Abstract
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.
Highlights
The pentagram map was introduced by R
The space Pn has a T -invariant Poisson structure, introduced in [20, 21]. The corank of this Poisson structure equals 2 or 4, according as n is odd or even, and the integrals are in involution
In this research announcement and the forthcoming detailed paper, we extend and generalize Glick’s work by including the pentagram map into a family of discrete completely integrable systems
Summary
The pentagram map was introduced by R. The corank of this Poisson structure equals 2 or 4, according as n is odd or even, and the integrals are in involution This provides Liouville integrability of the pentagram map on the space of twisted polygons. We move vertical arrows interchanging the right-most and the left-most position on the network in Fig. 5 using the fact that it is drawn on the torus These moves interchange the quadrilateral and octagonal faces of the graph thereby swapping the variables p and q. Using Proposition 3.1(i), we obtain the new values of (x, y); we shift the indices to conform with Fig. 4 This yields the map Tk, the main character of this note, described in the following proposition.
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