Poisson Geometry of Directed Networks in a Disk

Details Poisson Geometry of Directed Networks in a Disk Poisson Geometry of Directed Networks in a Disk

2008-05-22

arxiv.org/abs/0805.3541v2

Mathematics

Quantum Algebra

44 pages, 19 figures. Final version, to appear in Selecta Math. Many technical changes, including formulations of main results

Scientific paper

We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriately chosen R-matrix Poisson-Lie structure. We also prove that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.

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