Basic quasi-Hopf algebras of dimension [formula omitted

Abstract

We construct new finite dimensional basic quasi-Hopf algebras A ( q ) of dimension n 3 , n > 2 , parametrized by primitive roots of unity q of order n 2 , 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 H [ 0 ] = ( k [ Z / n Z ] , Φ ) , 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 ) .