Isomorphism invariants of enveloping algebras

TL;DR

This paper surveys the isomorphism problem for enveloping algebras of Lie algebras, exploring whether algebraic isomorphisms imply Lie algebra isomorphisms, including in the context of restricted Lie algebras over fields of positive characteristic.

Contribution

It compiles and discusses known results and open questions regarding when the structure of an enveloping algebra determines the underlying Lie algebra.

Findings

Isomorphism of enveloping algebras often implies Lie algebra isomorphism in characteristic zero.

In positive characteristic, the problem is more complex and less understood.

The paper highlights key results and open problems in the isomorphism problem for enveloping algebras.

Abstract

Let $L$ be a Lie algebra with its enveloping algebra $U(L)$ over a field. In this paper we survey results concerning the isomorphism problem for enveloping algebras: given another Lie algebra $H$ for which $U(L)$ and $U(H)$ are isomorphic as associative algebras, can we deduce that $L$ and $H$ are isomorphic Lie algebras? Over a field of positive characteristic we consider a similar problem for restricted Lie algebras, that is, given restricted Lie algebras $L$ and $H$ for which their restricted enveloping algebras are isomorphic as algebras, can we deduce that $L$ and $H$ are isomorphic?