### Abstract

For a braided finite tensor category C with unit object 1∈C, Lyubashenko considered a certain Hopf algebra F∈C endowed with a Hopf pairing ω:F⊗F→1 to define the notion of a ‘non-semisimple’ modular tensor category. We say that C is non-degenerate if the Hopf pairing ω is non-degenerate. In this paper, we show that C is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its Müger center is trivial, if and only if the linear map Hom_{C}(1,F)→Hom_{C}(F,1) induced by the pairing ω is invertible. As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in C is non-degenerate if and only if C is.

Original language | English |
---|---|

Article number | 106778 |

Journal | Advances in Mathematics |

Volume | 355 |

DOIs | |

Publication status | Published - 2019 Oct 15 |

### Fingerprint

### Keywords

- Briaded tensor category
- Finite tensor category
- Hopf algebra
- Modular tensor category

### ASJC Scopus subject areas

- Mathematics(all)