2011-10-14

Belavin-Drinfeld Classification and Cluster Structures on Simple Lie Groups

Mikhail Gekhtman, Notre Dame University

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