On Gauge Equivalence of Twisted Quantum Doubles

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 B(M₁⊕ M₂ ⊕ M₃) are infinite-dimensional where M₁,M₂,M₃ are three different simple Yetter-Drinfeld modules of D₈.