ITEP-TH-30/04 TCD-MATH-04-15

HMI-04-04

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

A. Gerasimov

Institute for Theoretical & Experimental Physics, 117259, Moscow, Russia Department of Pure and Applied Mathematics, Trinity College, Dublin 2, Ireland

Hamilton Mathematics Institute, TCD, Dublin 2, Ireland

S. Kharchev^{1},

Institute for Theoretical & Experimental Physics, 117259, Moscow, Russia

D. Lebedev^{2}

Institute for Theoretical & Experimental Physics, 117259, Moscow, Russia and Max-Planck-Institut fur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany,

S. Oblezin ^{3}

Institute for Theoretical & Experimental Physics, 117259, Moscow, Russia

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.

1E-mail: kharchev@itep.ru 2E-mail: lebedev@mpim-bonn.mpg.de 3E-mail: Sergey.Oblezin@itep.ru

1 Introduction

The Yangian Y (𝔤) for a semisimple complex Lie algebra 𝔤 was introduced by Drinfeld as a certain deformation of U(𝔤[t]) as a Hopf algebra [1], [2], [3] (see for recent review [4], and [5]). Recently a construction of the special class of infinite-dimensional representations of Y (𝔤𝔩(N)) and Y (𝔰𝔩(N)) based on the generalization of the Gelfand-Zetlin construction was introduced in [6] (see also [7]). In this paper we generalize this construction to Y (𝔤) for an arbitrary semisimple Lie algebra 𝔤. This generalization is based on the proposed generalization of the Drinfeld realization of Y (𝔤𝔩(N)) in terms of quantum minors to the case of Y (𝔤) for an arbitrary semisimple Lie algebra 𝔤. We also construct representations of Y (𝔟) where 𝔟 ⊂ 𝔤 is a Borel subalgebra of 𝔤. One should note that not all of the considered representations of Y (𝔟) may be lifted to the representations of Y (𝔤). We also describe the Poisson geometry behind the constructed representation by defining explicitly the symplectic leaves of the classical versions of the Yangians Y (𝔤) and Y (𝔟). The proposed description of the symplectic leaves reveals a deep connection with the moduli space of Gmonopoles, Lie(G) = 𝔤. The symplectic leaves as symplectic manifolds turn out to be open parts of the moduli space of monopoles with maximal symmetry breaking supplied with the symplectic structure introduced in [8] for G = SU(2), in [9] for G = SU(N), and in [10] for an arbitrary semisimple Lie group G. Thus our construction of the Yangian representations can be considered as a quantization of the moduli spaces of G-monopoles.

The results of this paper support the strong connection between quantum integrable systems and problems of the quantization of various moduli spaces. The demonstrated connection between the Yangian and the quantization of the moduli space of the monopoles is a particular example of this deep relation. We plan to consider its implications to the theory of quantum integrable theories elsewhere. Let us remark that the connection between the Atiyah-Hitchin symplectic structure on the moduli space of the SU(2) monopoles [8] with some particular integrable systems was noted previously in [11], [12].

Finally note that the explicit construction of the representations of Y (𝔤) discussed below appears to be similar to the constructions of the representations of a class of elliptic algebras proposed in [13], [14]. Nevertheless, the main result (Theorem 3.1) seems to be new.

The plan of the paper is as follows. In Section 2 we provide various descriptions of the Y (𝔤) in terms of the generators and relations. The construction of Y (𝔤) in terms of the generators A_{i}(u), B_{i}(u), C_{i}(u) for a general Lie algebra is proposed. In Section 3, we describe a particular class of representations of the Y (𝔤) and Y (𝔟) and give an explicit realization of its generators in terms of difference operators. The main result is formulated in the Theorem 3.1. In Section 4 we discuss the underlying Poisson geometry and provide a description of the corresponding symplectic leaves of the classical counterpart of the Yangian. It appears that there is an isomorphism between the open part of the symplectic leaves for Y (𝔟) and the open part of the moduli space of the G-monopoles with the maximal symmetry breaking.

