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

Abstract

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 $Ø(\G)$. 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. Besides, we prove that on $S{L}_3$ 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.

The proofs of the statements in Section 3 are contained in the companion paper arXiv:1307.1020. The results in Sections 4 and 5 are new. Restates a conjecture from arXiv:1101.0015 and proves it in a particular case

References in corpus (2)

• Exotic cluster structures on $S{L}_n$: the Cremmer-Gervais case
• Exotic cluster structures on $S{L}_5$

Cited by in corpus (7)

• Drinfeld double of $G{L}_n$ and generalized cluster structures
• Factoriality and class groups of cluster algebras
• Exotic cluster structures on $S{L}_n$: the Cremmer-Gervais case
• Upper cluster algebras and choice of ground ring
• Plethora of cluster structures on $G{L}_n$
• Building maximal green sequences via component preserving mutations
• Exotic cluster structures on $S{L}_n$ with Belavin-Drinfeld data of minimal size: II. Correspondence between cluster structures an BD triples