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arXiv · 0909.0361

Poisson structures compatible with the cluster algebra structure in Grassmannians

Michael Gekhtman·Michael Shapiro·Alexander Stolin·Alek Vainshtein
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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.

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Michael Gekhtman, Michael Shapiro, Alexander Stolin, Alek Vainshtein. 2009-09-02. Poisson structures compatible with the cluster algebra structure in Grassmannians. https://doi.org/10.1007/s11005-012-0547-8

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