Poisson Geometry of Directed Networks in an Annulus
Quantum Algebra
2016-05-19 v3
Abstract
As a generalization of Postnikov's construction (see arXiv: math/0609764), 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 arXiv: 0805.3541 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.
Keywords
Cite
@article{arxiv.0901.0020,
title = {Poisson Geometry of Directed Networks in an Annulus},
author = {Michael Gekhtman and Michael Shapiro and Alek Vainshtein},
journal= {arXiv preprint arXiv:0901.0020},
year = {2016}
}
Comments
28 pages, 10 figures. Theorem 4.1 in the previous version was wrong; it has been replaced by a weaker statement. As a result, the proof of Theorem 4.7 has been changed