Weyl groupoids of rank two and continued fractions

TL;DR

This paper explores the connection between continued fractions and Weyl groupoids of rank two, providing criteria to determine finiteness of root systems and establishing bounds for Cartan matrix entries.

Contribution

It introduces a novel relationship between continued fractions and Weyl groupoids of rank two, enabling easier classification of finite root systems and deriving bounds for Cartan matrices.

Findings

Criteria for finiteness of root systems in rank two Weyl groupoids

Obstructions and bounds for Cartan matrix entries

Explicit connection between continued fractions and Weyl groupoid structure

Abstract

A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices. Key words: Cartan matrix, continued fraction, Nichols algebra, Weyl groupoid