### 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 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Non-degeneracy conditions for braided finite tensor categories.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Non-degeneracy conditions for braided finite tensor categories

AU - Shimizu, Kenichi

PY - 2019/10/15

Y1 - 2019/10/15

N2 - 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 HomC(1,F)→HomC(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.

AB - 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 HomC(1,F)→HomC(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.

KW - Briaded tensor category

KW - Finite tensor category

KW - Hopf algebra

KW - Modular tensor category

UR - http://www.scopus.com/inward/record.url?scp=85071015284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071015284&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.106778

DO - 10.1016/j.aim.2019.106778

M3 - Article

AN - SCOPUS:85071015284

VL - 355

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 106778

ER -