Symplectic Maps from Cluster Algebras

Details Symplectic Maps from Cluster Algebras Symplectic Maps from Cluster Algebras

2011-05-16

arxiv.org/abs/1105.2985v2

SIGMA 7 (2011), 091, 12 pages

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

10.3842/SIGMA.2011.091

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.

Affiliated with

Also associated with

No associations

Rating

Symplectic Maps from Cluster Algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Symplectic Maps from Cluster Algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Symplectic Maps from Cluster Algebras will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-76711