Poisson maps as a tool for constructing cluster structures

We present a construction for cluster charts in simple Lie groups compatible with Poisson structures in the Belavin-Drinfeld classification. The latter are described by combinatorial data associated with the corresponding root system.

We use this data to construct a birational Poisson map from the group to itself that transform a Poisson bracket associated with a nontrivial Belavin-Drinfeld data into the standard one . It allows us to obtain a cluster chart as a pull-back of the Berenstein-Fomin-Zelevinsky cluster coordinates on the open double Bruhat cell.

This is a joint work with M. Shapiro and A. Vainshtein.