Poisson structures compatible with the cluster algebra structure in Grassmannians

TL;DR

This paper classifies all Poisson brackets compatible with the cluster algebra structure on Grassmannians, showing they correspond to specific R-matrix Poisson-Lie structures and endow Grassmannians with Poisson homogeneous space structures.

Contribution

It provides a complete classification of compatible Poisson brackets on Grassmannians and links them to R-matrix Poisson-Lie structures, revealing their geometric and algebraic significance.

Findings

All compatible Poisson brackets are described explicitly.

Each bracket makes the Grassmannian a Poisson homogeneous space.

Compatible brackets correspond to Belavin-Drinfeld R-matrices.

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 $SL_n$ 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.