#### Vol. 290, No. 1, 2017

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Braid groups and quiver mutation

### Joseph Grant and Robert J. Marsh

Vol. 290 (2017), No. 1, 77–116
##### Abstract

We describe presentations of braid groups of type $\mathit{ADE}$ 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