Acknowledgments: The authors are grateful to A. Levin and A. Rosly for useful discussions and to M. Finkelberg for the explanation of the results of [10], [15]. The research was partly supported by grants CRDF RM1-2545; INTAS 03-513350; grant NSh 1999.2003.2 for support of scientific schools, and by grants RFBR-040100646 (A. Gerasimov, D. Lebedev), and RFBR-040100642 (S. Kharchev, S. Oblezin). The research of A. Gerasimov was also partly supported by SFI Basic Research Grant. D. Lebedev would like to thank the Max-Planck-Institut für Mathematik for financial support and hospitality. S. Oblezin is deeply grateful to the Independent University of Moscow for the support.

2 The various presentations of Y (𝔤)

We start with the definition of the Yangian for a semisimple Lie algebra 𝔤 due to Drinfeld [1], [2] (see also [16]) in the form given in [17].

Let 𝔥 ⊂ 𝔟 ⊂ 𝔤 be a simple finite dimensional Lie algebra 𝔤 of rank ℓ over ℂ with a Cartan subalgebra 𝔥 and a Borel subalgebra 𝔟. Let a = ||a_{ij}||, i, j = 1, …, ℓ be the Cartan matrix of 𝔤, Γ be the set of vertices of the Dynkin diagram of 𝔤, {α_{i} ∈ 𝔥^{∗}, i ∈ Γ} be the set of simple roots and {α_{i}^{∨}, i ∈ Γ} be the set of the corresponding co-roots (a_{ij} = α_{i}^{∨}(α_{j})). There exist positive integers d_{1}, …, d_{ℓ} such that the matrix ||d_{i}a_{ij}|| is symmetric. Define the invariant bilinear form on 𝔥^{∗} by (α_{i}, α_{j}) = d_{i}a_{ij}, then a_{ij} = \frac{2(α_{i},α_{j})}{(α_{i},α_{i})}.

It is convenient to define the generators of the Yangian in terms of the generating series H_{i}(u) , E_{i}(u), and F_{i}(u) , i ∈ Γ:

\begin{matrix}H_{i}(u) = 1 + ∑_{s=0}^{∞} H_{i}^{(s)}u^{−s−1} , \\ E_{i}(u) = ∑_{s=0}^{∞} E_{i}^{(s)}u^{−s−1} ,  F_{i}(u) = ∑_{s=0}^{∞} F_{i}^{(s)}u^{−s−1} . \end{matrix}  (2.1)

Definition 2.1 The Yangian Y (𝔤) is the associative algebra with generators H_{i}^{(s)}, E_{i}^{(s)}, F_{i}^{(s)}, i ∈ Γ; s = 0, 1, … and the following defining relations

[H_{i}(u), H_{j}(v)] = 0 ,  (2.2)

\begin{matrix}[H_{i}(u), E_{j}(v)] = −\frac{ıℏ}{2} (α_{i}, α_{j}) \frac{[H_{i}(u), E_{j}(u) − E_{j}(v)]_{+}}{u − v} , \\ [H_{i}(u), F_{j}(v)] = \frac{ıℏ}{2} (α_{i}, α_{j}) \frac{[H_{i}(u), F_{j}(u) − F_{j}(v)]_{+}}{u − v} , \end{matrix}  (2.3)

[E_{i}(u), F_{j}(v)] = −ıℏ \frac{H_{i}(u) − H_{i}(v)}{u − v} δ_{i,j} ,  (2.4)

\begin{matrix}[E_{i}(u), E_{i}(v)] = −\frac{ıℏ}{2}(α_{i}, α_{i})\frac{(E_{i}(u) − E_{i}(v))^{2}}{u − v} , \\ [F_{i}(u), F_{i}(v)] = \frac{ıℏ}{2}(α_{i}, α_{i})\frac{(E_{i}(u) − E_{i}(v))^{2}}{u − v} , \\ [E_{i}(u), E_{j}(v)] = −\frac{ıℏ}{2}(α_{i}, α_{j})\frac{[E_{i}(u), E_{j}(u) − E_{j}(v)]_{+}}{u − v} − \frac{[E_{i}^{(0)}, E_{j}(u) − E_{j}(v)]}{u − v} , \\ [F_{i}(u), F_{j}(v)] = \frac{ıℏ}{2}(α_{i}, α_{j})\frac{[F_{i}(u), F_{j}(u) − F_{j}(v)]_{+}}{u − v} − \frac{[F_{i}^{(0)}, F_{j}(u) − F_{j}(v)]}{u − v} , \\ i ≠ j, a_{ij} ≠ 0; \end{matrix}  (2.5)

\begin{matrix}∑_{σ∈𝔖_{n}}[E_{i}(u_{σ(1)}), [E_{i}(u_{σ(2)}), …, [E_{i}(u_{σ(n)}), E_{j}(v)] …]] = 0 , \\ ∑_{σ∈𝔖_{n}}[F_{i}(u_{σ(1)}), [F_{i}(u_{σ(2)}), …, [F_{i}(u_{σ(n)}), F_{j}(v)] …]] = 0 , \\ i ≠ j, n = 1 − a_{ij} , \end{matrix}  (2.6)

where [a, b]_{+} := ab + ba.

