Globalization theorems for partial Hopf (co)actions, and some of their applications

TL;DR

This paper extends the theory of partial Hopf actions by establishing globalization theorems for both actions and coactions, enabling the transfer of results from group actions to Hopf algebra contexts and exploring duality principles.

Contribution

It proves a dual globalization theorem for partial Hopf coactions and applies these results to generalize duality theorems in Hopf algebra theory.

Findings

Existence of enveloping coactions for partial Hopf coactions

Extension of duality theorems to partial Hopf actions

Application of globalization to examples beyond partial group actions

Abstract

Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to prove the existence of enveloping actions, i.e., every partial Hopf action on a algebra A is induced by a Hopf action on a algebra B that contains A as a right ideal. This globalization theorem allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we prove a dual version of the globalization theorem: that every partial coaction of a Hopf algebra admits an enveloping coaction. We also show how this works on a series of examples which go beyond partial group actions. Finally, we explore some consequences of globalization theorems in order to present versions of the duality theorems of Cohen-Montgomery and Blattner-Montgomery for partial Hopf actions.