Symplectic Maps from Cluster Algebras

Symmetry Integrability and Geometry-Methods and Applications

article

Allan P. Fordy¹  Andrew Hone² 

[1] School of Mathematics, University of Leeds;School of Mathematics, Statistics and Actuarial Science, University of Kent

关键词: integrable maps;    Poisson algebra;    Laurent property;    cluster algebra;    algebraic entropy;    tropical;   

DOI  :  10.3842/SIGMA.2011.091

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown