Partial Hopf module categories

TL;DR

This paper introduces partial $H$-module categories, extending the theory of partial actions from algebras to categories, and explores their properties, including globalization, smash products, and specific cases like $k^G$ actions.

Contribution

It extends partial $H$-module algebra results to categories, introducing partial $H$-module categories and analyzing their structure and properties.

Findings

Extended globalization theorem to partial $H$-module categories

Constructed partial smash product and proved Morita equivalence

Described partial actions of $k^G$ in detail

Abstract

The effectiveness of the aplication of constructions in $G$-graded $k$-categories to the computation of the fundamental group of a finite dimensional $k$-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a $k$-linear category, motivated the investigation of $H$-module categories, i.e., actions of a Hopf algebra $H$ on a $k$-linear category. The $G$-graded case corresponds then to actions of the Hopf algebra $k^G$ on a $k$-linear category. In this work we take a step further and introduce partial $H$-module categories. We extend several results of partial $H$-module algebras to this context, such as the globalization theorem, the construction of the partial smash product and the Morita equivalence of this category with the smash product over a globalization. We also present a detailed description of partial actions of $k^G$.