Abstract:

It is well known that cluster structures and Poisson structures in the algebra of regular functions on a quasi-affine variety are closely related. In this talk, I will discuss this connection for Poisson structures on a simple simply connected complex Lie group G defined by a pair of classical R-matrices. The key element of the construction is a rational Poisson map from the group with a bracket defined by pair of suitably chosen standard R-matrices to the same group with an arbitrary pair of R-matrices. In the case, of G=SL_n one can build explicitly the corresponding cluster structure and prove its regularity and completeness.