Representations of the Drinfeld doubles of Pointed rank one Hopf algebras

TL;DR

This paper classifies all finite-dimensional indecomposable modules over the Drinfeld doubles of pointed rank one Hopf algebras, describing their Auslander-Reiten sequences and establishing the tame representation type of these algebras.

Contribution

It provides a complete classification of modules and explicit Auslander-Reiten sequences for the Drinfeld doubles of pointed rank one Hopf algebras, revealing their tame representation type.

Findings

Complete classification of indecomposable modules

Explicit description of Auslander-Reiten sequences

Proven tame representation type

Abstract

In this paper, we investigate the representations of the Drinfeld doubles $D(H_{\mathcal{D}})$ of pointed rank one Hopf algebras $H_{\mathcal{D}}$ over an algebraically closed field $\Bbbk$ of characteristic zero. We provide a complete classification of all finite-dimensional indecomposable $D(H_{\mathcal{D}})$-modules up to isomorphism and explicitly describe the Auslander-Reiten sequences in the category of finite-dimensional $D(H_{\mathcal{D}})$-modules. We show that $D(H_{\mathcal{D}})$ is of tame representation type.