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SIGMA 7 (2011), 091, 12 pages arXiv:1105.2985
https://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.
pdf (336 kb)
tex (17 kb)
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