Ribbon structures of the Drinfeld center of a finite tensor category

TL;DR

This paper classifies the ribbon structures of the Drinfeld center of finite tensor categories, extending known results from Hopf algebras and connecting to modular tensor categories.

Contribution

It generalizes the classification of ribbon elements from Hopf algebras to finite tensor categories and establishes conditions for modularity.

Findings

Classification of ribbon structures for Drinfeld centers

Extension of Kauffman-Radford's results to tensor categories

Identification of conditions for modular tensor categories

Abstract

We classify the ribbon structures of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$. Our result generalizes Kauffman and Radford's classification result of the ribbon elements of the Drinfeld double of a finite-dimensional Hopf algebra. As a consequence, we see that $\mathcal{Z}(\mathcal{C})$ is a modular tensor category in the sense of Lyubashenko if $\mathcal{C}$ is a spherical finite tensor category in the sense of Douglas, Schommer-Pries and Snyder.