Log-canonical poisson structures and non-commutative integrable systems

Log-canonical Poisson structures are a particularly simple type of bracket which are given by quadratic expressions in local coordinates. They appear in many places, including the study of cluster algebras. A Poisson bracket is "compatible" with a cluster algebra structure if the bracket is log-canonical with respect to each cluster. In joint work with John Machacek, we prove a structural result about such Poisson structures, which justifies the use and significance of such brackets in cluster theory. The result says that no rational coordinate-changes can transform these brackets into a simpler form. The pentagram map is a discrete dynamical system on the space of plane polygons first intro- duced by Schwartz in 1992. It was proved to be Liouville 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. These Poisson structures are log-canonical. 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 had a Lax representation. We reinterpret this Grassmann penta- gram map in terms of non-commutative algebra, in particular the double brackets of Van den Bergh, and generalize the approach of Gekhtman et al. to establish a non-commutative version of integrability. The integrability of the pentagram maps in both projective space and the Grass-mannian follow from this more general algebraic system by projecting to representation spaces.