On Hopf Algebras over quantum subgroups

TL;DR

This paper classifies certain finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, focusing on those with a specific coradical structure related to the smallest non-pointed non-cosemisimple Hopf algebra, and constructs new Nichols algebras and Hopf algebras.

Contribution

It provides a complete classification of Hopf algebras with a particular coradical and infinitesimal module structure, introducing new Nichols algebras and Hopf algebras of specific dimensions.

Findings

Classified all finite-dimensional Hopf algebras over the specified Hopf subalgebra.

Constructed new Nichols algebras of dimension 8.

Built new Hopf algebras of dimension 64.

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.