Mutations of the cluster algebra of type ${\mathbf{A}}_1^{(1)}$ and the periodic discrete Toda lattice

Abstract

A direct connection between two sequences of points, one of which is generated by seed mutations in the cluster algebra of type $A_1^{(1)}$ and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations in the cluster algebra of type $A_1^{(1)}$ as addition of points on an elliptic curve arising as the spectral curve of the Toda lattice.

19 pages, 4 figures

References in corpus (2)

• Investigation into the role of the Laurent property in integrability
• Solution of the genaralized periodic discrete Toda equation

Cited by in corpus (2)

• Generators of rank 2 cluster algebras of affine types via linearization of seed mutations
• Periodicity, linearizability and integrability in seed mutations of type $A_N^{(1)}$