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 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.

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

References in corpus (5)

• The quantum dilogarithm and representations quantum cluster varieties
• Condensation of Determinants
• Laurent Polynomials and Superintegrable Maps
• Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
• Recurrence Relations for Elliptic Sequences: every Somos 4 is a Somos $k$

Cited by in corpus (31)

• Investigation into the role of the Laurent property in integrability
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Some integrable maps and their Hirota bilinear forms
• Hankel determinant solutions to several discrete integrable systems and the Laurent Property
• Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
• On Reductions of the Hirota-Miwa Equation
• Integrability of reductions of the discrete KdV and potential KdV equations
• A step towards cluster superalgebras
• Matrix factorizations and pentagon maps
• A family of linearisable recurrences with the Laurent property
• Discrete Painlevé equations from Y-systems
• Dressing the Dressing Chain
• Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation
• Dynamics of the birational maps arising from $F_0$ and $d{P}_3$ quivers
• From integrable equations to Laurent recurrences
• Algebraic entropy of sign-stable mutation loops
• Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice
• Earthquake theorem for cluster algebras of finite type
• A family of integrable and non-integrable difference equations arising from cluster algebras
• From frieze patterns to cluster categories
• Linear relations for Laurent polynomials and lattice equations
• A two dimensional lattice equation as an extension of the Heideman-Hogan recurrence
• Conserved quantities of Q-systems from dimer integrable systems
• Dynamics and periodicity in a family of cluster maps
• Continued fractions for $q$-deformed real numbers, $\{-1,0,1\}$-Hankel determinants, and Somos-Gale-Robinson sequences
• Period 2 quivers and their T- and Y-systems
• Periodicity, linearizability and integrability in seed mutations of type $A_N^{(1)}$
• Rooted Clusters for Graph LP Algebras
• Difference equations arising from cluster algebras
• Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property