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Discrete integrable systems and Poisson algebras from cluster maps

Fordy, Allan P., Hone, Andrew N.W. (2014) Discrete integrable systems and Poisson algebras from cluster maps. Communications in Mathematical Physics, 325 (2). pp. 527-584. ISSN 0010-3616. (doi:10.1007/s00220-013-1867-y) (KAR id:41486)

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We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric 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 Liouville–Arnold sense.

Item Type: Article
DOI/Identification number: 10.1007/s00220-013-1867-y
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Andrew Hone
Date Deposited: 20 Jun 2014 23:43 UTC
Last Modified: 10 Dec 2022 18:34 UTC
Resource URI: (The current URI for this page, for reference purposes)
Hone, Andrew N.W.:
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