Belavin-Drinfeld classification and
cluster structures on simple Lie groups

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). I will explaining how different parts of the conjecture are related to each other and present a supporting evidence that includes SL(n), n<5, as well as the standard Poisson-Lie structure on any simple G.