Locally acyclic cluster algebras

Abstract

This paper studies cluster algebras locally, by identifying a special class of localizations which are themselves cluster algebras. A `locally acyclic cluster algebra' is a cluster algebra which admits a finite cover (in a geometric sense) by acyclic cluster algebras. Many important results about acyclic cluster algebras extend to local acyclic cluster algebras (such as finite generation, integrally closure, and equaling their upper cluster algebra), as well as results which are new even for acyclic cluster algebras (such as regularity when the exchange matrix has full rank). We develop several techniques for determining whether a cluster algebra is locally acyclic. We show that cluster algebras of marked surfaces with at least two boundary marked points are locally acyclic, providing a large class of examples of cluster algebras which are locally acyclic but not acyclic. We also work out several specific examples in detail.

44 pages, 16 figures

References in corpus (1)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths

Cited by in corpus (30)

• Positive Geometries and Canonical Forms
• Introduction to Cluster Algebras. Chapters 1-3
• A=U for Locally Acyclic Cluster Algebras
• Cluster Configuration Spaces of Finite Type
• Cohomology of cluster varieties. I. Locally acyclic case
• Skein algebras and cluster algebras of marked surfaces
• Cluster structures in Schubert varieties in the Grassmannian
• On cluster algebras from once punctured closed surfaces
• Cluster algebra structures on Poisson nilpotent algebras
• Domain wall formation and magnon localization in twisted chromium trihalides
• Cluster algebraic interpretation of infinite friezes
• Factoriality and class groups of cluster algebras
• Singularities of locally acyclic cluster algebras
• $\mathcal{A}=\mathcal{U}$ for cluster algebras from moduli spaces of $G$-local systems
• A Combinatorial Formula for Certain Elements of Upper Cluster Algebras
• Upper cluster algebras and choice of ground ring
• Reddening Sequences for Banff Quivers and the Class $\mathcal{P}$
• Schemes of modules over gentle algebras and laminations of surfaces
• Generic Caldero-Chapoton functions with coefficients and applications to surface cluster algebras
• Extending Upper Cluster Algebras
• On the Generalized Cluster Algebras of Geometric Type
• On the relationship between green-to-red sequences, local-acyclicity, and upper cluster algebras
• Cluster algebras generated by projective cluster variables
• Computing upper cluster algebras
• Combinatorics of cluster structures in Schubert varieties
• Acyclic cluster algebras from a ring theoretic point of view
• On class groups of upper cluster algebras
• Defects in Spin Chains via Cluster Categories
• Machine Learning Mutation-Acyclicity of Quivers
• Red sizes of quivers