Semisimplicity of certain representation categories

TL;DR

This paper explores the relationship between module subcategories and bimodules, establishing conditions for semisimplicity, and applies these results to quantum group representations, demonstrating semisimplicity for specific cases.

Contribution

It introduces a correspondence between module subcategories and bimodules, linking semisimplicity to Peter-Weyl decompositions, and applies this to quantum double representations.

Findings

Semisimplicity is characterized by Peter-Weyl decomposition.

Established correspondence between subcategories and bimodules.

Proved semisimplicity for certain quantum group representations.

Abstract

We exhibit a correspondence between subcategories of modules over an algebra and sub-bimodules of the dual of that algebra. We then prove that the semisimplicity of certain such categories is equivalent to the existence of a Peter-Weyl decomposition of the corresponding sub-bimodule. Finally, we use this technique to establish the semisimplicity of certain finite-dimensional representations of the quantum double of sl_2 for generic q.