Integrable clusters

TL;DR

This paper explores quantum cluster algebras where cluster variables commute, demonstrating that this property remains invariant under mutations and linking it to the proven sign coherence conjecture.

Contribution

It establishes the preservation of commutation among cluster variables under mutations and connects this property to the sign coherence conjecture.

Findings

Commutation among cluster variables is preserved by mutations.

The property is equivalent to the sign coherence conjecture.

Provides insights into quantum cluster algebra structures.

Abstract

The goal of this note is to study quantum clusters in which cluster variables (not coefficients) commute which each other. It turns out that this property is preserved by mutations. Remarkably, this is equivalent to the celebrated sign coherence conjecture recently proved by M. Gross, P. Hacking, S. Keel and M. Kontsevich