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Algebra i Analiz, 2010, Volume 22, Issue 2, Pages 14–104 (Mi aa1177)  

This article is cited in 4 scientific papers (total in 4 papers)

Research Papers

Cluster $\mathcal X$-varieties for dual Poisson–Lie groups. I

R. Brahami

Institut Mathématiques de Bourgogne, Dijon, France

Abstract: We associate a family of cluster $\mathcal X$-varieties with the dual Poisson–Lie group $G^*$ of a complex semi-simple Lie group $G$ of adjoint type given with the standard Poisson structure. This family is described by the $W$-permutohedron associated with the Lie algebra $\mathfrak g$ of $G$, vertices being labeled by cluster $\mathcal X$-varieties and edges by new Poisson birational isomorphisms on appropriate seed $\mathcal X$-tori, called saltation. The underlying combinatorics is based on a factorization of the Fomin–Zelevinsky twist maps into mutations and other new Poisson birational isomorphisms on seed $\mathcal X$-tori, called tropical mutations (because they are obtained by a tropicalization of the mutation formula), associated with an enrichment of the combinatorics on double words of the Weyl group $W$ of $G$.

Keywords: cluster combinatorics, Poisson structure, tropical mutation, saltations.

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English version:
St. Petersburg Mathematical Journal, 2011, 22:2, 183–250

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Received: 22.09.2009
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Citation: R. Brahami, “Cluster $\mathcal X$-varieties for dual Poisson–Lie groups. I”, Algebra i Analiz, 22:2 (2010), 14–104; St. Petersburg Math. J., 22:2 (2011), 183–250

Citation in format AMSBIB
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\by R.~Brahami
\paper Cluster $\mathcal X$-varieties for dual Poisson--Lie groups.~I
\jour Algebra i Analiz
\yr 2010
\vol 22
\issue 2
\pages 14--104
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2668124}
\zmath{https://zbmath.org/?q=an:1225.22011}
\transl
\jour St. Petersburg Math. J.
\yr 2011
\vol 22
\issue 2
\pages 183--250
\crossref{https://doi.org/10.1090/S1061-0022-2011-01138-0}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Gekhtman M., Shapiro M., Stolin A., Vainshtein A., “Poisson structures compatible with the cluster algebra structure in Grassmannians”, Lett. Math. Phys., 100:2 (2012), 139–150  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Gekhtman M., Shapiro M., Vainshtein A., “Generalized cluster structure on the Drinfeld double of GL n”, C. R. Math., 354:4 (2016), 345–349  crossref  mathscinet  zmath  isi  elib  scopus
    3. Schrader G., Shapiro A., “Quantum Groups, Quantum Tori, and the Grothendieck-Springer Resolution”, Adv. Math., 321 (2017), 431–474  crossref  mathscinet  zmath  isi  scopus
    4. Gekhtman M., Shapiro M., Vainshtein A., “Drinfeld Double of Gln and Generalized Cluster Structures”, Proc. London Math. Soc., 116:3 (2018), 429–484  crossref  mathscinet  zmath  isi  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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