Poisson Geometry of Directed Networks in an Annulus

Details Poisson Geometry of Directed Networks in an Annulus Poisson Geometry of Directed Networks in an Annulus

2008-12-31

arxiv.org/abs/0901.0020v3

Mathematics

Quantum Algebra

28 pages, 10 figures. Theorem 4.1 in the previous version was wrong; it has been replaced by a weaker statement. As a result,

Scientific paper

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.

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