On nondiagonal finite quasi-qantum groups over finite abelian groups

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 ${}_{\text{k}G}^{\text{k}G}{\mathcal{Y}\mathcal{D}}^{\mathrm{\Phi }}$ with $\mathrm{\Phi }$ a nonabelian $3$-cocycle on a finite abelian group $G\mathrm{.}$ 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.