The Arens-Michael envelope of a smash product

TL;DR

This paper characterizes the Arens-Michael envelope of a smash product of a Hopf algebra and an H-module algebra, providing explicit descriptions and examples, and highlighting the importance of the Hopf algebra structure.

Contribution

It explicitly describes the Arens-Michael envelope of a smash product involving a Hopf algebra and an H-module algebra, and shows the limitations for non-Hopf bialgebras.

Findings

The Arens-Michael envelope of A#H can be described via the envelopes of H and a completion of A.

The result holds for Hopf algebras but fails for non-Hopf bialgebras, as shown by the example of Manin's quantum plane.

The paper provides explicit constructions and counterexamples to illustrate the theory.

Abstract

Given a Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H in terms of the Arens-Michael envelope of H and a certain completion of A. We also give an example (Manin's quantum plane) showing that the result fails for non-Hopf bialgebras.