Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 091, 12 pages      arXiv:1105.2985      http://dx.doi.org/10.3842/SIGMA.2011.091
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Symplectic Maps from Cluster Algebras

Allan P. Fordy a and Andrew Hone b
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

Received May 16, 2011, in final form September 16, 2011; Published online September 22, 2011

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.

Key words: integrable maps; Poisson algebra; Laurent property; cluster algebra; algebraic entropy; tropical.

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