On Nichols algebras over basic Hopf algebras

Abstract

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field k 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.