One of the most studied kinds of finite-dimensional Hopf algebras is the family of pointed ones: it means that the coradical is the algebra of the group-like elements. When the group is abelian, all such examples are known following the so-called Lifting Method by Andruskiewitsch-Schneider and include deformations of small quantum groups, their super analogues and some exceptional examples of Nichols algebras. When the group is not abelian, the classification is not known yet. Even more, the first step of the Lifting Method (the computation of all finite-dimensional Nichols algebras) has not been completed: the classification has been performed by Heckenberger-Vendramin when the elements in degree one form a non-simple Yetter-Drinfeld module, and consist of low rank exceptions and large rank families.
In this talk we will present finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated with a simple Lie algebra together with a Dynkin diagram automorphism.
We will show conversely that every finite-dimensional pointed Hopf algebra over a non-abelian group with a non-simple infinitesimal braiding is of this form for large rank families. The proof follows the steps of the Lifting Method. Indeed we prove that the large rank families are cocycle twists of Nichols algebras constructed by Lentner 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.
The talk is based on joint work with Simon Lentner and Guillermo Sanmarco.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
