On nondiagonal finite quasi-quantum 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 (kGYD Phi)-Y-kG 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.