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Abstract: We investigate Poisson properties of Postnikov-s map from the space of edgeweights of a planar directed network into the Grassmannian. We show that thismap is Poisson if the space of edge weights is equipped with a representativeof a 6-parameter family of universal quadratic Poisson brackets and theGrasmannian is viewed as a Poisson homogeneous space of the general lineargroup equipped with an appropriately chosen R-matrix Poisson-Lie structure. Wealso prove that Poisson brackets on the Grassmannian arising in this way arecompatible with the natural cluster algebra structure.



Author: Michael Gekhtman University of Notre Dame, Michael Shapiro Michigan State University, Alek Vainshtein University of Haifa

Source: https://arxiv.org/



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