Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

M. Gekhtman; M. Shapiro; A. Stolin; A. Vainshtein

doi:10.1007/s11005-012-0547-8

Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

M. Gekhtman, M. Shapiro, A. Stolin, A. Vainshtein

Research output: Contribution to journal › Article › peer-review

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
Pages (from-to)	139-150
Number of pages	12
Journal	Letters in Mathematical Physics
Volume	100
Issue number	2
DOIs	https://doi.org/10.1007/s11005-012-0547-8
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

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10.1007/s11005-012-0547-8

Cite this

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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.",

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Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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