On Hopf Algebras over quantum subgroups

Abstract

Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic to the smallest non-pointed non-cosemisimple Hopf algebra $\mathcal{K}$ and the corresponding infinitesimal module is an indecomposable object in ${}_{\mathcal{K}}^{\mathcal{K}}\mathcal{YD}$ (we assume that the diagrams are Nichols algebras). As a byproduct, we obtain new Nichols algebras of dimension 8 and new Hopf algebras of dimension 64.

First version: 34 pages, with an Appendix. Second version: 26 pages with minor changes in exposure; we also remove Theorem 5.1. Third version: 26 pages; accepted version to appear in J. Pure Appl. Algebra

Cited by in corpus (7)

• Nichols algebras that are quantum planes
• Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$
• Finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina
• Finite-dimensional Hopf algebras over the smallest non-pointed basic Hopf algebra
• Finite-dimensional Nichols algebras over dual Radford algebras
• On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^{\ast }$ of Kashina
• On Hopf algebras whose coradical is a cocentral abelian cleft extension