# Tensor algebras in finite tensor categories

## TL;DR

This paper develops classification methods for actions of finite-dimensional Hopf algebras on tensor algebras within finite tensor categories, extending to fusion categories and illustrating with specific examples.

## Contribution

It introduces a framework for classifying tensor algebra actions in finite tensor categories, including semisimple and group-theoretical cases, with new classification results.

## Key findings

- Classification of Hopf algebra actions preserving filtrations on tensor algebras
- Extension of classification to pointed and group-theoretical fusion categories
- Application to the representation category of the Kac-Paljutkin Hopf algebra

## Abstract

This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions which preserve the ascending filtration on tensor algebras, and the second for actions which preserve the descending filtration on completed tensor algebras. Extending to more general fusion categories, we illustrate our classification result for tensor algebras in the pointed fusion categories ${\sf Vec}_{G}^{\omega}$ and in group-theoretical fusion categories, especially for the representation category of the Kac-Paljutkin Hopf algebra.