Normal Hopf subalgebras of semisimple Drinfeld doubles

TL;DR

This paper characterizes all normal Hopf subalgebras within semisimple Drinfeld doubles using fusion category techniques, revealing structural properties and extensions.

Contribution

It provides a complete description of normal Hopf subalgebras in semisimple Drinfeld doubles and applies Goursat's lemma analogue to fusion categories.

Findings

All normal Hopf subalgebras of semisimple Drinfeld doubles are classified.

The Drinfeld double of any abelian extension remains an abelian extension.

Fusion subcategory analysis yields structural insights into Hopf algebra extensions.

Abstract

A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.