Poisson Manifolds Associated with Group Actions and Classical Triangular r-Matrices

Let P be a Poisson G-space and Λ a classical triangular r-matrix. Using the Poisson reduction, we construct a new Poisson structure PΛ on P. For this new Poisson structure PΛ, we construct its symplectic groupoid, describe its symplectic leaves, and classify its symplectic realizations. The deformation quantization of PΛ is also discussed.