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

References in corpus (6)

• Calabi-Yau algebras
• Seiberg Duality for Quiver Gauge Theories
• Auslander algebras and initial seeds for cluster algebras
• Tilting theory and cluster combinatorics
• Cluster mutation via quiver representations
• Laurent expansions in cluster algebras via quiver representations

Cited by in corpus (12)

• Brane Tilings and Reflexive Polygons
• BPS Quivers and Spectra of Complete N=2 Quantum Field Theories
• On Arnold's 14 `exceptional' N=2 superconformal gauge theories
• Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions
• A Comprehensive Survey of Brane Tilings
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• On cluster algebras arising from unpunctured surfaces II
• Mutation of cluster-tilting objects and potentials
• Mutations for quivers with potentials: Oberwolfach talk, April 2007
• $Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT
• On the combinatorics of rigid objects in 2-Calabi-Yau categories
• A formula for $F$-Polynomials in terms of $C$-Vectors and Stabilization of $F$-Polynomials