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.Comment: 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. Algebr