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Mosc. Math. J., 2012, Volume 12, Number 2, Pages 293–312 (Mi mmj468)  

This article is cited in 7 scientific papers (total in 7 papers)

Cluster structures on simple complex Lie groups and Belavin–Drinfeld classification

M. Gekhtmana, M. Shapirob, A. Vainshteinc

a Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
b Department of Mathematics, Michigan State University, East Lansing, MI 48823
c Department of Mathematics & Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905, Israel

Abstract: We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson–Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin–Drinfeld classification of Poisson-Lie structures on $\mathcal{G}$ corresponds to a cluster structure in $\mathcal{O}(\mathcal{G})$. We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for $SL_n$, $n<5$, and for any $\mathcal{G}$ in the case of the standard Poisson–Lie structure.

Key words and phrases: Poisso–Lie group, cluster algebra, Belavin–Drinfeld triple.

DOI: https://doi.org/10.17323/1609-4514-2012-12-2-293-312

Full text: http://www.ams.org/.../abst12-2-2012.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 53D17, 13F60
Received: December 29, 2010
Language:

Citation: M. Gekhtman, M. Shapiro, A. Vainshtein, “Cluster structures on simple complex Lie groups and Belavin–Drinfeld classification”, Mosc. Math. J., 12:2 (2012), 293–312

Citation in format AMSBIB
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\by M.~Gekhtman, M.~Shapiro, A.~Vainshtein
\paper Cluster structures on simple complex Lie groups and Belavin--Drinfeld classification
\jour Mosc. Math.~J.
\yr 2012
\vol 12
\issue 2
\pages 293--312
\mathnet{http://mi.mathnet.ru/mmj468}
\crossref{https://doi.org/10.17323/1609-4514-2012-12-2-293-312}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2978758}
\zmath{https://zbmath.org/?q=an:1259.53075}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309365900006}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Gekhtman M., Shapiro M., Vainshtein A., “Cremmer-Gervais Cluster Structure on Sln”, Proc. Natl. Acad. Sci. U. S. A., 111:27 (2014), 9688–9695  crossref  zmath  isi  scopus
    2. Eisner I., “Exotic Cluster Structures on Sl5”, J. Phys. A-Math. Theor., 47:47 (2014), 474002  crossref  zmath  isi  scopus
    3. M. Gekhtman, M. Shapiro, A. Vainshtein, “Generalized cluster structure on the Drinfeld double of $GL_n$”, C. R. Math., 354:4 (2016), 345–349  crossref  isi  elib  scopus
    4. I. Eisner, “Exotic cluster structures on $SL_n$ with Belavin–Drinfeld data of minimal size, I. The structure”, Isr. J. Math., 218:1 (2017), 391–443  crossref  mathscinet  zmath  isi  scopus
    5. I. Eisner, “Exotic cluster structures on $SL_n$ with Belavin–Drinfeld data of minimal size, II. Correspondence between cluster structures and Belavin–Drinfeld triples”, Isr. J. Math., 218:1 (2017), 445–487  crossref  mathscinet  zmath  isi  scopus
    6. M. Gekhtman, M. Shapiro, A. Vainshtein, “Exotic cluster structures on $SL_n$: the Cremmer–Gervais case”, Mem. Am. Math. Soc., 246:1165 (2017), 1+  crossref  mathscinet  isi  scopus
    7. 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
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