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Algebra and Geometry Seminar: Pointed Hopf algebras over non-abelian groups

Author: Lona | Image: Lona

Guillermo Sanmarco (ISU)

The quantum group associated with a complex Lie algebra is a Hopf algebra for which the coradical (i.e., the biggest cosemisimple part) is an abelian group algebra. From that point of view, the quantum group is a pointed Hopf algebra over an abelian group. In the early 2000s, Andruskiewitsch-Schneider designed a strategy to obtain a classification of pointed Hopf algebras by studying two invariants: the coradical and certain braided subalgebra. This strategy led to a complete classification of finite-dimensional pointed Hopf algebras with abelian coradical.

We will begin this talk with a brief review of the main ingredients that led to the classification in the abelian group case. The braided objects aforementioned are generalizations of the positive parts of small quantum groups, known as Nichols algebras. As happens for quantum groups, we will see that these Nichols algebras are equipped with several combinatorial structures, which are fundamental for the classification.

Then we will focus on the classification problem in the non-abelian group realm, where the main obstruction is the lack of a complete understanding of the Nichols algebras. However, a big family of finite-dimensional Nichols algebras over non-abelian groups were recently classified by Heckenberger-Vendramin, again using combinatorial structures reminiscent of those available for quantum groups. We will report on joint work with Angiono and Lentner, where we classified all Hopf algebras over these Nichols algebras. We will explain how we could translate several results known for the abelian case to our setting using the folding construction for Nichols algebras due to Lentner, which relates to equivariantization/de-equivariantization for Hopf algebras.