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

Let g be an n-dimensional Lie algebra over a field k of characteristic zero and let W be a g-module such that dimW⩾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).