Cluster algebras and discrete integrability

Hone, Andrew N. W.; Lampe, Philipp; Kouloukas, Theodoros E.

doi:10.48550/arxiv.1903.08335

Cited by 5 publications

(9 citation statements)

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“…Although relations between integrable systems and cluster algebras have intensively studied for a dozen years, as far as the authors know, non-integrable systems concerning cluster algebras have not been studied so much. Nevertheless, such non-integrable systems are more universal than integrable ones; the difference equation ( 10) is non-integrable for infinitely many β ∈ N except four cases (β = 1, 2, 3, 4) (see also [17]). Moreover, as we have shown in this paper, non-integrable systems concerning cluster algebras are expected to have fine properties such as computability of the algebraic entropy.…”

Section: Discussionmentioning

confidence: 99%

“…where we use δ t+1 β = (β − 1)δ t β (see (17)). Although we reduce the second order difference equation for δ t β in (18) from the iterates of ϕ β , the reason why the recursion relation of δ t β turns into such three term relation is not so clear.…”

Section: Propositionmentioning

confidence: 99%

“…They examined these families in detail, with many explicit examples given, and showed how they led to discrete integrable systems. Also, in [17], Hone, Lampe and Kouloukas investigated the algebraic entropy of coefficient-free cluster algebras, and classified nonlinear recurrences of exchange relations for certain cluster algebras with skew-symmetric exchange matrices. They reduced piecewise linear recurrence relations for the degrees of the iterates of the seed mutations.…”

Section: Propositionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Nobe,

Matsukidaira

2019

Preprint

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The one-parameter family of second order nonlinear difference equations each of which is given byis explored. Since the equation above is arising from seed mutations of a rank 2 cluster algebra, its solution is periodic only when β ≤ 3. In order to evaluate the dynamics with β ≥ 4, the algebraic entropy of the birational map equivalent to the difference equation is investigated; it vanishes when β = 4 but is positive when β ≥ 5. This fact suggests that the difference equation with β ≤ 4 is integrable but that with β ≥ 5 is not. It is moreover shown that the difference equation with β ≥ 4 fails the singularity confinement test. This fact is consistent with linearizability of the equation with β = 4 and reinforces non-integrability of the equation with β ≥ 5.

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Section: Discussionmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Nobe,

Matsukidaira

2019

Preprint

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“…N has linear degree growth 21 . In two dimension, it is well known that linear degree growth of a map iteration leads to linearization of the map 19 ; however, as far as the authors know, it is not clear whether the map exhibiting linear degree growth is linearizable or not in higher dimensions.…”

Section: Linearizationmentioning

confidence: 99%

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

Nobe,

Matsukidaira

2020

Preprint

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In the network of seed mutations arising from a certain initial seed, an appropriate path emanating from the initial seed is intendedly chosen, noticing periodicity of the exchange matrices in the path each of which is assigned to the generalized Cartan matrix of type A(1) N . Then dynamical property of the seed mutations along the path, which is referred to as of type A(1) N , is intensively investigated. The coefficients assigned to the path form certain N monomials that posses periodicity with period N under the seed mutations and enable to obtain the general terms of the coefficients. The cluster variables assigned to the path of type A(1) N also form certain N Laurent polynomials possessing the same periodicity as the monomials generated by the coefficients. These Laurent polynomials lead to sufficiently number of conserved quantities of the dynamical system derived from the cluster mutations along the path. Furthermore, by virtue of the Laurent polynomials with periodicity, the dynamical system is non-autonomously linearized and its general solution is concretely constructed. Thus the seed mutations along the path of type A(1) N exhibit discrete integrability.

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An algebraic approach to discrete time integrability

Doikou¹,

Findlay²

2021

J. Phys. A: Math. Theor.

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We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrödinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz–Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two-dimensional quantum lattice is thus also examined.

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Cluster algebras and discrete integrability

Cited by 5 publications

References 0 publications

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

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

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

An algebraic approach to discrete time integrability

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