The Arens-Michael envelope of a solvable Lie algebra is a homological epimorphism

TL;DR

This paper proves that for finite-dimensional complex Lie algebras, the Arens-Michael envelope of their universal enveloping algebra is a homological epimorphism precisely when the algebra is solvable, completing a characterization.

Contribution

It establishes the sufficiency of solvability for the Arens-Michael envelope to be a homological epimorphism, complementing previous necessity results.

Findings

The Arens-Michael envelope is a homological epimorphism if and only if the Lie algebra is solvable.

Completes the characterization of when the envelope has this property.

Provides a full criterion linking algebraic structure to homological properties.

Abstract

The Arens-Michael envelope of the universal enveloping algebra of a finite-dimensional complex Lie algebra is a homological epimorphism if and only if the Lie algebra is solvable. The necessity was proved by Pirkovskii in [Proc. Amer. Math. Soc. 134, 2621--2631, 2006]. We prove the sufficiency.