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arXiv · 0805.3541

Poisson Geometry of Directed Networks in a Disk

Michael Gekhtman·Michael Shapiro·Alek Vainshtein
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Abstract

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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Michael Gekhtman, Michael Shapiro, Alek Vainshtein. 2008-05-22. Poisson Geometry of Directed Networks in a Disk. https://arxiv.org/abs/0805.3541

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