The Representations of Quantum Double of Dihedral Groups

TL;DR

This paper classifies all finite dimensional indecomposable modules over the quantum double of dihedral groups, providing a comprehensive understanding of their structure in the context of algebraic and quantum group theory.

Contribution

It describes and classifies all finite dimensional indecomposable modules over the quantum double of dihedral groups, a novel comprehensive structural analysis.

Findings

Complete classification of indecomposable modules

Explicit descriptions of module structures

Advancement in understanding quantum doubles of dihedral groups

Abstract

Let $k$ be an algebraically closed field of odd characteristic $p$, and let $D_n$ be the dihedral group of order $2n$ such that $p\mid 2n$. Let $D(kD_n)$ denote the quantum double of the group algebra $kD_n$. In this paper, we describe the structures of all finite dimensional indecomposable left $D(kD_n)$-modules, equivalently, of all finite dimensional indecomposable Yetter-Drinfeld $kD_n$-modules, and classify them.