TY - JOUR
T1 - Cremmer–Gervais cluster structure on <em>SL</em><sub><em>n</em></sub>
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 9688
LP - 9695
DO - 10.1073/pnas.1315283111
VL - 111
IS - 27
AU - Gekhtman, Michael
AU - Shapiro, Michael
AU - Vainshtein, Alek
Y1 - 2014/07/08
UR - http://www.pnas.org/content/111/27/9688.abstract
N2 - Coexistence of diverse mathematical structures supported on the same variety leads to deeper understanding of its features. If the manifold is a Lie group, endowing it with a Poisson structure that respects group multiplication (Poisson–Lie structure) is instrumental in a study of classical and quantum-mechanical systems with symmetries. In turn, a Poisson structure on a variety can be compatible with a cluster structure—a useful combinatorial tool that organizes generators of the ring of regular functions into a collection of overlapping clusters connected via rational transformations. We conjectured that this is the case for an important class of Poisson–Lie groups. The paper verifies this conjecture for a group of invertible matrices equipped with a nonstandard Poisson–Lie structure.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.
ER -