Serre Theorem for involutory Hopf algebras

TL;DR

This paper establishes conditions under which certain categories of modules over involutory Hopf algebras are Serre categories, linking semisimplicity of tensor products to individual module semisimplicity.

Contribution

It proves that categories of finite-dimensional modules, comodules, or Yetter-Drinfel'd modules over involutory Hopf algebras are Serre categories when module dimensions are invertible in the base ring.

Findings

Semisimplicity of tensor products implies individual semisimplicity under certain conditions.

Categories of modules with invertible dimension form Serre categories.

Results apply to modules, comodules, and Yetter-Drinfel'd modules over 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.