Poisson structures compatible with the cluster algebra structure in Grassmannians
Abstract
We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian Gk(n) and show that any such bracket endows Gk(n) with a structure of a Poisson homogeneous space with respect to the natural action of SLn equipped with an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin-Drinfeld classification. Moreover, every compatible Poisson structure can be obtained this way.
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