Pure Maths Colloquium: Braid groups and quiver mutation
16 May 2016 16:00 in CM221
Joint work with Joseph Grant.
The braid group is a classical object in mathematics: the elements are the ways of twisting a fixed number of strands, up to isotopy, with the binary operation given by concatenation. Although it is defined topologically, the braid group has a beautiful presentation as an abstract group, given by Artin. This presentation can be associated to a Dynkin diagram of type A. In this way, a generalised braid group, or Artin braid group, can be associated to every Dynkin diagram.
A quiver is a directed graph, and an orientation of a Dynkin diagram is referred to as a Dynkin quiver. As part of the definition of a cluster algebra, Fomin and Zelevinsky introduced the notion of quiver mutation, where a quiver is changed only locally. A quiver which is mutation-equivalent to a Dynkin quiver is said to be mutation-Dynkin. Our main result is a presentation of an Artin braid group for any mutation-Dynkin quiver. We show how these presentations can be understood topologically in types A and D using a disk and a disk with a cone point of order two (i.e. an orbifold) respectively.
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