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We consider a class of map, recently derived in the context of clustermutation. In this paper we start with a brief review of the quiver context, butthen move onto a discussion of a related Poisson bracket, along with thePoisson algebra of a special family of functions associated with these maps. Abi-Hamiltonian structure is derived and used to construct a sequence of Poissoncommuting functions and hence show complete integrability. Canonicalcoordinates are derived, with the map now being a canonical transformation witha sequence of commuting invariant functions. Compatibility of a pair of thesefunctions gives rise to Liouville's equation and the map plays the role of aB\"acklund transformation.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras