Dilations of partial representations of Hopf algebras

TL;DR

This paper develops a dilation theory for partial representations of Hopf algebras, establishing categorical equivalences and exploring the relationship between partial and global modules.

Contribution

It introduces a dilation concept for partial Hopf algebra representations and connects partial modules with global modules and smash product modules.

Findings

Categorical equivalences between partial and global modules

Construction of a smash product with a Hopfish algebra structure

Insights into the interaction between partial and global representation theories

Abstract

We introduce the notion of a dilation for a partial representation (i.e. a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (i.e.a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of partial $H$-modules, a category of (global) $H$-modules endowed with a projection satisfying a suitable commutation relation and the category of modules over a (global) smash product constructed upon $H$, from which we deduce the structure of a Hopfish algebra on this smash product. These equivalences are used to study the interactions between partial and global representation theory.