Morita equivalence and globalization for partial Hopf actions on nonunital algebras

TL;DR

This paper studies partial Hopf algebra actions on nonunital algebras, exploring their globalization, Morita equivalence, and implications for module categories, extending existing theories to broader algebraic contexts.

Contribution

It introduces Morita equivalence for partial Hopf actions and establishes conditions for the existence and uniqueness of globalizations, extending prior work to nonunital algebras.

Findings

Partial actions correspond to nonunital algebras in partial representation categories.

Sufficient conditions for the existence and uniqueness of minimal globalizations.

Morita equivalence of partial actions implies Morita equivalence of their globalizations.

Abstract

In this work we investigate partial actions of a Hopf algebra H on nonunital algebras and the associated partial smash products. We show that our partial actions correspond to nonunital algebras in the category of partial representations of H. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of k-categories.