Non-degeneracy conditions for braided finite tensor categories

Abstract

For a braided finite tensor category $\mathcal{C}$ with unit object $1\in \mathcal{C}$, Lyubashenko considered a certain Hopf algebra $\mathbb{F}\in \mathcal{C}$ endowed with a Hopf pairing $\omega :\mathbb{F}\otimes \mathbb{F}\to 1$ to define the notion of a `non-semisimple' modular tensor category. We say that $\mathcal{C}$ is non-degenerate if the Hopf pairing $\omega$ is non-degenerate. In this paper, we show that $\mathcal{C}$ 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 $\Omega :{\mathrm{Hom}}_{\mathcal{C}}(1,\mathbb{F})\to {\mathrm{Hom}}_{\mathcal{C}}(\mathbb{F},1)$ induced by the pairing $\omega$ is invertible. As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in $\mathcal{C}$ is non-degenerate if and only if $\mathcal{C}$ is.

28 pages, some figures. List of changes from v1: (1) An example of C such that the linear map $\Omega$ is surjective but not bijective. (2) Remarks on the rank of $\Omega$. (3) Detailed explanation on the construction of small quantum groups. There are several other minor changes. References are also updated

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