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

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

2004-09-02

arxiv.org/abs/math/0409031v2

Mathematics

Algebraic Geometry

16 pages, LaTex2e, some misprints are fixed

Scientific paper

10.1007/s00220-005-1417-3

A new class of infinite dimensional representations of the Yangians $Y(\frak{g})$ and $Y(\frak{b})$ corresponding to a complex semisimple algebra $\frak{g}$ and its Borel subalgebra $\frak{b}\subset\frak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\frak{g})$, $\frak{g}=\frak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\frak{g}$. 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.

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