Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras

Abstract

We describe and investigate Arens–Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, we explicitly describe their Arens–Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens–Michael envelopes are homological epimorphisms (i.e., localizations in the sense of J. Taylor). For that purpose we introduce and study the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum (2 × 2)-matrices, and universal enveloping algebras of some Lie algebras of small dimensions.