Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

M. Gekhtman, M. Shapiro, A. Stolin, A. Vainshtein
2012 Letters in Mathematical Physics  
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.
doi:10.1007/s11005-012-0547-8 fatcat:ajr5cx6wdjh3plruw7shncpo5u