Dynamical Yang-Baxter equation and quantization of certain Poisson brackets

Abstract

In this paper we develop a q-analogue of the results obtained in [KST]. Let G be a Lie group, g = LieG, U o G a connected closed Lie subgroup such that the corresponding subalgebra u o g is reductive in g (i.e., there exists an u-invariant subspace m o g such that g = u a m), and S 2 (u a u) a (m a m) a symmetric tensor. Take a solution o 2 g a g of the classical Yang-Baxter equation such that o + o 21 = S and consider the corresponding Poisson Lie group structure oo on G. In [KMS] we established a one-to-one correspondence between the moduli space of classical dynamical r-matrices for the pair (g;u) with the symmetric part S 2 and the set of all structures of Poisson homogeneous (G;oo)-space on G=U. We emphasize that the Þrst example of such a correspondence was found by Lu in [L]. In [KST] we proposed a partial quantization of the results of [KS] and [KMS]. We explained how starting from the dynamical twist for a pair (Ug;h) (where g is a Lie algebra, h is its abelian subalgebra, and Ug is the universal enveloping algebra of g) one can get a G-equivariant star-product on G=H (where H o G are connected Lie groups corresponding to h o g). In the case when g is a complex simple Lie algebra and h is its Cartan subalgebra we gave a representation-theoretic explanation of our results in terms of Verma modules. The aim of the paper is to obtain an analogue of these results for quantized universal enveloping algebras. Therefore, we quantize some Poisson homogeneous structures appeared in [K]. The following idea lies in the background of our quantization. There is a one-to-one correspondence between Poisson homogeneous structures and dynamical r-matrices [KMS, L]. Then there should be a relation between quantization of the Poisson homogeneous structures and quantization of the dynamical r-matrices. Notice that results in this direction were obtained in the recent papers [DM] and [AL]. However, we emphasize the connection between star-products and dynamical twists very explicitly (cf. [AL]). Let us explain the structure of this paper in more detail. In Section 2 we remind the reader of some results of [KST] and give a conceptual explanation of