Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

The present paper is a first step toward establishing connections between solutions of the classical Yang-Baxter equations and cluster algebras. We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian G _k(n) and show that any such bracket endows G _k(n) with a structure of a Poisson homogeneous space with respect to the natural action of SL _n 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.