The Gelfand-Kirillov conjecture for semi-direct products of Lie algebras

Abstract

Let g be an n-dimensional Lie algebra over a field k of characteristic zero and let W be a g-module of dimension at least n. Sufficient conditions are given in order for the semi-direct product g + W to satisfy the Gelfand-Kirillov conjecture. This implies that this conjecture holds for an important class of Frobenius Lie algebras. Special attention is devoted to the case where g = sl(2,k).