On Gauge Equivalence of Twisted Quantum Doubles

TL;DR

This paper investigates when twisted quantum doubles of finite abelian groups are gauge equivalent to ordinary doubles, providing criteria and classifications that advance understanding of their algebraic structures and applications.

Contribution

It offers new conditions for gauge equivalence of twisted quantum doubles and classifies when such doubles are genuine, enhancing the theory of quantum groups and coquasi-Hopf algebras.

Findings

Identifies conditions for gauge equivalence to ordinary quantum doubles.

Classifies when twisted doubles of cyclic groups are genuine.

Shows certain Nichols algebras are infinite-dimensional.

Abstract

We study the quantum double of a finite abelian group $G$ twisted by a $3$-cocycle and give a sufficient condition when such a twisted quantum double will be gauge equivalent to a ordinary quantum double of a finite group. Moreover, we will determine when a twisted quantum double of a cyclic group is genuine. As an application, we contribute to the classification of coradically graded finite-dimensional pointed coquasi-Hopf algebras over abelian groups. As a byproduct, we show that the Nichols algebras $\mathcal{B}(M_1\oplus M_2 \oplus M_3)$ are infinite-dimensional where $M_1,M_2,M_3$ are three different simple Yetter-Drinfeld modules of $D_8$.