Year: 2011
Type: preprint
Abstract: The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. Recently, M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras. In this paper, we extend and generalize Glick's work by including the pentagram map into a family of discrete completely integrable systems. Our main tool is Poisson geometry of weighted directed networks on surfaces developed by M. Gekhtman, M. Shapiro, and A. Vainshtein. The ingredients necessary for complete integrability -- invariant Poisson brackets, integrals of motion in involution, Lax representation -- are recovered from combinatorics of the networks. Our integrable systems depend on one discrete parameter $k>1$. The case $k=3$ corresponds to the pentagram map. For $k>3$, we give our integrable systems a geometric interpretation as pentagram-like maps involving deeper diagonals. If $k=2$ and the ground field is $\C$, we give a geometric interpretation in terms of circle patterns.
Source: arXiv (Cornell University)
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: G2786222436, Collaborative Research: Cluster Algebras Approach to Poisson-Lie Groups and Higher Genus Directed Networks, A Study in Non-holonomic Dynamics, Collaborative Research: Cluster Algebras Approach to Poisson-Lie Groups and Higher Genus Directed Networks, G6826065938, G781899229, Collaborative Research: Cluster Algebras, Canonical Bases, and Nets on Surfaces of Higher Genus
OpenAlex ID: W2061661253