Mutations of the cluster algebra of type A₁ ⁽¹⁾ and the periodic discrete Toda lattice

A direct connection between two sequences of points, one of which is generated by seed mutations of the cluster algebra of type A1 ⁽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 of the cluster algebra of type A1 ⁽¹⁾ as an addition of points on an elliptic curve.