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Non-degeneracy conditions for braided finite tensor categories

Research paper by Kenichi Shimizu

Indexed on: 21 Feb '16Published on: 21 Feb '16Published in: Mathematics - Quantum Algebra



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 paring $\omega: \mathbb{F} \otimes \mathbb{F} \to 1$ to define a `non-semisimple' modular tensor category. We say that $\mathcal{C}$ is non-degenerate if the Hopf paring $\omega$ is non-degenerate. Our main result is 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 M\"uger centralizer is trivial, if and only if the linear map $\mathrm{Hom}_{\mathcal{C}}(1, \mathbb{F}) \to \mathrm{Hom}_{\mathcal{C}}(\mathbb{F}, 1)$ induced by $\omega$ 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.