# Non-degeneracy conditions for braided finite tensor categories

Research paper by **Kenichi Shimizu**

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

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.