Centers of Braided Tensor Categories

TL;DR

This paper explores the structure of centers in braided tensor categories, linking them to comodules over a cocommutative Hopf algebra, and applies findings to classify irreducible Yetter-Drinfeld modules over certain weak Hopf algebras.

Contribution

It establishes an isomorphism between the center of a braided category and comodules over Majid's automorphism braided group, generalizing previous results.

Findings

Center of $
$ is isomorphic to left $B$-comodules in $
$

Decomposition of $B$ induces a decomposition of comodules and subcategories

Explicit characterization of irreducible Yetter-Drinfeld modules over semisimple quasi-triangular weak Hopf algebras

Abstract

Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of $\mathcal{C}$ is isomorphic to the category of left $B$-comodules in $\mathcal{C}$, and the decomposition of $B$ into a direct sum of indecomposable $\mathcal{C}$-subcoalgebras leads to a decomposition of $B$-$\operatorname*{Comod}_{\mathcal{C}}$ into a direct sum of indecomposable $\mathcal{C}$-module subcategories. As an application, we present an explicit characterization of the structure of irreducible Yetter-Drinfeld modules over semisimple quasi-triangular weak Hopf algebras. Our results generalize those results on finite groups and on quasi-triangular Hopf algebras.