Partial representations of connected and smash product Hopf algebras

TL;DR

This paper investigates partial representations of connected and smash product Hopf algebras, establishing conditions under which they are global and describing the structure of associated partial Hopf algebras.

Contribution

It proves that all partial representations of connected Hopf algebras are global and characterizes partial Hopf algebras for certain classes of smash product Hopf algebras.

Findings

Partial representations of connected Hopf algebras are always global.

Description of partial Hopf algebra when the first tensorand is connected.

Identification of $H_{par}$ as a weak Hopf algebra from a Hopf category under certain conditions.

Abstract

We show that every partial representation of a connected Hopf algebra is global. Some interesting classes of partial representations of smash product Hopf algebras are studied, and a description of the partial "Hopf" algebra if the first tensorand is connected is given. If $H$ is cocommutative and has finitely many grouplikes, this allows to see $H_{par}$ as the weak Hopf algebra coming from a Hopf category.