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 matrix-valued 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.
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Michael Gekhtman, Michael Shapiro, Alek Vainshtein, Poisson geometry of directed networks in an annulus. J. Eur. Math. Soc. 14 (2012), no. 2, pp. 541–570DOI 10.4171/JEMS/311