Poisson Geometry of Directed Networks in a Disk
arXiv:0805.3541
Abstract
We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if 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. We also prove that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.
44 pages, 19 figures. Final version, to appear in Selecta Math. Many technical changes, including formulations of main results, new references added, one section moved to a subsequent paper