Morita equivalence and globalization for partial Hopf actions on nonunital algebras

Abstract

In this work, we investigate partial actions of a Hopf algebra $H$ on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both $𝕜$-algebras with local units and $𝕜$-categories. 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.