On the existence of deformed Lie-Poisson structures for quantized groups

The geometrical description of deformation quantization based on quantum duality principle makes it possible to introduce deformed Lie-Poisson structure. It serves as a natural analogue of classical Lie bialgebra for the case when the initial object is a quantized group. The explicit realization of the deformed Lie-Poisson structure is a difficult problem. We study the special class of such constructions characterized by quite a simple form of tanjent vector fields. It is proved that in such a case it is sufficient to find four Lie compositions that form two deformations of the first order and four Lie bialgebras. This garantees the existence of two families of deformed Lie-Poisson structures due to the intrinsic symmetry of the initial compositions. The explicit example is presented.