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Deformations of cluster mutations and invariant presymplectic forms

Hone, Andrew N.W., Kouloukas, Theodoros E. (2023) Deformations of cluster mutations and invariant presymplectic forms. Journal of Algebraic Combinatorics, 57 (3). pp. 763-791. ISSN 0925-9899. (doi:10.1007/s10801-022-01203-5) (KAR id:89747)

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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.

Item Type: Article
DOI/Identification number: 10.1007/s10801-022-01203-5
Uncontrolled keywords: Cluster algebra, Quiver, Presymplectic form, Laurent property, Integrable map
Subjects: Q Science > QA Mathematics (inc Computing science) > QA165 Combinatorics
Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: Engineering and Physical Sciences Research Council (
Royal Society (
Depositing User: Andrew Hone
Date Deposited: 12 Aug 2021 16:25 UTC
Last Modified: 20 Apr 2023 08:27 UTC
Resource URI: (The current URI for this page, for reference purposes)
Hone, Andrew N.W.:
Kouloukas, Theodoros E.:
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