Module categories over affine supergroup schemes

TL;DR

This paper classifies indecomposable exact module categories over tensor categories associated with affine supergroup schemes, extending to finite supergroups and providing classifications of twists and triangular Hopf algebras.

Contribution

It provides a comprehensive classification of module categories over supergroup scheme tensor categories, including finite cases and applications to Hopf algebras.

Findings

Classification of indecomposable exact module categories over ${ m sCoh}_{ m f}( ext{supergroup})$

Complete classification of twists for supergroup algebras

Classification of finite-dimensional triangular Hopf algebras with Chevalley property

Abstract

Let $k$ be an algebraically closed field of characteristic $0$ or $p>2$. Let $\mathcal{G}$ be an affine supergroup scheme over $k$. We classify the indecomposable exact module categories over the tensor category ${\rm sCoh}_{\rm f}(\mathcal{G})$ of (coherent sheaves of) finite dimensional $\mathcal{O}(\mathcal{G})$-supermodules in terms of $(\mathcal{H},\Psi)$-equivariant coherent sheaves on $\mathcal{G}$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\sRep(\mathcal{G})$. When $\mathcal{G}$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\sRep(\mathcal{G})$. In particular, we obtain a classification of twists for the supergroup algebra $k\mathcal{G}$ of a finite supergroup scheme $\mathcal{G}$, and then combine it with \cite[Corollary 4.1]{EG3} to classify finite dimensional triangular Hopf algebras with the Chevalley property over $k$.