Weyl groupoids with at most three objects

TL;DR

This paper classifies all finite Weyl groupoids with up to three objects, revealing mostly standard cases with a few exceptional ones, by adapting root system concepts to category theory.

Contribution

It introduces Cartan schemes and classifies finite Weyl groupoids with at most three objects, extending the theory to a categorical framework.

Findings

Most Weyl groupoids are standard

Only 9 exceptional cases identified

Complete classification achieved

Abstract

We adapt the generalization of root systems of the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields that there exist infinitely many standard, but only 9 exceptional cases. Key words: Nichols algebra, reflection, root system, Weyl groupoid