Co-primeness preserving higher dimensional extension of q-discrete Painleve I, II equations

TL;DR

This paper constructs higher-dimensional extensions of q-discrete Painleve I and II equations using cluster algebras, demonstrating their integrability through co-primeness properties for specific cases.

Contribution

It introduces a novel method to generate higher-order q-discrete Painleve equations via periodic cluster algebras, establishing their integrability.

Findings

q-discrete Painleve I and II equations derived from cluster algebras

Higher order analogues satisfy co-primeness integrability criterion

Explicit connection between exchange matrices and Painleve equations

Abstract

We construct the q-discrete Painleve I and II equations and their higher order analogues by virtue of periodic cluster algebras. Using particular (k,k) exchange matrices, we show that the cluster algebras corresponding to k=4 and 5 give the q-discrete Painleve I and II equations respectively. For k=6,7,..., we have the higher order discrete equations that satisfy an integrable criterion, the co-primeness property.