Pointed Hopf algebras over non-abelian groups with non-simple standard braidings

TL;DR

This paper classifies and constructs finite-dimensional pointed Hopf algebras over non-abelian groups with complex braiding structures, linking them to Lie theory and automorphisms, and describes their categorical extensions.

Contribution

It provides a complete classification of such Hopf algebras, linking large rank families to Nichols algebra foldings and automorphisms, and constructs explicit liftings and deformations.

Findings

Large rank families are cocycle twists of Nichols algebras from Cartan type foldings.

Every lifting is a cocycle deformation of a graded Hopf algebra.

Constructs graded extensions of quantum group representations by diagram automorphisms.

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 non-abelian group with non-simple 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 non-abelian 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 one 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.