Nichols algebras over groups with finite root system of rank two II

TL;DR

This paper classifies certain non-abelian groups based on the finiteness of Nichols algebras generated by simple Yetter-Drinfeld modules, revealing constraints on their dimensions and employing Weyl groupoids as a key tool.

Contribution

It provides a complete classification of non-abelian groups with finite-dimensional Nichols algebras of rank two under specific conditions, introducing new structural insights.

Findings

Dimensions of modules V and W are at most six

Classification of non-abelian groups with finite Nichols algebras

Use of Weyl groupoid as a classification tool

Abstract

We classify all non-abelian groups G such that there exists a pair (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional under two assumptions: the square of the braiding between V and W is not the identity, and G is generated by the support of V and W. As a corollary, we prove that the dimensions of such V and W are at most six. As a tool we use the Weyl groupoid of (V,W).