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

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