Finite Weyl groupoids of rank three

Abstract

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.

Finite Weyl groupoids

“…A very general definition of the Weyl groupoid of a Nichols algebra and the necessary structural results have been obtained by Andruskiewitsch, Schneider, and the second author in [1]. An axiomatic approach to Weyl groupoids via Cartan schemes and root systems was developed in a series of papers by Yamane and the authors [7,8,10,16]. For historical and practical reasons, the emphasis was put on connected simply connected Cartan schemes such that the real roots form a finite irreducible root system.…”

“…The connection to simplicial arrangements was established successively in [6,10,15]. If the real roots of a connected Cartan scheme form a finite irreducible root system, then they define a simplicial arrangement similarly as in the construction in [17,Section 1.15] for Coxeter groups.…”

Geometric combinatorics of Weyl groupoids

“…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.…”

“…A priori the geometric structure of this arrangement of hyperplanes is not clear. It was observed in [7] that for Weyl groupoids of rank three this arrangement is simplicial and therefore can be seen as an arrangement of lines in the projective plane cutting out triangles. Interestingly, the classification of such arrangements is not yet completed [11].…”

“…Interestingly, the classification of such arrangements is not yet completed [11]. It was noted in [7] that most known exceptional arrangements, in particular the largest one, can be explained via Weyl groupoids. In Sect.…”