Poisson structures compatible with the cluster algebra structure in Grassmannians
Quantum Algebra
2016-05-25 v2 Symplectic Geometry
Abstract
We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian and show that any such bracket endows with a structure of a Poisson homogeneous space with respect to the natural action of 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.
Cite
@article{arxiv.0909.0361,
title = {Poisson structures compatible with the cluster algebra structure in Grassmannians},
author = {Michael Gekhtman and Michael Shapiro and Alexander Stolin and Alek Vainshtein},
journal= {arXiv preprint arXiv:0909.0361},
year = {2016}
}
Comments
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