Non-degeneracy conditions for braided finite tensor categories
Non-degeneracy conditions for braided finite tensor categories
复制标题
DOI：
10.1016/j.aim.2019.106778
复制
发表时间：
2016-02
影响因子：
1.7
通讯作者：
K. Shimizu
中科院分区：
文献类型：
--
作者：
K. Shimizu
对于单位对象为1∈ C的辫状有限张量范畴C，Lyubashenko考虑了一个赋有Hopf对ω：F <$F→ 1的Hopf代数F∈ C来定义“非半单”模张量范畴的概念。我们说C是非退化的，如果Hopf对ω是非退化的。本文证明了C是非退化的当且仅当它在Etingof，Nikshych和Ostrik意义下是可分解的，当且仅当它的Müger中心是平凡的，当且仅当由对ω诱导的线性映射Hom C（1，F）→ Hom C（F，1）是可逆的.作为应用，我们证明了C中Hopf代数上的Yetter-Drinfeld模范畴是非退化的当且仅当C是.
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.