Weyl groupoids of rank two and continued fractions

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.

Partitions of unity in $$\mathrm {SL}(2,\mathbb Z)$$ SL ( 2 , Z ) , negative continued fractions, and dissections of polygons

“…They were used by Conway and Coxeter [7]; see also [4,11] and many other sources. In particular, the totally positive solutions of Problem II are precisely the solutions obtained by a sequence of operations (1.2); see "Appendix" section.…”

“…A direct proof of the Conway-Coxeter theorem in terms of 2 × 2-matrices is given in [4,11]. For a simple direct proof, see also [16].…”

“…Example 7.7 Gluing a triangle to the above triangulated pentagon An operation (1.2) applied to a quiddity consists in gluing a triangle to a triangulated ngon, so that the Conway-Coxeter theorem implies the following statement (see also [11], Theorem 5.5). solution (a 1 , a 2 , a 3 ) = (1, 1, 1) by a sequence of operations (1.2).…”

Geometric combinatorics of Weyl groupoids

“…We mainly follow the notation in [8,9]. The foundations of the general theory have been developed in [13].…”

“…The theory was further extended by a series of papers of M. Cuntz and the first author, and a satisfactory classification result of finite Weyl groupoids of rank two and three was achieved [7,8]. Interestingly, not all finite Weyl groupoids obtained via the classification are related to known Nichols algebras.…”