Basic quasi-Hopf algebras of dimension n3

We construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n3, n>2, parametrized by primitive roots of unity q of order n2, with radical of codimension n, which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in [3]. These quasi-Hopf algebras are not twist equivalent to a Hopf algebra, and may be regarded as quasi-Hopf analogs of Taft Hopf algebras. By [4], our construction is equivalent to the construction of new finite tensor categories whose simple objects form a cyclic group of order n, and which are not tensor equivalent to a representation category of a Hopf algebra. We also prove that if H is a finite dimensional radically graded quasi-Hopf algebra with image, where n is prime and Φ is a nontrivial associator, such that H[1] is a free left module over H[0] of rank 1 (it is always free), then H is isomorphic to A(q).