# Finite Weyl groupoids of rank three

## TL;DR

This paper presents an algorithm to classify finite Weyl groupoids of rank three, identifying all such structures and their roots, and linking them to Nichols algebras of diagonal type.

## Contribution

It introduces a finite classification algorithm for rank three Cartan schemes with finite root systems and connects these to Nichols algebra classifications.

## Key findings

- Exactly 55 such Cartan schemes exist up to equivalence.
- The number of real roots ranges from 6 to 37.
- Identifies Weyl groupoids relevant to Nichols algebra classification.

## Abstract

We continue our study of Cartan schemes and their Weyl groupoids. The results in this paper provide an algorithm to determine connected simply connected Cartan schemes of rank three, where the real roots form a finite irreducible root system. The algorithm terminates: Up to equivalence there are exactly 55 such Cartan schemes, and the number of corresponding real roots varies between 6 and 37. We identify those Weyl groupoids which appear in the classification of Nichols algebras of diagonal type.