Factorial cluster algebras

TL;DR

This paper proves that factorial cluster algebras have no non-trivial units, all variables are irreducible, and provides criteria for when they are polynomial rings, linking factoriality with upper bounds.

Contribution

It establishes that factorial cluster algebras coincide with their upper bounds and offers a criterion for factoriality, advancing understanding of their algebraic structure.

Findings

Cluster algebras have no non-trivial units.

All cluster variables are irreducible.

Factorial cluster algebras coincide with their upper bounds.

Abstract

We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.