A simplicial complex of Nichols algebras

TL;DR

This paper introduces a new simplicial complex framework for Nichols algebras, revealing novel structures and root systems, and explaining their relation to Weyl groupoids and Satake diagrams.

Contribution

It develops a simplicial complex approach to Nichols algebras, uncovering new Weyl groupoids and explaining root systems in broader contexts.

Findings

Constructed a simplicial complex decorated by Nichols algebras.

Identified Weyl groupoids not arising from finite groups.

Realized all root systems of finite Weyl groupoids of rank > 3.

Abstract

We translate the concept of restriction of an arrangement in terms of Hopf algebras. In consequence, every Nichols algebra gives rise to a simplicial complex decorated by Nichols algebras with restricted root systems. As applications, some of these Nichols algebras provide Weyl groupoids which do not arise for Nichols algebras over finite groups and in fact we realize all root systems of finite Weyl groupoids of rank greater than three. Further, our result explains the root systems of the folded Nichols algebras over nonabelian groups and of generalized Satake diagrams.