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Simple Yetter-Drinfeld modules over Generalized Liu algebras

Xiangjun Zhen, Gongxiang Liu, Jing Yu

Abstract

Let $H$ be a generalized Liu algebra over an algebraically closed field $k$ of characteristic zero. We prove that all simple Yetter-Drinfeld modules over $H$ are finite-dimensional and present an explicit classification of these modules. Moreover, we completely determine which of them admit a finite-dimensional Nichols algebra.

Simple Yetter-Drinfeld modules over Generalized Liu algebras

Abstract

Let be a generalized Liu algebra over an algebraically closed field of characteristic zero. We prove that all simple Yetter-Drinfeld modules over are finite-dimensional and present an explicit classification of these modules. Moreover, we completely determine which of them admit a finite-dimensional Nichols algebra.
Paper Structure (17 sections, 31 theorems, 109 equations)

This paper contains 17 sections, 31 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Braided vector spaces and Yetter-Drinfeld modules
  4. Nichols Algebras
  5. Bosonization
  6. 2-Cocycle Deformations
  7. Generalized Liu algebras
  8. Comodule Structures over Simple Yetter-Drinfeld Modules
  9. Existence of Standard Bases for Simple Yetter-Drinfeld Modules
  10. The Comatrix Relative to a Standard Basis
  11. Further Analysis of the Comatrix
  12. Classification of Simple Yetter-Drinfeld Modules over $H$
  13. Simple Yetter-Drinfeld modules over $H$
  14. Connection to simple Yetter-Drinfeld modules over $A(d,m,\gamma,1)$
  15. Finite-dimensional Nichols algebras over the simple modules $V(\alpha,\beta,x^rg^i)$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $H = B(n,w,\gamma)$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 61 more