Non-Commutative Integrability of the Grassmann Pentagram Map
From MaRDI portal
Publication:6308870
DOI10.1016/J.AIM.2020.107309arXiv1810.11742MaRDI QIDQ6308870
Publication date: 27 October 2018
Abstract: The pentagram map is a discrete integrable system first introduced by Schwartz in 1992. It was proved to be intregable by Schwartz, Ovsienko, and Tabachnikov in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associated to certain networks embedded in a disc or annulus, and its relation to cluster algebras. Later, Gekhtman et al. and Tabachnikov reinterpreted the pentagram map in terms of these networks, and used the associated Poisson structures to give a new proof of integrability. In 2011, Mari Beffa and Felipe introduced a generalization of the pentagram map to certain Grassmannians, and proved it was integrable. We reinterpret this Grassmann pentagram map in terms of noncommutative algebra, in particular the double brackets of Van den bergh, and generalize the approach of Gekhtman et al. to establish a noncommutative version of integrability.
Poisson manifolds; Poisson groupoids and algebroids (53D17) Applications of Lie algebras and superalgebras to integrable systems (17B80) Poisson algebras (17B63) Cluster algebras (13F60) Integrable difference and lattice equations; integrability tests (39A36) Completely integrable discrete dynamical systems (37J70)
This page was built for publication: Non-Commutative Integrability of the Grassmann Pentagram Map
Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6308870)
