The isomorphism problem for universal enveloping algebras of four-dimensional solvable Lie algebras

This paper is a contribution to the isomorphism problem for universal enveloping algebras of finite-dimensional Lie algebras. The main result is that this problem has a positive solution in the class of solvable Lie algebras of dimension at most four over fields of characteristic~0. We also prove, over an arbitrary field, that the isomorphism type of a metabelian Lie algebra whose derived subalgebra has codimension one is determined by its universal enveloping algebra.