Cluster algebras of infinite rank as colimits

TL;DR

This paper demonstrates that infinite rank cluster algebras can be constructed as colimits of finite rank ones, proving positivity for infinite cases and analyzing morphisms between them.

Contribution

It formalizes the structure of infinite rank cluster algebras as colimits and explores the properties of rooted cluster morphisms, including a partial classification.

Findings

Infinite rank cluster algebras are colimits of finite rank algebras.

Positivity conjecture holds for infinite rank cluster algebras.

Rooted cluster morphisms without specializations are ideal, but not all are.

Abstract

We formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of the posivity conjecture for skew-symmetric cluster algebras (of finite rank) by Lee and Schiffler, it follows as a direct consequence that the positivity conjecture holds for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a ring homomorphism between cluster algebras to give rise to a rooted cluster morphism without specializations. Assem, Dupont and Schiffler proposed the problem of a classification of ideal rooted cluster morphisms. We provide a partial solution by showing that every rooted cluster morphism without specializations is ideal, but in general rooted cluster morphisms are not ideal.