Serre Theorem for involutory Hopf algebras

Abstract

We call a monoidal category $\mathcal{C}$ a Serre category if for any $C$, $D\in \mathcal{C}$ such that $C\ot D$ is semisimple, $C$ and $D$ are semisimple objects in $\mathcal{C}$. Let $H$ be an involutory Hopf algebra, $M$, $N$ two $H$-(co)modules such that $M\otimes N$ is (co)semisimple as a $H$-(co)module. If $N$ (resp. $M$) is a finitely generated projective $k$-module with invertible Hattory-Stallings rank in $k$ then $M$ (resp. $N$) is (co)semisimple as a $H$-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel'd modules over $H$ the dimension of which is invertible in $k$ are Serre categories.

arXiv commenta new version: 8 pages