Module categories over affine group schemes

TL;DR

This paper classifies indecomposable module categories over the tensor category of coherent sheaves on affine group schemes, leading to new insights into twists of group algebras and finite-dimensional Hopf algebras in positive characteristic.

Contribution

It provides a classification of indecomposable exact module categories over affine group schemes and their representation categories, including new results on twists and Hopf algebras in positive characteristic.

Findings

Classification of indecomposable exact module categories over $ ext{Coh}_f(G)$

Identification of twists for $k[G]$ of finite group schemes

Construction of new finite-dimensional noncommutative, noncocommutative triangular Hopf algebras

Abstract

Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of finite dimensional $k-$vector spaces on $G$, in terms of $(H,\psi)-$equivariant coherent sheaves on $G$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\Rep(G)$. When $G$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\Rep(G)$. In particular, we obtain a classification of twists for the group algebra $k[G]$ of a finite group scheme $G$. Applying this to $u(\mathfrak {g})$, where $\mathfrak {g}$ is a finite dimensional $p-$Lie algebra over $k$ with positive characteristic, produces (new) finite dimensional noncommutative and noncocommutative triangular Hopf algebras in positive characteristic. We also introduce and study group scheme theoretical categories, and study isocategorical finite group schemes.