Partial Representations of Hopf Algebras

TL;DR

This paper introduces the concept of partial representations of Hopf algebras, linking them to partial actions and establishing a monoidal category structure for partial modules via a Hopf algebroid.

Contribution

It defines partial representations of Hopf algebras, constructs the associated Hopf algebroid, and develops a monoidal category framework for partial modules, including specific examples.

Findings

Partial representations are characterized via a universal Hopf algebroid.

The category of partial modules is endowed with a monoidal structure.

Examples include the category of partial $\

Abstract

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which has the universal property that each partial representation of $H$ can be factorized by an algebra morphism from $H_{par}$. We define also the category of partial modules over a Hopf algebra $H$, which is the category of modules over its associated Hopf algebroid $H_{par}$. The Hopf algebroid structure of $H_{par}$ enables us to enhance the category of partial $H$ modules with a monoidal structure and such that the algebra objects in this category are the usual partial actions. Some examples of categories of partial $H$ modules are explored. In particular we can describe fully the category of partially $\mathbb{Z}_2$-graded modules.