Poisson structures compatible with the cluster algebra structure in Grassmannians

Details Poisson structures compatible with the cluster algebra structure in Grassmannians Poisson structures compatible with the cluster algebra structure in Grassmannians

2009-09-02

arxiv.org/abs/0909.0361v2

Mathematics

Quantum Algebra

Minor corrections: formulation of Proposition 2.2 made more precise; as a result, proofs of Proposition 2.2 and Theorem 4.3 sl

Scientific paper

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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