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 (g) and Y (b) corresponding to a complex semisimple algebra g and its Borel subalgebra b ⊂ g is constructed. It is based on the generalization of the Drinfeld realization of Y (g), g = gl(N ) in terms of quantum minors to the case of an arbitrary semisimple Lie algebra 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 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.

On category O for affine Grassmannian slices and categorified tensor products

“…Proof This follows from results of : Equation can be inverted to express the generators in terms of the , and the relations in Lemma 4.2 imply those from Definition .…”

On a class of representations of quantum groups

“…In this section we remind the explicit construction of a class of representations of the Yangian in terms of difference operators given in [17].…”

“…[2], [24], [25]). Our construction is the new one and has deep relations with the Quantum Inverse Scattering Method [14] as well as with the natural parameterization of the moduli spaces of monopoles [17]. The detailed discussion of the connection with the monopoles on R 2 × S 1 will be described in [18].…”

“…In this note we further generalize the construction of [14], [16], [17] to obtain a realization of a class of the infinite-dimensional representations of the finite-dimensional quantum groups U q (g) and affine quantum groups U q (ĝ) c=0 with zero level c = 0 for an arbitrary semi-simple Lie algebra g. As an intermediate step, we construct the embedding of quantum groups into the algebra of rational functions on the quantum multi-dimensional torus. We postpone the technical details to another occasion [18] and concentrate on the explicit expressions for the representations of the quantum groups.…”

“…Thus in [17] the construction of some class of the representations of the Yangian Y (g) for an arbitrary finite-dimensional semisimple Lie algebra g was given. These representations arise as a quantization of the symplectic leaves of the classical counterpart of the Yangian.…”