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On the Classification of Finite Quasi-Quantum Groups over Abelian Groups

Hua-Lin Huang, Gongxiang Liu, Yuping Yang, Yu Ye

Abstract

Using a variety of methods developed in the theory of finite-dimensional quasi-Hopf algebras, we classify all finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups. As a consequence, we partially confirm the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.

On the Classification of Finite Quasi-Quantum Groups over Abelian Groups

Abstract

Using a variety of methods developed in the theory of finite-dimensional quasi-Hopf algebras, we classify all finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups. As a consequence, we partially confirm the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.
Paper Structure (13 sections, 35 theorems, 106 equations)

This paper contains 13 sections, 35 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pointed coquasi-Hopf algebras
  4. Nichols algebras of Yetter-Drinfeld modules
  5. Reduction
  6. Nichols algebras
  7. Nichols algebras of rank $3$
  8. The general case
  9. Classification
  10. The proof of Theorem \ref{['t3.9']}
  11. A special case
  12. A proof of Theorem \ref{['t3.9']}
  13. Finite quasi-quantum groups over abelian groups

Key Result

Lemma 2.4

HLYY2 The twisting $B(V)^J$ of $B(V)$ is a Nichols algebra in $^{\mathbbm{k} G}_{\mathbbm{k} G}\mathcal{Y}\mathcal{D}^{\Phi\ast \partial J}$ and $B(V)^J\cong B(V^J)$.

Theorems & Definitions (62)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7: HLYY2, Lemma 4.4
  • Lemma 2.8
  • proof
  • Proposition 2.9
  • ...and 52 more