Acyclic cluster algebras from a ring theoretic point of view

TL;DR

This paper explores the ring theoretic properties of cluster algebras, providing conditions for when they are unique factorization domains and applying these to classify certain types.

Contribution

It introduces a sufficient condition for cluster algebras to be UFDs based on primary decompositions, advancing the understanding of their algebraic structure.

Findings

All cluster variables are irreducible elements.

Necessary conditions for UFD include irreducibility and coprimality.

A criterion to determine UFD property in Dynkin types A, D, E.

Abstract

The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster algebra to be a unique factorization domain, namely the irreducibility and the coprimality of the initial exchange polynomials. We present a sufficient condition for a cluster algebra to be a unique factorization domain in terms of primary decompositions of certain ideals generated by initial cluster variables and initial exchange polynomials. As an application, the criterion enables us to decide which coefficient-free cluster algebras of Dynkin type A, D or E are unique factorization domains. Moreover, it yields a normal form for irreducible elements in cluster algebras that satisfy the condition. Proof techniques include methods from commutative algebra. In addition, we state a conjecture about the range of application of the criterion.