Tensor Algebras in Finite Tensor Categories
Name
1906.02828.pdf
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Accepted version
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462.57 KB
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d09bb4a62456089ebe207c35a7c4c665
Author(s)
Etingof, Pavel
Kinser, Ryan
Walton, Chelsea
Date Issued
2019
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Version
Author's final manuscript
Abstract
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 that preserve the ascending filtration on tensor algebras and the second for actions that 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 $\textsf{Vec}_{G}^{\omega }$ and in group-theoretical fusion categories, especially for the representation category of the Kac–Paljutkin Hopf algebra.
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 that preserve the ascending filtration on tensor algebras and the second for actions that 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 $\textsf{Vec}_{G}^{\omega }$ and in group-theoretical fusion categories, especially for the representation category of the Kac–Paljutkin Hopf algebra.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1093/IMRN/RNZ332
