Uniqueness of braidings of quasitriangular Lie bialgebras and lifts of classical r-matrices

It is known that any quantization of a quasitriangular Lie bialgebra g gives rise to a braiding on the dual Poisson-Lie formal group G*. We show that this braiding always coincides with the Weinstein-Xu braiding. We show that this braiding is the “time one automorphism” of a Hamiltonian vector field, corresponding to a certain formal function on G* × G*, the “lift of r”, which can be expressed in terms of r by universal formulas. The lift of r coincides with the classical limit of the rescaled logarithm of any R-matrix quantizing it.