Module categories over affine group schemes

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 ${\mathrm{Coh}}_f(G)$ 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 geometrical module categories over Rep$(G)$. When $G$ is finite, this yields the classification of

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.