Positivity and tameness in rank 2 cluster algebras

TL;DR

This paper explores the connection between positivity and tameness in rank 2 cluster algebras, revealing that a basis of positive elements exists only in finite or affine types, contradicting previous conjectures.

Contribution

It establishes a precise criterion linking positivity bases to the algebra's type, challenging existing conjectures in the field.

Findings

Positivity basis exists only in finite or affine types

Contradicts Fock and Goncharov's conjecture

Provides new classification criteria for rank 2 cluster algebras

Abstract

We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if and only if it is of finite or affine type. This statement disagrees with a conjecture by Fock and Goncharov.