On a symmetry of M\"uger's centralizer for the Drinfeld double of a semisimple Hopf algebra

TL;DR

This paper establishes a formula connecting Muger's centralizer in the representation category of a factorizable Hopf algebra to Hopf kernels, providing a complete description for certain fusion subcategories of Drinfeld doubles.

Contribution

It introduces a new formula linking Muger's centralizer to Hopf kernels, advancing understanding of fusion subcategories in Drinfeld doubles of semisimple Hopf algebras.

Findings

Derived a formula relating Muger's centralizer to Hopf kernels.

Provided a complete description of Muger's centralizer for specific fusion subcategories.

Enhanced the theoretical framework for studying representations of factorizable Hopf algebras.

Abstract

In this paper we prove a formula that relates M\"uger's centralizer in the category of representations of a factorizable Hopf algebra to the notion of Hopf kernel of a representation of the dual Hopf algebra. Using this relation we obtain a complete description for M\"uger's centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a Drinfeld double of a semisimple Hopf algebra.