Cluster structures on simple complex lie groups and the Belavin-Drinfeld classification

Abstract

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 $\mathcal{G}$ corresponds to a cluster structure in $\mathcal{O}(\mathcal{G})$. We prove 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.