| Generalizing the theory of symplectic geometry which plays an important role in both classical and quantum mechanics, Poisson geometry popularized by Weinstein has recently gained a lot of attention from people in various disciplines. In particular, the theory of Poisson Lie groups has flourished in connection with the fast-growing new area, called quantum groups. In this lecture, after reviewing basic definitions and properties of Poisson manifolds, we will discuss some interesting interplay between Poisson geometry and C*-algebraic quantization. In particular, we will consider the construction of deformation quantizations of a Poisson Lie group within the C*-algebraic framework formulated by Rieffel and Woronowicz respectively for deformation quantizations of a Poisson structure on a manifold and the group structure of a compact matrix group. We will describe how the underlying symplectic foliation structure is reflected in the C*-algebra structure after quantization, and explain the surprising result that although there are deformation quantizations which preserve the underlying symplectic foliation structure and there are deformation quantizations which preserve the underlying group structure, there does not exist any deformation quantization that preserves both structures. |