Finite Weyl groupoids of rank three. (English) Zbl 1246.20037
The symmetry of a classical root system associated with a Cartan matrix is described by the corresponding Weyl group. There are more general root systems which are associated to a family of Cartan matrices (the so-called ‘Cartan schemes’). The corresponding symmetry is not a group but a groupoid, called the Weyl groupoid of the root system.
In the paper under review the authors obtain a complete list of all simply connected Cartan schemes of rank three for which real roots form a finite irreducible root system. To prove this the authors provide an algorithm which determines all root systems and eventually terminates. It turns out that, up to equivalence, there are \(55\) such Cartan schemes. As an application, Weyl groupoids which appear in the classification of Nichols algebras of diagonal type are determined.
In the paper under review the authors obtain a complete list of all simply connected Cartan schemes of rank three for which real roots form a finite irreducible root system. To prove this the authors provide an algorithm which determines all root systems and eventually terminates. It turns out that, up to equivalence, there are \(55\) such Cartan schemes. As an application, Weyl groupoids which appear in the classification of Nichols algebras of diagonal type are determined.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 17B22 | Root systems |
| 16T30 | Connections of Hopf algebras with combinatorics |
| 52C30 | Planar arrangements of lines and pseudolines (aspects of discrete geometry) |
Keywords:
Weyl groupoids; Cartan schemes of rank three; finite irreducible root systems; Nichols algebras; algorithmsReferences:
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