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

## TL;DR

This paper investigates whether the universal enveloping algebra uniquely determines four-dimensional solvable Lie algebras, proving this for certain classes over arbitrary fields and addressing characteristic-specific issues.

## Contribution

It establishes that for metabelian Lie algebras with codimension-one derived subalgebra, the universal enveloping algebra determines the Lie algebra over any field.

## Key findings

- Universal enveloping algebra determines certain four-dimensional solvable Lie algebras.
- Solved the isomorphism problem for four-dimensional solvable Lie algebras over characteristic zero.
- Identified challenges in prime characteristic cases.

## Abstract

This paper is a contribution to the isomorphism problem for universal enveloping algebras of finite-dimensional Lie algebras. We focus on solvable Lie algebras of small dimensions over fields of arbitrary characteristic. We 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. As an application of the results in this paper, we solve the isomorphism problem for solvable Lie algebras of dimension four over fields of characteristic zero and also point out the problems that occur in prime characteristic.