Discrete Integrable Systems and Poisson Algebras From Cluster Maps

Abstract

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure.Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.

Integrability of reductions of the discrete Korteweg–de Vries and potential Korteweg–de Vries equations

“…For the purposes of this paper, the main advantage of considering the cluster algebra is that it provides a natural presymplectic structure for the tau-functions. A presymplectic form that is compatible with cluster mutations was presented in Gekhtman et al [20], and in Fordy & Hone [22], it was explained how this presymplectic structure is preserved by the recurrences considered in Fordy & Marsh [33].…”

“…, τ d+1 , τ 1 ) associated with the exchange relation (3.13) for index k = 1, where the exponents are given by the entries b 1,j in the first row of B. Moreover, given a recurrence relation of this type, the conditions (3.14) and (3.15) allow the rest of the matrix B to be constructed from the exponents corresponding to the first row, and these conditions are also necessary and sufficient for a logcanonical presymplectic form ω, as in (3.16) below, to be preserved (see lemma 2.3 in Fordy & Hone [22]). In general, this two-form is closed, but it may be degenerate.…”

“…By lemma 2.7 in Fordy & Hone [22] (see also §6 therein), symplectic coordinates are obtained by taking a complete set of invariants for these scaling symmetries. With a suitable choice of coordinates, denoted below by y n , the associated symplectic maps can be written in the form of recurrences, like so: Owing to the presence of the periodic coefficients, the latter maps are not of standard Quispel-Roberts-Thompson (QRT) type [3], but they reduce to symmetric QRT maps when α n = constant.…”

Some integrable maps and their Hirota bilinear forms

“…The other results in section 3 are based on the connection between bilinear equations and cluster algebras, as explained in [8], which leads to a Poisson structure for the lifted DTKQ equation (19). For N even, both bilinear equations (5) and (6) reveal the connection with reductions of Hirota's discrete KdV equation; while for N odd, the second bilinear equation (7) leads to a link with reductions of a discrete time Toda equation, as well as an associated Bäcklund transformation (or BT, in the sense of [25]).…”

“…Sequences generated by certain bilinear recurrences of Somos type also admit further bilinear relations of higher order (see [33], for example), with the coefficients being first integrals, and this has been used to obtain first integrals for four-term Somos-6 and Somos-7 recurrences, in [19] and [8], respectively. Another approach to finding first integrals, based on reduction of conservation laws for the discrete KP or BKP equations, was used in [28].…”