Quivers with potentials and their representations I: Mutations

TL;DR

This paper explores quivers with potentials and their mutations, providing a representation-theoretic interpretation that generalizes classical reflection functors, with applications in physics, algebra, and cluster theory.

Contribution

It introduces a framework for quiver mutations with potentials, extending classical concepts and connecting to various mathematical and physical theories.

Findings

Representation-theoretic interpretation of quiver mutations

Generalization of Bernstein-Gelfand-Ponomarev reflection functors

Connections to superpotentials, Calabi-Yau algebras, and cluster algebras

Abstract

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.