## Abstract

The present paper is a first step toward establishing connections between solutions of the classical Yang-Baxter equations and cluster algebras. 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.

Original language | English |
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Pages (from-to) | 139-150 |

Number of pages | 12 |

Journal | Letters in Mathematical Physics |

Volume | 100 |

Issue number | 2 |

DOIs | |

State | Published - May 2012 |

### Bibliographical note

Funding Information:M. G. was supported in part by NSF Grant DMS #0801204. M. S. was supported in part by NSF Grants DMS #0800671 and PHY #0555346. A. S. was supported in part by KVVA. A. V. was supported in part by ISF Grant #1032/08. The authors are grateful to A. Zelevinsky for useful comments.

## Keywords

- Grassmannian
- Poisson-Lie group
- cluster algebra

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics