Locally acyclic cluster algebras

TL;DR

This paper introduces the concept of locally acyclic cluster algebras, extending properties of acyclic cluster algebras to a broader class, and provides techniques and examples for identifying them.

Contribution

It defines locally acyclic cluster algebras, extends key properties from acyclic cases, and offers methods and examples for their identification.

Findings

Many properties of acyclic cluster algebras extend to local acyclic cluster algebras

Cluster algebras from marked surfaces with ≥2 boundary points are locally acyclic

Examples of locally acyclic but not acyclic cluster algebras are provided

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.