Multiplier Hopf Algebras: Globalization for partial actions

TL;DR

This paper extends the theory of partial actions of Hopf algebras to nonunitary algebras using multiplier Hopf algebras, establishing globalization theorems and a bijection with group partial actions.

Contribution

It introduces globalization results for partial (co)actions of multiplier Hopf algebras on nonunitary algebras and links these to group partial actions.

Findings

Globalization theorems for multiplier Hopf algebra actions

Bijection between group partial actions and multiplier Hopf algebra actions

Extension of partial action theory to nonunitary algebras

Abstract

In partial action theory, a pertinent question is whenever given a partial (co)action of a Hopf algebra A on an algebra R, it is possible to construct an enveloping (co)action. The authors Alves and Batista, in [2],have shown that this is always possible if R has unit. We are interested in investigating the situation where both algebras A and R are nonunitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [11], which is called multiplier Hopf algebra. Therefore, we will consider partial (co)actions of multipliers Hopf algebras on algebras not necessarily unitary and we will present globalization theorems for these structures. Moreover, Dockuchaev, Del Rio and Sim\'on, in [5], have shown when group partial actions on nonunitary algebras are globalizable. Based in [5], we will establish a bijection between group partial actions on an algebra R not necessarily unitary and partial actions of a multiplier Hopf algebra on the algebra R.