抄録
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.
| 元の言語 | English |
|---|---|
| 記事番号 | 106778 |
| ジャーナル | Advances in Mathematics |
| 巻 | 355 |
| DOI | |
| 出版物ステータス | Published - 2019 10 15 |
ASJC Scopus subject areas
- Mathematics(all)
これを引用
Non-degeneracy conditions for braided finite tensor categories. / Shimizu, Kenichi.
：: Advances in Mathematics, 巻 355, 106778, 15.10.2019.研究成果: 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 -