On Hopf algebras whose coradical is a cocentral abelian cleft extension

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras Bξ depending on 3rd roots of unity. Here, B1 is isomorphic to the well-known Fomin-Kirillov algebra, and Bξ≃Bξ2 as graded algebras but B1 is not isomorphic to Bξ as algebra for ξ≠1. As a byproduct we obtain new Hopf algebras of dimension 216.