Non-Commutative Integrability of the Grassmann Pentagram Map

TL;DR

This paper extends the integrability of the pentagram map to a noncommutative setting using algebraic structures like double brackets, broadening the understanding of its geometric and algebraic properties.

Contribution

It introduces a noncommutative framework for the Grassmann pentagram map, generalizing previous commutative approaches and establishing noncommutative integrability.

Findings

Reinterpreted the Grassmann pentagram map using noncommutative algebra.

Established a noncommutative version of integrability for the map.

Extended algebraic tools to analyze geometric discrete integrable systems.

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.