Deformed cluster maps of type $A_{2N}$

TL;DR

This paper constructs and analyzes deformations of integrable cluster maps of type A_{2N}, extending previous work and providing the first infinite class of such maps with demonstrated integrality for N ≤ 3.

Contribution

It introduces a new method for deforming and studying integrable cluster maps of type A_{2N} using a local expansion on quivers, extending the class of known integrable systems.

Findings

Constructed deformations of cluster maps for type A_{2N}

Demonstrated integrality of these deformations for N ≤ 3

Provided a new approach using local expansion on quivers

Abstract

We extend recent work of the third author and Kouloukas by constructing deformations of integrable cluster maps corresponding to the Dynkin types $A_{2N}$, lifting these to higher-dimensional maps possessing the Laurent property and demonstrating integrality of the deformations for $N\leq 3$. This provides the first infinite class of examples (in arbitrarily high rank) of such maps and gives information on the associated discrete integrable systems. Key to our approach is a ``local expansion'' operation on quivers which allows us to construct and study mutations in type $A_{2N}$ from those in type $A_{2(N-1)}$.