  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.
