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This article is cited in 18 scientific papers (total in 18 papers)
Symplectic Maps from Cluster Algebras
Allan P. Fordya, Andrew Honeb a School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK
Abstract:
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.
Keywords:
integrable maps; Poisson algebra; Laurent property; cluster
algebra; algebraic entropy; tropical.
Received: May 16, 2011; in final form September 16, 2011; Published online September 22, 2011
Citation:
Allan P. Fordy, Andrew Hone, “Symplectic Maps from Cluster Algebras”, SIGMA, 7 (2011), 091, 12 pp.
Linking options:
https://www.mathnet.ru/eng/sigma649 https://www.mathnet.ru/eng/sigma/v7/p91
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