JournalsjemsVol. 14, No. 2pp. 541–570

Poisson geometry of directed networks in an annulus

  • Michael Gekhtman

    University of Notre Dame, USA
  • Michael Shapiro

    Michigan State University, East Lansing, USA
  • Alek Vainshtein

    University of Haifa, Israel
Poisson geometry of directed networks in an annulus cover
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Abstract

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.

Cite this article

Michael Gekhtman, Michael Shapiro, Alek Vainshtein, Poisson geometry of directed networks in an annulus. J. Eur. Math. Soc. 14 (2012), no. 2, pp. 541–570

DOI 10.4171/JEMS/311