Cremmer-Gervais cluster structure on SLn

Michael Gekhtman, Michael Shapiro, Alek Vainshtein

Research output: Contribution to journalArticlepeer-review


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 script O sign(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.

Original languageEnglish
Pages (from-to)9688-9695
Number of pages8
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number27
StatePublished - 8 Jul 2014


  • Belavin-Drinfeld triple
  • Poisson-Lie group

ASJC Scopus subject areas

  • General


Dive into the research topics of 'Cremmer-Gervais cluster structure on SLn'. Together they form a unique fingerprint.

Cite this