文章基本信息

标题：Cremmer–Gervais cluster structure on SLn
本地全文：下载
作者：Michael Gekhtman ; Michael Shapiro ; Alek Vainshtein 等
期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2014
卷号：111
期号：27
页码：9688-9695
DOI：10.1073/pnas.1315283111
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：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 [IMG]f1.gif" ALT="Formula" BORDER="0"> corresponds to a cluster structure in [IMG]f2.gif" ALT="Formula" BORDER="0">. We have shown before that this conjecture holds for any [IMG]f1.gif" ALT="Formula" BORDER="0"> in the case of the standard Poisson-Lie structure and for all Belavin-Drinfeld classes in [IMG]f3.gif" ALT="Formula" BORDER="0">, [IMG]f4.gif" ALT="Formula" BORDER="0">. In this paper we establish it for the Cremmer-Gervais Poisson-Lie structure on [IMG]f3.gif" ALT="Formula" BORDER="0">, which is the least similar to the standard one. Besides, we prove that on [IMG]f5.gif" ALT="Formula" BORDER="0"> 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.
关键词：Poisson–Lie group ; Belavin–Drinfeld triple