Quivers with potentials and their representations I: Mutations

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.

58 pages; v.2: a typo fixed, references rearranged in the correct alphabetical order, an acknowledgment added; v.3: more typos fixed, minor editorial improvements; v.4: minor editorial changes, final version to appear in Selecta Math