Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

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

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)139-150
Number of pages12
JournalLetters in Mathematical Physics
Issue number2
StatePublished - 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.


  • Grassmannian
  • Poisson-Lie group
  • cluster algebra

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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