Discrete Integrable Systems and Poisson Algebras From Cluster Maps
Abstract
We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutationperiodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skewsymmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure. Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the LiouvilleArnold sense.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 January 2014
 DOI:
 10.1007/s002200131867y
 arXiv:
 arXiv:1207.6072
 Bibcode:
 2014CMaPh.325..527F
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 Mathematics  Quantum Algebra;
 17B63;
 37K10;
 37P05
 EPrint:
 49 pages, 3 figures. Reduced to satisfy journal page restrictions. Sections 2.4, 4.5, 6.3, 7 and 8 removed. All other results remain, with minor editing