Enveloping Actions for Partial Hopf Actions

TL;DR

This paper extends the theory of partial group actions to partial Hopf algebra actions, establishing enveloping actions, Morita contexts, dualization to coactions, and linking partial representations with actions.

Contribution

It generalizes the globalization theorem for partial Hopf actions and constructs a Morita context, advancing the understanding of partial symmetries in algebraic structures.

Findings

Existence of enveloping actions for partial Hopf actions

Construction of a Morita context between partial and global smash products

Dualization of the globalization theorem to partial coactions

Abstract

Motivated by partial group actions on unital algebras, in this article we extend many results obtained by Exel and Dokuchaev to the context of partial actions of Hopf algebras, according to Caenepeel and Jansen. First, we generalize the theorem about the existence of an enveloping action, also known as the globalization theorem. Second, we construct a Morita context between the partial smash product and the smash product related to the enveloping action. Third, we dualize the globalization theorem to partial coactions and finally, we define partial representations of Hopf algebras and show some results relating partial actions and partial representations.