Representation category of the quantum double of ${\rm SL}(2,q)$

TL;DR

This paper explores the detailed structure of the representation category of the quantum double of ${ m SL}(2,q)$ for specific prime powers, focusing on tensor products and braiding properties.

Contribution

It provides a comprehensive analysis of the tensor and braiding structures in the representation category of the quantum double of ${ m SL}(2,q)$ for certain prime powers.

Findings

Determined tensor product decompositions.

Classified braiding structures.

Enhanced understanding of quantum double categories.

Abstract

For each odd prime power $q$ with $q\ge 5$ and $4\mid q-1$, we investigate the structure of the representation category of the quantum double of ${\rm SL}(2,q)$, determining its tensor products and braidings.