Deformations of cluster mutations and invariant presymplectic forms

Hone, Andrew and Kouloukas, Theodoros (2023) Deformations of cluster mutations and invariant presymplectic forms. Journal of Algebraic Combinatorics . ISSN 0925-9899

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Deformations of cluster mutations and invariant presymplectic forms
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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 A2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A3 and A4, 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 clus- ter 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.

Keywords:Cluster algebras, Presymplectic forms, Laurent property, Integrability
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:52939
Deposited On:17 Jan 2023 14:47

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