Braid groups and quiver mutation

Joseph Grant and Bethany Rose Marsh

Vol. 290 (2017), No. 1, 77–116

DOI: 10.2140/pjm.2017.290.77

Abstract

We describe presentations of braid groups of type $ADE$ and show how these presentations are compatible with mutation of quivers. In types $A$ and $D$ these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.

Keywords

mutation, braid groups, cluster algebras, Ginzburg dg algebra, spherical twist

Mathematical Subject Classification 2010

Primary: 13F60, 16G20, 20F36

Secondary: 16E35, 16E45, 18E30

Milestones

Received: 28 March 2016

Revised: 13 January 2017

Accepted: 22 February 2017

Published: 7 July 2017