Higher pentagram maps, weighted directed networks, and cluster dynamics. (English) Zbl 1278.37047

Gekhtman, Michael; Shapiro, Michael; Tabachnikov, Serge; Vainshtein, Alek

Electron. Res. Announc. Math. Sci. 19, 1-17 (2012).

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

Keywords:

pentagram map; cluster dynamics; discrete integrable system