Root systems and Weyl groupoids for Nichols algebras

TL;DR

This paper introduces a new combinatorial framework of root systems and Weyl groupoids for Nichols algebras, enhancing understanding of their structure and providing new results for Nichols algebras over certain groups.

Contribution

It defines a root system and Weyl groupoid for semisimple Yetter-Drinfeld modules, linking to generalized root systems and offering novel insights into Nichols algebras.

Findings

New root system and Weyl groupoid structures for Nichols algebras.

Application to Nichols algebras over finite non-abelian simple groups.

Enhanced understanding of Nichols algebra classification.

Abstract

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing framework of generalized root systems associated to a family of Cartan matrices, and provides novel insight into Nichols algebras. We demonstrate the power of our construction with new results on Nichols algebras over finite non-abelian simple groups and symmetric groups. Key words: Hopf algebra, quantum group, root system, Weyl group