Description:
We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structutures 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 reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SLn, n<5, and for any G in the case of the standard Poisson-Lie structure.