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

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 $\mathcal{O}(K{)}^{\mathrm{cop}}{\#}_{\sigma }^{\tau }\mathbf{k}[G/H]$, where $K$ and $H$ are normal subgroup schemes of $G$ that centralize each other and $B:\mathbf{k}[H]\to \mathcal{O}(K)$ 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 $0$.

37 pages