Poisson Geometry of Directed Networks in an Annulus

As a generalization of Postnikov’s construction [P], we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced for the space of edge weights in [GSV3] induce a family of Poisson structures on rational-valued matrix functions and on the space of loops in the Grassmannian. In the former case, this family includes, for a particular kind of networks, the Poisson bracket associated with the trigonometric R-matrix.