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Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

M. Gekhtman ; M. Shapiro ; Alexander Stolin (Institutionen för matematiska vetenskaper) ; A. Vainshtein
Letters in Mathematical Physics (0377-9017). Vol. 100 (2012), 2, p. 139-150.
[Artikel, refereegranskad vetenskaplig]

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 SLn 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.

Nyckelord: Grassmannian, Poisson-Lie group, cluster algebra, lie groups



Denna post skapades 2012-07-05. Senast ändrad 2012-07-10.
CPL Pubid: 160106

 

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