Poisson structures for difference equations

Evripidou, Charalampos A.; Quispel, G.; Roberts, John A. G.

doi:10.1088/1751-8121/aae746

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…3. The slightly different (but closely related) problem of when an ordinary difference equation preserves a log-canonical Poisson bracket was considered in [7].…”

Section: And Defining Dsmentioning

confidence: 99%

Deformations of cluster mutations and invariant presymplectic forms

Hone

Kouloukas

2022

J Algebr Comb

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We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type $$A_2$$ A 2 : this deforms to the Lyness family of integrable symplectic maps in the plane. For types $$A_3$$ A 3 and $$A_4$$ A 4 , we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

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“…3. The slightly different (but closely related) problem of when an ordinary difference equation preserves a log-canonical Poisson bracket was considered in [7].…”

Section: And Defining Dsmentioning

confidence: 99%

Deformations of cluster mutations and invariant presymplectic forms

Hone

Kouloukas

2022

J Algebr Comb

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show abstract

“…We shall consider an example of this with a generalized mutation in section 3. The slightly different (but closely related) problem of when an ordinary difference equation preserves a log-canonical Poisson bracket was considered in [7].…”

Section: Lyness Maps and Zamolodchikov Periodicitymentioning

confidence: 99%

Deformations of cluster mutations and invariant presymplectic forms

Hone¹,

Kouloukas²

2021

Preprint

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We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A 2 : this deforms to the Lyness family of integrable symplectic maps in the plane. For types A 3 and A 4 we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation.

show abstract

Classification of variational multiplicative fourth-order difference equations

Gubbiotti

2022

Journal of Difference Equations and Applications

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Poisson structures for difference equations

Cited by 3 publications

References 27 publications

Deformations of cluster mutations and invariant presymplectic forms

Deformations of cluster mutations and invariant presymplectic forms

Deformations of cluster mutations and invariant presymplectic forms

Classification of variational multiplicative fourth-order difference equations

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