title: Mutation-periodic quivers, integrable maps and associated Poisson algebras.
abstract: We consider a class of map, recently derived in the context of cluster mutation. In this paper, we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville's equation and the map plays the role of a Bäcklund transformation.
authors.affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, UK. amt6apf@leeds.ac.uk
authors.forename: Allan P
authors.initials: AP
authors.lastname: Fordy