Fordy, Allan P., Hone, Andrew N.W. (2011) Symplectic Maps from Cluster Algebras. Symmetry, Integrability and Geometry: Methods and Applications, 7 (091). pp. 112. EISSN 18150659. (doi:10.3842/SIGMA.2011.091) (KAR id:41487)
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Official URL: http://dx.doi.org/10.3842/SIGMA.2011.091 
Abstract
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skewsymmetric 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 nonintegrable 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 
Resource URI:  https://kar.kent.ac.uk/id/eprint/41487 (The current URI for this page, for reference purposes) 
Hone, Andrew N.W.:  https://orcid.org/0000000197807369 
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