The classification of Nichols algebras over groups with finite root system of rank two

Abstract

We classify all groups $G$ and all pairs $(V,W)$ of absolutely simple Yetter–Drinfeld modules over $G$ such that the support of $V\oplus W$ generates $G$, $c_{W,V}{c}_{V,W}\ne \mathrm{id}$, and the Nichols algebra of the direct sum of $V$ and $W$ admits a finite root system. As a byproduct, we determine the dimensions of such Nichols algebras, and several new families of finite-dimensional Nichols algebras are obtained. Our main tool is the Weyl groupoid of pairs of absolutely simple Yetter–Drinfeld modules over groups.