Braid groups and quiver mutation. (English) Zbl 1375.13032
Coxeter groups split into two distinct classes: those of finite type, corresponding to the Dynkin diagrams of type \(ADE\), and those of infinite type. Each such a group is a quotient of a corresponding Artin braid group and those of Dynkin type have a different character to others.
The authors describe presentations of braid groups of type \(ADE\) and show how these presentations are compatible with mutation of quivers. In types \(A\) and \(B\) these presentations can be understood geometrically. Further, a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras [V. Ginzburg, Calabi-Yau algebras”, arXiv:math/0612139] of quivers with potential, is given as well.
The authors describe presentations of braid groups of type \(ADE\) and show how these presentations are compatible with mutation of quivers. In types \(A\) and \(B\) these presentations can be understood geometrically. Further, a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras [V. Ginzburg, Calabi-Yau algebras”, arXiv:math/0612139] of quivers with potential, is given as well.
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 13F60 | Cluster algebras |
| 20F36 | Braid groups; Artin groups |
| 16E35 | Derived categories and associative algebras |
| 16E45 | Differential graded algebras and applications (associative algebraic aspects) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16G20 | Representations of quivers and partially ordered sets |
References:
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
