A classification of Nichols algebras of semi-simple Yetter-Drinfeld modules over non-abelian groups

TL;DR

This paper classifies all finite-dimensional Nichols algebras arising from certain simple Yetter-Drinfeld modules over non-abelian groups, using Dynkin diagram analogs and Weyl groupoids, extending understanding in Hopf algebra theory.

Contribution

It provides a complete classification of braid-indecomposable tuples of simple Yetter-Drinfeld modules over non-abelian groups with finite-dimensional Nichols algebras, introducing new diagrammatic tools.

Findings

Classified all relevant Nichols algebras over non-abelian groups.

Developed analogs of Dynkin diagrams for these classifications.

Computed the dimensions of the classified Nichols algebras.

Abstract

Over fields of arbitrary characteristic we classify all braid-indecomposable tuples of at least two absolutely simple Yetter-Drinfeld modules over non-abelian groups such that the group is generated by the support of the tuple and the Nichols algebra of the tuple is finite-dimensional. Such tuples are classified in terms of analogs of Dynkin diagrams which encode much information about the Yetter-Drinfeld modules. We also compute the dimensions of these finite-dimensional Nichols algebras. Our proof uses the Weyl groupoid of a tuple of simple Yetter-Drinfeld modules.