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

Let be an n-dimensional Lie algebra over a field k of characteristic zero and let W be a -module such that dimW n. Sufficient conditions are given in order for the semi-direct product 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 .