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 $G_k(n)$ 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 $S{L}_n$ 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.

Minor corrections: formulation of Proposition 2.2 made more precise; as a result, proofs of Proposition 2.2 and Theorem 4.3 slightly modified; a misprint in the reference list corrected; an acknowledgment added

References in corpus (2)

• Total positivity, Grassmannians, and networks
• Cluster X-varieties for dual Poisson-Lie groups II

Cited by in corpus (7)

• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Drinfeld double of $G{L}_n$ and generalized cluster structures
• Exotic cluster structures on $S{L}_n$: the Cremmer-Gervais case
• Rigid Indecomposable Modules in Grassmannian Cluster Categories
• Toric Poisson Ideals in Cluster Algebras
• Indecomposable Modules in the Grassmannian Cluster Category $\mathrm{CM}(B_{5,10})$
• Cluster Algebras, Symplectic Leaves and Quantum Groups