Bilinear equations and $q$-discrete Painlev\'e equations satisfied by variables and coefficients in cluster algebras

TL;DR

This paper constructs specific cluster algebras whose variables and coefficients satisfy integrable bilinear equations, including discrete mKdV, Toda, and q-discrete Painlevé equations, revealing deep connections between cluster algebra structures and integrable systems.

Contribution

It introduces new cluster algebra frameworks that encode integrable bilinear difference equations and explores their reductions to q-discrete Painlevé equations.

Findings

Cluster variables satisfy discrete integrable equations.

Reduction of quivers corresponds to difference equation reductions.

Framework links cluster algebras with integrable systems.

Abstract

We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to q-discrete Painlev\'e equations. These cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation-period property. We will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation.