Reflection groupoids of rank two and Cluster algebras of type $A$

Abstract

We extend the classification of finite Weyl groupoids of rank two. Then we generalize these Weyl groupoids to `reflection groupoids' by admitting non-integral entries of the Cartan matrices. This leads to the unexpected observation that the spectrum of the cluster algebra of type $A_{n-3}$ completely describes the set of finite reflection groupoids of rank two with $2n$ objects.

18 pages

References in corpus (1)

• The Weyl-Brandt groupoid of a Nichols algebra of diagonal type

Cited by in corpus (1)

• Finite Weyl groupoids of rank three