Nets on surfaces, cluster algebras,
and compatible Poisson structures

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.

Finally, 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.

Based on joint work with M.Gekhtman and M.Shapiro.