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Symplectic Maps from Cluster Algebras

Fordy, Allan P., Hone, Andrew N.W. (2011) Symplectic Maps from Cluster Algebras. Symmetry, Integrability and Geometry: Methods and Applications, 7 (091). pp. 1-12. E-ISSN 1815-0659. (doi:10.3842/SIGMA.2011.091) (KAR id:41487)

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We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map.

Item Type: Article
DOI/Identification number: 10.3842/SIGMA.2011.091
Uncontrolled keywords: integrable maps; Poisson algebra; Laurent property; cluster algebra; algebraic entropy; tropical.
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Andrew Hone
Date Deposited: 20 Jun 2014 23:58 UTC
Last Modified: 16 Nov 2021 10:16 UTC
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