v 2 8 N ov 1 99 5 QUANTIZATION OF POISSON ALGEBRAIC GROUPS AND POISSON HOMOGENEOUS SPACES

This paper consists of two parts. In the first part we show that any Poisson algebraic group over a field of characteristic zero and any Poisson Lie group admits a local quantization. This answers positively a question of Drinfeld and generalizes the results of [BFGP] and [BP]. In the second part we apply our techniques of quan-tization to obtain some nontrivial examples of quantization of Poisson homogeneous spaces. 1. Quantization of Poisson algebraic and Lie groups. Below we will freely use the notation from our previous paper [EK]. 1.1. Poisson algebras and manifolds. Let k be a field of characteristic 0, B be a commutative algebra over k. A commutative algebra B equipped with a Poisson bracket is called a Poisson algebra. Let B, C be Poisson algebras. Then B ⊗ C has a natural structure of a Poisson algebra, defined by {b 1 ⊗ c 1 , b 2 ⊗ c 2 } = b 1 b 2 ⊗ {c 1 , c 2 } + {b 1 , b 2 } ⊗ c 1 c 2. Let X be an algebraic variety over k. Denote by O X the structure sheaf of X. X is called a Poisson algebraic variety if the sheaf O X is equipped with the structure of a sheaf of Poisson algebras. Similarly one defines the notions of a smooth or complex analytic Poisson manifold. If X, Y are Poisson manifolds, then the product X × Y has a natural structure of a Poisson manifold. This is a consequence of the fact that the tensor product of Poisson algebras is a Poisson algebra. 1.2. Poisson groups. Let a be a finite-dimensional Lie algebra over k. Then the ring k[[a]] of formal power series on a is a commutative topological Hopf algebra. We will denote by A the " spectrum " of k[[a]] We call A the formal group associated