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 $\mathbb{k}$ 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.

v4: 45 pages, minor changes. To appear in Math. Z. v3: 45 pages. Substantial improvement of the previous one; we show that a finite-dimensional Hopf algebra with basic Hopf coradical is a lifting of a Nichols algebra of a semisimple YD module. This follows from the previous version plus two new results