Nets on surfaces, cluster algebras and compatible Poisson structures

Alek Vainshtein

In the first part of the talk I will consider Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. This map turns out to be Poisson, provided the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriately chosen R-matrix Poisson-Lie structure. I will explain that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.

In the second part, I will extend Postnikov's model to the case of networks embedded into an annulus, which leads to a map into the space of loops in the Grassmannian. Natural Poisson brackets on edge weights in this case are intimately connected to trigonometric R-matrix brackets on matrix-valued rational functions. Finally, I will utilize a certain class of networks in an annulus to introduce a cluster algebra structure on the coordinate ring of the space of normalized rational functions in one variable. In this case, the Poisson bracket compatible with the cluster algebra structure coincides with the quadratic Poisson bracket studied earlier in the context of Toda flows on minimal orbits, and cluster transformations serve as Bäcklund-Darboux transformations between different minimal Toda flows.