Non-commutative integrability of the Grassmann pentagram map

Abstract

The pentagram map is a discrete dynamical system first introduced by Schwartz in 1992. It was proved to be integrable 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, Marí Beffa and Felipe introduced a generalization of the pentagram map to certain Grassmannians, and proved it had a Lax representation. 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.