Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings

Abstract

We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite-dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch–Schneider. Our starting point is the classification of finite-dimensional Nichols algebras over nonabelian groups by Heckenberger–Vendramin, which consist of low-rank exceptions and large-rank families. We prove that the large-rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie-theoretic descriptions of the large-rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.