The isomorphism problem for universal enveloping algebras of nilpotent Lie algebras

TL;DR

This paper investigates whether the universal enveloping algebra uniquely determines a nilpotent Lie algebra, establishing this for dimensions up to 6 over fields with characteristic not 2 or 3, and providing counterexamples otherwise.

Contribution

It proves the isomorphism problem is solvable for nilpotent Lie algebras of dimension up to 6 in certain characteristics, advancing understanding of algebraic structures and their invariants.

Findings

Universal enveloping algebra determines nilpotent Lie algebra up to dimension 6

Characteristic restrictions are necessary for the isomorphism to hold

Counterexamples exist in characteristic 2 and 3

Abstract

In this paper we study the isomorphism problem for the universal enveloping algebras of nilpotent Lie algebras. We prove that if the characteristic of the underlying field is not~2 or~3, then the isomorphism type of a nilpotent Lie algebra of dimension at most~6 is determined by the isomorphism type of its universal enveloping algebra. Examples show that the restriction on the characteristic is necessary.