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 G corresponds to a cluster structure in O(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 G in the case of the standard Poisson-Lie structure.