Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556.
Department of Mathematics, Michigan State University, East Lansing, MI 48823, and.
Proc Natl Acad Sci U S A. 2014 Jul 8;111(27):9688-95. doi: 10.1073/pnas.1315283111. Epub 2014 Jun 30.
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.
我们研究了简单复李群上正则函数环中的自然簇结构以及与这些簇结构相容的 Poisson-Lie 结构。根据我们的主要猜想，G 上的 Poisson-Lie 结构的 Belavin-Drinfeld 分类中的每个类对应于 O(G) 中的一个簇结构。我们之前已经证明，对于 G 的任何情况，标准 Poisson-Lie 结构和 SLn（n<5）中的所有 Belavin-Drinfeld 类，这个猜想都是成立的。在本文中，我们对 SLn 上的 Cremmer-Gervais Poisson-Lie 结构进行了证明，它与标准结构的相似度最低。此外，我们证明了在 SL3 上，与标准簇结构不同，与 Cremmer-Gervais 簇结构相对应的簇代数和上簇代数并不重合。最后，我们表明，Cremmer-Gervais 簇结构的正区域包含在全正矩阵集合中。
