On Nichols algebras over basic Hopf algebras

TL;DR

This paper advances the classification of finite-dimensional Hopf algebras with basic Hopf coradicals by linking them to Nichols algebras of semisimple Yetter-Drinfeld modules, introducing new examples and classification methods.

Contribution

It demonstrates that such Hopf algebras are liftings of Nichols algebras and provides a framework for classifying these Nichols algebras, including new examples.

Findings

Finite-dimensional Hopf algebras with basic coradicals are liftings of Nichols algebras.

New examples of Nichols algebras with finite Gelfand-Kirillov dimension.

A classification approach for Nichols algebras over basic Hopf algebras.

Abstract

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field $\Bbbk$ of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter-Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand-Kirillov dimension.