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Moscow Mathematical Journal, 2003, Volume 3, Number 3, Pages 899–934
DOI: https://doi.org/10.17323/1609-4514-2003-3-3-899-934 (Mi mmj115) 		 
This article is cited in 153 scientific papers (total in 153 papers)

Cluster algebras and Poisson geometry

M. Gekhtmana, M. Z. Shapirobc, A. D. Vainshteind

a Department of Mathematics, University of Notre Dame
b Michigan State University
c Stockholm University
d University of Haifa
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DOI: https://doi.org/10.17323/1609-4514-2003-3-3-899-934
Abstract: We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians $G(k,n)$ over $\mathbb R$.
Key words and phrases: Cluster algebras, Poisson brackets, toric action, symplectic leaves, real Grassmannians, Sklyanin bracket.
Received: September 5, 2002
Bibliographic databases:
MSC: 53D17, 14M15, 05E15
Language: English
Citation: M. Gekhtman, M. Z. Shapiro, A. D. Vainshtein, “Cluster algebras and Poisson geometry”, Mosc. Math. J., 3:3 (2003), 899–934
Citation in format AMSBIB
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\by M.~Gekhtman, M.~Z.~Shapiro, A.~D.~Vainshtein
\paper Cluster algebras and Poisson geometry
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 3
\pages 899--934
\mathnet{http://mi.mathnet.ru/mmj115}
\crossref{https://doi.org/10.17323/1609-4514-2003-3-3-899-934}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2078567}
\zmath{https://zbmath.org/?q=an:1057.53064}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000208594300008}
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    • This publication is cited in the following 153 articles:
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