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Higher pentagram maps, weighted directed networks, and cluster dynamics. (English) Zbl 1278.37047
Summary: The pentagram map was extensively studied in a series of papers by V. Ovsienko, R. Schwartz and S. Tabachnikov. It was recently interpreted by M. Glick as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick’s construction to include the pentagram map into a family of geometrically meaningful discrete integrable maps.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
13F60 Cluster algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
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