ar X iv : 0 80 5 . 35 41 v 2 [ m at h . Q A ] 2 0 Fe b 20 09 Poisson Geometry of Directed Networks in a Disk
نویسنده
چکیده
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. Mathematics Subject Classification (2000). 53D17, 14M15.
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