Exotic cluster structures on $S{L}_n$: the Cremmer-Gervais case

Abstract

This is the second paper in the series of papers dedicated to the study of 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 $Ø(\G)$. 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 $S{L}_n$, $n<5$. In this paper we establish it for the Cremmer-Gervais Poisson-Lie structure on $S{L}_n$, which is the least similar to the standard one.

86 pages, 23 figures; final version; Proposition 2.1 is corrected, which leads to corrections in Theorem 3.6; Proposition 2.2 is stated in a different form; all main results remain unchanged

References in corpus (1)

• Cremmer--Gervais cluster structure on $S{L}_n$

Cited by in corpus (3)

• Drinfeld double of $G{L}_n$ and generalized cluster structures
• Cremmer--Gervais cluster structure on $S{L}_n$
• Exotic cluster structures on $S{L}_n$ with Belavin-Drinfeld data of minimal size: II. Correspondence between cluster structures an BD triples