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

TL;DR

The paper resolves when the Arens–Michael envelope of the universal enveloping algebra $U(\mathfrak{g})$ yields a homological epimorphism, proving this occurs if and only if the finite-dimensional complex Lie algebra $\mathfrak{g}$ is solvable. The author develops a framework of analytic smash products and relative homological epimorphisms, replacing Ore extensions with iterated smash products and leveraging the unique extension property for derivations (UDE). The sufficiency proof proceeds by decomposing $\mathfrak{g}$ into an iterated semidirect sum of 1-dimensional subalgebras, obtaining a compatible tower of analytic smash-product decompositions and applying relative-HE results at each step. This work extends prior partial results, providing a concrete constructive path from solvability to homological epimorphism for the Arens–Michael envelope with potential implications for functional-analytic approaches to algebras of analytic functionals and noncommutative localization.

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.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Proposition 2.6
• proof