In [19] Lu showed that any dynamical r-matrix for the pair (g, u) naturally induces a Poisson homogeneous structure on G/U . She also proved that if g is complex simple, u is its Cartan subalgebra and r is quasitriangular, then this correspondence is in fact 1-1. In the present paper we find some general conditions under which the Lu correspondence is 1-1. Then we apply this result to describe all triangular Poisson homogeneous structures on G/U for a simple complex group G and its reductive subgroup U containing a Cartan subgroup.Later Lu [19] found a connection (which is essentially a 1-1 correspondence) between dynamical r-matrices for the pair (g, u) (where u is a Cartan subalgebra of the complex simple finite-dimensional algebra g), and Poisson homogeneous G-structures on G/U. Here U ⊂ G are connected Lie groups corresponding to u ⊂ g, and G is equipped with the standard quasitriangular (with Ω = 0) Poisson-Lie structure.Lu also noticed that this connection can be generalized to the case u is a subspace in a Cartan subalgebra (with some "regularity" condition). The dynamical r-matrices for the latter case were classified by Schiffmann in [20]. In this case connections between dynamical r-matrices and certain Lagrangian subalgebras can be derived directly from [20]. Now let G be a connected complex semisimple Lie group, and let U be its connected subgroup. Suppose u ⊂ g are the corresponding Lie algebras. In the present paper we consider connections between Poisson homogeneous structures on G/U related to the triangular Poisson-Lie structures on G (i.e., with Ω = 0), where U is a reductive subgroup containing a Cartan subgroup of G, and triangular dynamical r-matrices for the pair (g, u).In fact, our results are based on a general result on relations between classical dynamical r-matrices and Poisson homogeneous structures (see Theorem 12), which is valid also in the quasitriangular case. Notice that the results of Sections 2 and 3 can be used to describe a 1-1 correspondence between dynamical r-matrices for the pair (g, u), where u ⊂ g is a Cartan subalgebra, and Poisson homogeneous G-structures on G/U, where G is equipped with any quasitriangular (with Ω = 0) Poisson-Lie structure (of course the latter result is due to Lu). Our approach is based on some strong classification results for dynamical r-matrices given recently by Etingof and Schiffmann in [9].The paper is organized as follows. In Section 2 we describe a correspondence between the (moduli space of) dynamical r-matrices for a pair (g, u) and Poisson homogeneous G-structures on G/U proving that under certain assumptions it is a bijection. In Section 3 we consider a procedure of twisting for Lie bialgebras and examine its impact on the double D(g) and Poisson homogeneous spaces for corresponding Poisson-Lie groups. Then we use the twisting to weaken some restrictions needed in Section 2. In Section 4 we consider the basic example of our paper: g is semisimple, u ⊂ g is a reductive Lie subalgebra that contains some Cartan subalgebra of g, and the Li...

In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra g. This problem reduces to the classification of all Lie bialgebra structures on g(K), where K = C(( )). The associated classical double is of the form g(K) ⊗ K A, where A is one of the following: K[ε], where ε 2 = 0, K ⊕ K or K[j], where j 2 = . The first case relates to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin-Drinfeld cohomology associated to any non-skewsymmetric r-matrix from the Belavin-Drinfeld list [1]. We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on g(K) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.Mathematics Subject Classification (2010): 17B37, 17B62.

The aim of this paper is to provide an overview of the results about classification of quantum groups that were obtained in [10], [11].Mathematics Subject Classification (2010): 17B37, 17B62.

In this paper we continue to study Belavin-Drinfeld cohomology introduced in [6] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra g. Here we compute Belavin-Drinfeld cohomology for all nonskewsymmetric r-matrices from the Belavin-Drinfeld list for simple Lie algebras of type B, C, and D.Mathematics Subject Classification (2010): 17B37, 17B62.

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