Liftings of graded quasi-Hopf algebras with radical of prime codimension

Abstract

Let p be a prime, and let RG ( p ) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C , whose radical has codimension p . The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p ; that is, A with gr ( A ) in RG ( p ) , where gr ( A ) is the associated graded algebra taken with respect to the radical filtration on A . The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p . Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H ( 2 ) , H ± ( p ) , A ( q ) , or H ( 32 ) , constructed in [5,8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.