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 K₃. This includes a family of 12-dimensional Nichols algebras $\left\{{\mathcal{B}}_{\xi }\right\}$ depending on 3rd roots of unity. Here, ${\mathcal{B}}_1$ is isomorphic to the well-known Fomin-Kirillov algebra, and ${\mathcal{B}}_{\xi }\simeq {\mathcal{B}}_{{\xi }^2}$ as graded algebras but ${\mathcal{B}}_1$ is not isomorphic to ${\mathcal{B}}_{\xi }$ as algebra for $\xi \ne 1$. As a byproduct we obtain new Hopf algebras of dimension 216.

Keywords:

• Abelian extensions
• Hopf algebras
• fusion rules
• Nichols algebras
• Yetter-Drinfeld modules

2020 Mathematics Subject Classification:

• 16T05

Additional information

Funding

This work was supported, in part, by CONICET, ANPCyT (PICT 2018-00858), Secyt-UNLP (Argentina), and NSERC Discovery Grant 371994-2019 (Canada).