Partial actions of weak Hopf algebras: smash products, globalization and Morita theory

TL;DR

This paper introduces the concept of partial actions of weak Hopf algebras on algebras, developing tools like partial smash products and globalization, and explores their Morita equivalences and applications to groupoid actions.

Contribution

It unifies various notions of partial actions under weak Hopf algebras and constructs foundational tools like partial smash products and globalization, establishing Morita equivalences and bijections.

Findings

Partial actions unify group, Hopf, and groupoid actions.

Construction of partial smash product and globalization tools.

Morita equivalence between globalized partial actions.

Abstract

In this paper we introduce the notion of partial action of a weak Hopf algebra on algebras, unifying the notions of partial group action [11], partial Hopf action ([2],[3],[9]) and partial groupoid action [4]. We construct the fundamental tools to develop this new subject, namely, the partial smash product and the globalization of a partial action, as well as, we establish a connection between partial and global smash products via the construction of a surjective Morita context. In particular, in the case that the globalization is unital, these smash products are Morita equivalent. We show that there is a bijective correspondence between globalizable partial groupoid actions and symmetric partial groupoid algebra actions, extending similar result for group actions [9]. Moreover, as an application we give a complete description of all partial actions of a weak Hopf algebra on its ground field, which suggests a method to construct more general examples.