On a class of representations of quantum groups

This paper is a short account of the construction of a new class of infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups U-q(q), Yangians Y(g) and affine quantum groups at zero level U-q((g) over cap)(c=0) corresponding to an arbitrary finite-dimensional semisimple Lie algebra g. At an intermediate step we construct an embedding of the quantum groups into the algebra of rational functions on the quantum multi-dimensional torus. The explicit parameterization of the quantum groups used in this paper turns out to be closely related to the parameterization of the moduli spaces of monopoles. As a result the proposed constructions of the representations provide a quantization of the moduli spaces of monopoles on R-3 and R-2 x S-1.