On partial representations of pointed Hopf algebras

TL;DR

This paper explores the structure of partial representations of pointed Hopf algebras, showing how the associated algebra decomposes into ideals linked to a groupoid, extending known results from finite groups to pointed Hopf algebras.

Contribution

It demonstrates that for pointed Hopf algebras with a finite group of grouplikes, the partial Hopf algebra decomposes into ideals indexed by a groupoid, generalizing previous finite group results.

Findings

H_{par} decomposes into a direct sum of unital ideals

The decomposition is indexed by the components of an associated groupoid

Extends finite group results to pointed Hopf algebras

Abstract

Partial representations of Hopf algebras were motivated by the theory of partial representations of groups. Alves, Batista e Vercruysse introduced partial representations of a Hopf algebra and showed that, as in the case of partial groups actions, a partial $H$-action on an algebra $A$ leads to a partial representation on the algebra of linear endomorphisms of $A$, and a left module $M$ over the partial smash product of $A$ by $H$ carries also a partial representation of $H$ on its algebra of linear endomorphisms. Moreover, partial representations of $H$ correspond to left modules over a Hopf algebroid $H_{par}$. It is known from a result by Dokuchaev, Exel and Piccione that when $H$ is the algebra of a finite group $G$, then $H_{par}$ is isomorphic to the algebra of a finite groupoid determined by $G$. In this work we show that if $H$ is a pointed Hopf algebra with finite group $G$ of grouplikes then $H_{par}$ can be written as a direct sum of unital ideals indexed by the components of the same groupoid associated to the group $G$.