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.
Bibliographical noteFunding Information:
M. G. was supported in part by NSF Grant DMS #0801204. M. S. was supported in part by NSF Grants DMS #0800671 and PHY #0555346. A. S. was supported in part by KVVA. A. V. was supported in part by ISF Grant #1032/08. The authors are grateful to A. Zelevinsky for useful comments.
- Poisson-Lie group
- cluster algebra
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics