# Poisson geometry of directed networks in an annulus

Michael Gekhtman; Michael Shapiro; Vainshtein, Alek

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 2, page 541-570
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topGekhtman, Michael, Shapiro, Michael, and Vainshtein, Alek. "Poisson geometry of directed networks in an annulus." Journal of the European Mathematical Society 014.2 (2012): 541-570. <http://eudml.org/doc/277392>.

@article{Gekhtman2012,

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.},

author = {Gekhtman, Michael, Shapiro, Michael, Vainshtein, Alek},

journal = {Journal of the European Mathematical Society},

keywords = {network; annulus; Grassmannian; Poissson bracket; $R$-matrix; Poisson geometry; network; annulus; weight; Grassmannnian; R-matrix},

language = {eng},

number = {2},

pages = {541-570},

publisher = {European Mathematical Society Publishing House},

title = {Poisson geometry of directed networks in an annulus},

url = {http://eudml.org/doc/277392},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Gekhtman, Michael

AU - Shapiro, Michael

AU - Vainshtein, Alek

TI - Poisson geometry of directed networks in an annulus

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 2

SP - 541

EP - 570

AB - 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.

LA - eng

KW - network; annulus; Grassmannian; Poissson bracket; $R$-matrix; Poisson geometry; network; annulus; weight; Grassmannnian; R-matrix

UR - http://eudml.org/doc/277392

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.