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Hopf algebras with the dual Chevalley property of discrete corepresentation type

Jing Yu, Gongxiang Liu

TL;DR

The paper advances the classification of Hopf algebras with the dual Chevalley property by leveraging link quivers to characterize discrete corepresentation type in both finite- and infinite-dimensional settings. It establishes equivalences between discreteness and the quiver being a disjoint union of basic cycles when the coradical of the link-indecomposable component is finite-dimensional, and it provides a tripartite division for the infinite coradical case, detailing the structure of the link-indecomposable component. A central contribution is the explicit infinite-dimensional example $H(e_{\\pm1}, f_{\\pm1}, u, v)$, which is non-pointed yet has discrete corepresentation type, illustrating new phenomena beyond pointed Hopf algebras. The results unify and extend finite-dimensional classifications and point to a broader landscape of DCT Hopf algebras via the link quiver framework.

Abstract

We try to classify Hopf algebras with the dual Chevalley property of discrete corepresentation type over an algebraically closed field $\Bbb{k}$ with characteristic 0. For such Hopf algebra $H$, we characterize the link quiver of $H$ and determine the structures of the link-indecomposable component $H_{(1)}$ containing $\Bbb{k}1$. Besides, we construct an infinite-dimensional non-pointed non-cosemisimple link-indecomposable Hopf algebra $H(e_{\pm 1}, f_{\pm 1}, u, v)$ with the dual Chevalley property of discrete corepresentation type.

Hopf algebras with the dual Chevalley property of discrete corepresentation type

TL;DR

The paper advances the classification of Hopf algebras with the dual Chevalley property by leveraging link quivers to characterize discrete corepresentation type in both finite- and infinite-dimensional settings. It establishes equivalences between discreteness and the quiver being a disjoint union of basic cycles when the coradical of the link-indecomposable component is finite-dimensional, and it provides a tripartite division for the infinite coradical case, detailing the structure of the link-indecomposable component. A central contribution is the explicit infinite-dimensional example , which is non-pointed yet has discrete corepresentation type, illustrating new phenomena beyond pointed Hopf algebras. The results unify and extend finite-dimensional classifications and point to a broader landscape of DCT Hopf algebras via the link quiver framework.

Abstract

We try to classify Hopf algebras with the dual Chevalley property of discrete corepresentation type over an algebraically closed field with characteristic 0. For such Hopf algebra , we characterize the link quiver of and determine the structures of the link-indecomposable component containing . Besides, we construct an infinite-dimensional non-pointed non-cosemisimple link-indecomposable Hopf algebra with the dual Chevalley property of discrete corepresentation type.
Paper Structure (10 sections, 24 theorems, 134 equations)

This paper contains 10 sections, 24 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Based ring
  4. Link quiver
  5. discrete corepresentation type
  6. Hopf algebras with the dual Chevalley property of discrete corepresentation type
  7. Case (1)
  8. Case (2)
  9. Case (3)
  10. Example

Key Result

Theorem 1.1

Let $H$ be a non-cosemisimple Hopf algebra over $\Bbbk$ with the dual Chevalley property and $H_{(1)}$ be its link-indecomposable component containing $\Bbbk1$. If the coradical of $H_{(1)}$ is finite-dimensional, then the following statements are equivalent:

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 47 more