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We study natural cluster structures in the rings of regular functions onsimple complex Lie groups and Poisson-Lie structures compatible with thesecluster structures. According to our main conjecture, each class in theBelavin-Drinfeld classification of Poisson-Lie structures on $\G$ correspondsto a cluster structure in $\O(\G)$. We prove a reduction theorem explaining howdifferent parts of the conjecture are related to each other. The conjecture isestablished for $SL_n$, $n<5$, and for any $\G$ in the case of the standardPoisson-Lie structure.

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