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Mutation-periodic quivers, integrable maps and associated Poisson algebras. (English) Zbl 1219.17020

Summary: 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.

MSC:

17B63 Poisson algebras
13F60 Cluster algebras
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
53D99 Symplectic geometry, contact geometry

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