Mutations of the cluster algebra of type ${A}_{1}^{(1)}$ 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.

Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1)^{(1)}$ arising from cluster algebra

“…coefficients y i ). Thanks to those two properties, some mutation-periodic quivers become sources of discrete integrable systems ( [7,8,9,11,18,21]) or q-Painlevé equations (4n + 3)…”

Periodicity, linearizability and integrability in seed mutations of type $A^{(1)}_N$

“…Sufficiently many conserved quantities of each dynamical system naturally follow from the monomials and the Laurent polynomials, both of which have the same periodicity, respectively. Moreover, the dynamical system of cluster variables is non-autonomously linearized by virtue of the periodic Laurent polynomials similar to the rank 2 cases investigated in the previous papers 12,17 . Due to the linearizability, the general solution to the dynamical system is concretely constructed, and it gives the general terms of the cluster variables.…”

“…Unfortunately, almost all of the infinitely many dynamical systems thus related with cluster algebras do not have integrable structures; nevertheless, we can find abundant integrable systems among them. In fact, since the introduction of cluster algebras by Fomin and Zelevinsky in 2002 1 we have found plenty of cluster algebras related with discrete/quantum integrable systems such as discrete soliton equations, integrable maps on algebraic curves, discrete/q-Painlevé equations and Y -systems [3][4][5][6][7][8][9][10][11][12][13] . Thus we see that appropriate paths in the network of seeds in adequate cluster algebras are strongly related with integrable systems, and hence it is expected that we find unknown integrable systems among cluster algebras.…”

“…Especially, a rank 2 cluster algebra itself is assigned to a 2 × 2 GCM since it has the unique non-trivial path in the network of seeds, and the cluster algebra is referred to as of the GCM type. In the preceding papers 12,16,17 , the authors investigated a certain family of rank 2 cluster algebras from the viewpoint of discrete integrability. The family consists of infinitely many cluster algebras some of which have integrable structures and are respectively assigned to the GCMs of finite and affine types.…”

A family of integrable and non-integrable difference equations arising from cluster algebras

“…It has been open for a decade but finally established by Lee and Schiffler in 2013 [22] for skew-symmetric cases and by Gross, Hacking, Keel and Kontsevich in 2014 [15] for cluster algebras of geometric type. Positivity of cluster algebras strongly promotes application of cluster algebras to combinatorial representation theory, tropical geometry and discrete integrable systems [9,30,13,27,12,15,25,24,17].…”