Cluster structures on simple complex Lie groups and Belavin–Drinfeld classification

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 $\mathcal{G}$ corresponds to a cluster structure in $\mathcal{O}(\mathcal{G})$. We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for $SL_n$, $n<5$, and for any $\mathcal{G}$ in the case of the standard Poisson–Lie structure.