Linear relations and integrability for cluster algebras from affine quivers

TL;DR

This paper studies the linear relations and integrability properties of cluster variables from affine D and E type quivers, revealing their linear recurrences and Liouville integrability as discrete dynamical systems.

Contribution

It demonstrates that cluster variables satisfy linear recurrences with periodic coefficients and proves Liouville integrability of the associated symplectic map.

Findings

Cluster variables satisfy linear recurrences with periodic coefficients.

The frieze sequence reduces to a symplectic map.

Liouville integrability of the system is established.

Abstract

We consider frieze sequences corresponding to sequences of cluster mutations for affine D and E type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.