On nondiagonal finite quasi-qantum groups over finite abelian groups

TL;DR

This paper explores the structure and classification of nondiagonal finite quasi-quantum groups over finite abelian groups, focusing on Nichols algebras in twisted Yetter-Drinfeld categories with nonabelian 3-cocycles.

Contribution

It provides a complete classification of certain finite-dimensional coquasi-Hopf algebras and clarifies Nichols algebras in twisted categories with nonabelian cocycles.

Findings

Complete classification of finite-dimensional coradically graded pointed coquasi-Hopf algebras.

Clarification of Nichols algebras for simple twisted Yetter-Drinfeld modules of nondiagonal type.

Partial confirmation of the generation conjecture for pointed finite tensor categories.

Abstract

In this paper, we initiate the study of nondiagonal finite quasi-quantum groups over finite abelian groups. We mainly study the Nichols algebras in the twisted Yetter-Drinfeld module category $_{\k G}^{\k G}\mathcal{YD}^\Phi$ with $\Phi$ a nonabelian $3$-cocycle on a finite abelian group $G.$ A complete clarification is obtained for the Nichols algebra $B(V)$ in case $V$ is a simple twisted Yetter-Drinfeld module of nondiagonal type. This is also applied to provide a complete classification of finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups of odd order and confirm partially the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.