On Hopf algebras whose coradical is a cocentral abelian cleft extension

Abstract

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 K_{n}, 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 K_{3}. This includes a family of 12-dimensional Nichols algebras $B_{\xi }$ depending on 3rd roots of unity. Here, $B_1$ is isomorphic to the well-known Fomin-Kirillov algebra, and $B_{\xi }\simeq B_{{\xi }^2}$ as graded algebras but $B_1$ is not isomorphic to $B_{\xi }$ as algebra for $\xi \ne 1$. As a byproduct we obtain new Hopf algebras of dimension 216.

27 pages. In this version we include the description of the finite-dimensional Nichols algebras of the simple modules over K_3, which could be completed thanks to a recent result of Heckenberger, Mehir and Vendramin that appeared after the first version of this manuscript. To appear in Communications in Algebra

References in corpus (2)

• On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^{\ast }$ of Kashina
• Classification of finite-dimensional Hopf algebras over dual Radford algebras