Tensor algebras in finite tensor categories

Understand

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.