Poisson geometry of directed networks in an annulus

Gekhtman, 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 -