Quivers with potentials and their representations II: Applications to cluster algebras

Abstract

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit.

44 pages; version 2: revised according to the referee's suggestions; version 3: final version, typos corrected, references corrected and updated.

References in corpus (2)

• Quiver varieties and cluster algebras
• On cluster algebras with coefficients and 2-Calabi-Yau categories

Cited by in corpus (11)

• Quiver Grassmannians associated with string modules
• Mutations of group species with potentials and their representations. Applications to cluster algebras
• Quivers with potentials associated to triangulated surfaces, Part II: Arc representations
• Derived equivalences from mutations of quivers with potential
• Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups
• Donaldson-Thomas Transformation of Grassmannian
• Quantum F-polynomials in Classical Types
• Euler characteristic of quiver Grassmannians and Ringel-Hall algebras of string algebras
• F-polynomials in Quantum Cluster Algebras
• Maximal green sequences of skew-symmetrizable 3x3 matrices
• Cluster algebras and symmetric matrices