On the Drinfeld double of a finite group scheme and its representation category

TL;DR

This paper classifies certain Hopf algebra quotients of the Drinfeld double of a finite group scheme, describing their structure and categorical properties in terms of group scheme data.

Contribution

It provides a classification of Hopf algebra quotients of the Drinfeld double and analyzes their tensor subcategories and objects, generalizing previous characteristic zero results.

Findings

Classified Hopf algebra quotients of D(G) using group scheme data.

Described tensor subcategories and their properties within the representation category.

Extended categorical results to positive characteristic fields.

Abstract

We classify equivalence classes of Hopf algebra quotient pairs $(D,\theta)$ of the Drinfeld double $D(G)$ of a finite group scheme $G$ over an algebraically closed field $\mathbf{k}$ of characteristic $p\ge 0$, in terms of group scheme-theoretical data. We prove that such Hopf algebra quotients $D$ are Hopf algebra extensions $\mathscr{O}(K)^{\mathrm{cop}}\#_{\sigma}^{\tau} \mathbf{k}[G/H]$, where $K$ and $H$ are normal subgroup schemes of $G$ that centralize each other and $B:\mathbf{k}[H]\to \mathscr{O}(K)$ is a $G$-equivariant Hopf algebra map, and describe the surjective Hopf algebra map $\theta:D(G)\twoheadrightarrow D$. Using this classification, we determine the tensor subcategories of the center $\mathscr{Z}(G):=\Rep(D(G))$ of $G$, describe their centralizers, determine when they are symmetric or non-degenerate, and give a description of their simple and projective objects using \cite{GS}. Our categorical results generalize those found in \cite{NNW} in characteristic $0$.