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. 2014 Jul 8;111(27):9688-95.
doi: 10.1073/pnas.1315283111. Epub 2014 Jun 30.

Cremmer-Gervais cluster structure on SLn

Michael Gekhtman  1 , Michael Shapiro  2 , Alek Vainshtein  3
Affiliations

Cremmer-Gervais cluster structure on SLn

Michael Gekhtman et al. Proc Natl Acad Sci U S A. .

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 G corresponds to a cluster structure in O(G). We have shown before that this conjecture holds for any G in the case of the standard Poisson-Lie structure and for all Belavin-Drinfeld classes in SLn, n<5. In this paper we establish it for the Cremmer-Gervais Poisson-Lie structure on SLn, which is the least similar to the standard one. Besides, we prove that on SL3 the cluster algebra and the upper cluster algebra corresponding to the Cremmer-Gervais cluster structure do not coincide, unlike the case of the standard cluster structure. Finally, we show that the positive locus with respect to the Cremmer-Gervais cluster structure is contained in the set of totally positive matrices.

Keywords: Belavin–Drinfeld triple; Poisson–Lie group.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Quiver QCG(5).
See this image and copyright information in PMC

References

    1. Gekhtman M, Shapiro M, Vainshtein A. Cluster structures on simple complex Lie groups and Belavin–Drinfeld classification. Moscow Math J. 2012;12(2):293–312.
    1. Gekhtman M, Shapiro M, Vainshtein A. Cluster algebras and Poisson geometry. Moscow Math J. 2003;3(3):899–934.
    1. Gekhtman M, Shapiro M, Vainshtein A. Generalized Bäcklund-Darboux transformations of Coxeter-Toda flows from a cluster algebra perspective. Acta Math. 2011;206(2):245–310.
    1. Gekhtman M, Shapiro M, Vainshtein A. Cluster Algebras and Poisson Geometry. Mathematical Surveys and Monographs, 167. Providence, RI: American Mathematical Society; 2010.
    1. Berenstein A, Fomin S, Zelevinsky A. Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math J. 2005;126(1):1–52.

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