Braid groups and quiver mutation

TL;DR

This paper explores the relationship between braid groups of type ADE and quiver mutations, providing geometric and categorical interpretations that deepen understanding of their algebraic structures.

Contribution

It introduces new presentations of braid groups compatible with quiver mutations and offers geometric and categorical insights into their structure.

Findings

Presentations of braid groups of type ADE compatible with quiver mutation.

Geometric interpretation for types A and D via triangulated surfaces.

Categorical interpretation using spherical twists in derived categories.

Abstract

We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. 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.