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Representations of the Drinfeld doubles of Pointed rank one Hopf algebras

Hua Sun, Huixiang Chen, Yinhuo Zhang

TL;DR

The work addresses the representation theory of Drinfeld doubles of pointed rank-one Hopf algebras by classifying all finite-dimensional indecomposable modules over $D(H_D)$ when the group datum is abelian. It combines explicit constructions of simple and projective modules with Auslander-Reiten theory to obtain complete indecomposable classifications and AR sequences, showing that $D(H_D)$ is of tame representation type. The authors delineate the Loewy structure of projectives, describe all indecomposables of Loewy length two (splitting into the cases $m>1$ and $m=1$), and provide detailed families parameterized by $(l, ext{lambda}, ext{eta})$, along with their AR sequences. These results enhance the understanding of the representation theory of quasi-triangular Hopf algebras and contribute to the broader study of braided tensor categories arising from Drinfeld doubles.

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.

Representations of the Drinfeld doubles of Pointed rank one Hopf algebras

TL;DR

The work addresses the representation theory of Drinfeld doubles of pointed rank-one Hopf algebras by classifying all finite-dimensional indecomposable modules over when the group datum is abelian. It combines explicit constructions of simple and projective modules with Auslander-Reiten theory to obtain complete indecomposable classifications and AR sequences, showing that is of tame representation type. The authors delineate the Loewy structure of projectives, describe all indecomposables of Loewy length two (splitting into the cases and ), and provide detailed families parameterized by , along with their AR sequences. These results enhance the understanding of the representation theory of quasi-triangular Hopf algebras and contribute to the broader study of braided tensor categories arising from Drinfeld doubles.

Abstract

In this paper, we investigate the representations of the Drinfeld doubles of pointed rank one Hopf algebras over an algebraically closed field of characteristic zero. We provide a complete classification of all finite-dimensional indecomposable -modules up to isomorphism and explicitly describe the Auslander-Reiten sequences in the category of finite-dimensional -modules. We show that is of tame representation type.
Paper Structure (8 sections, 44 theorems, 170 equations)

This paper contains 8 sections, 44 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Pointed Hopf algebras of rank one and their doubles
  3. Simple modules and projective modules
  4. Simple modules
  5. Indecomposable projective modules
  6. Indecomposable modules with Loewy length two
  7. The case of $m>1$
  8. The case of $m=1$

Key Result

Proposition 2.1

KR The Drinfeld double $D(H_{\mathcal{D}})$ is generated as an algebra by $G$, $x$, $\Gamma$ and $\xi$ subject to the relations defining $H_{\mathcal{D}}$ and $H^{*\rm cop}_{\mathcal{D}}$ and the following relations:

Theorems & Definitions (74)

  • Proposition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Remark 3.4
  • Corollary 3.5
  • proof
  • Proposition 3.6
  • Corollary 3.7
  • Proposition 3.8
  • ...and 64 more