On a class of representations of the Yangian and moduli space of monopoles

Abstract

A new class of infinite dimensional representations of the Yangians $Y(\mathfrak{g})$ and $Y(\mathfrak{b})$ corresponding to a complex semisimple algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}\subset \mathfrak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\mathfrak{g})$, $\mathfrak{g}=\mathfrak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\mathfrak{g}$. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of $G$-monopoles defined as the components of the space of based maps of ${\mathbb{P}}^1$ into the generalized flag manifold $X=G/B$. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles.

16 pages, LaTex2e, some misprints are fixed