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Symmetry, Integrability and Geometry: Methods and Applications, 2011, Volume 7, 091, 12 pp.
DOI: https://doi.org/10.3842/SIGMA.2011.091
(Mi sigma649)
 

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
References:
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
Bibliographic databases:
Document Type: Article
Language: English
Citation: Allan P. Fordy, Andrew Hone, “Symplectic Maps from Cluster Algebras”, SIGMA, 7 (2011), 091, 12 pp.
Citation in format AMSBIB
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\paper Symplectic Maps from Cluster Algebras
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\papernumber 091
\totalpages 12
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  • This publication is cited in the following 18 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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