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Cremmer--Gervais cluster structure on $SL_n$

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 $SL_n$, $n<5$. In this paper we establish it for the Cremmer-Gervais Poisson-Lie structure on $SL_n$, which is the least similar to the standard one. Besides, we prove that on $SL_3$ 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.