Denote Y (𝔟) ⊂ Y (𝔤) the subalgebra generated by H_{i}(u), E_{i}(u), i ∈ Γ.

There is another closely related set of generators of the Yangian which were given by Drinfeld [3] in the case Y (𝔰𝔩(ℓ + 1)). Below we propose the generalization of this description to Y (𝔤) for an arbitrary semisimple Lie algebra 𝔤. In sequel we will use the following

convention: ∏_{s=j}^{k} f_{s} = 1, for any f_{s} if k ≤ j which help to write down the formulas in a

compact way.

Lemma 2.1 Any generation series H_{i}(u) of the form (2.1) can be represented in the following form

H_{i}(u) = \frac{∏_{s≠i} ∏_{r=1}^{−a_{si}} A_{s}(u − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}))}{A_{i}(u)A_{i}(u − \frac{ıℏ}{2}(α_{i}, α_{i}))} , i = 1, …, ℓ.  (2.7)

where A_{i}(u), i = 1, …, ℓ are formal series A_{i}(u) = 1 + ∑_{s=0}^{∞} A_{i}^{(s)}u^{−s−1}.

Proof: Expanding the left and right hand sides of (2.7) in powers u^{−1} and equating the coefficients in u^{−r−1} one can check that

H_{i}^{(r)} = ∑_{s=1}^{ℓ} a_{si}A_{s}^{(r)} + f(A_{i}^{(0)}, ⋯, A_{i}^{(r−1)}).  (2.8)

Since the Cartan matrix is invertible, the statement follows by the induction procedure

Let us introduce the generating series C_{i}(u) and B_{i}(u):

B_{i}(u) = d_{i}^{1/2}A_{i}(u)E_{i}(u), C_{i}(u) = d_{i}^{1/2}F_{i}(u)A_{i}(u) , i ∈ Γ ,  (2.9)

where d_{i} = (α_{i}, α_{i})/2. Due to (2.7), the change of the generators H_{i}(u), E_{i}(u), F_{i}(u) to A_{i}(u), C_{i}(u), B_{i}(u) is invertible. It is convenient to introduce an additional set of series D_{i}(u). For any i ∈ Γ let

D_{i}(u) = A_{i}(u)H_{i}(u) + C_{i}(u)A_{i}^{−1}(u)B_{i}(u) ,  (2.10)

where H_{i}(u) are expressed in terms of A_{i}(u) in (2.7). Straightforward calculations lead to the following statement.

Proposition 2.1 For any i = 1, …, ℓ the series A_{i}(u), B_{i}(u), C_{i}(u), D_{i}(u) satisfy the relations

[A_{i}(u), A_{j}(v)] = 0 ,  (2.11)

\begin{matrix}[A_{i}(u), B_{j}(v)] = [A_{i}(u), C_{j}(v)] = 0 ,  (i ≠ j) , \\ [B_{i}(u), B_{j}(v)] = [C_{i}(u), C_{j}(v)] = 0 ,  (a_{ij} = 0, i ≠ j) , \\ [B_{i}(u), B_{i}(v)] = [C_{i}(u), C_{i}(v)] = 0 , \\ [B_{i}(u), C_{j}(v)] = 0 ,  (i ≠ j) , \end{matrix}  (2.12)

(u − v)[A_{i}(u), B_{i}(v)] = ıℏd_{i}(B_{i}(u)A_{i}(v) − B_{i}(v)A_{i}(u)) ,  (2.13)
(u − v)[A_{i}(u), C_{i}(v)] = ıℏd_{i}(A_{i}(u)C_{i}(v) − A_{i}(v)C_{i}(u)) ,  (2.14)
(u − v)[B_{i}(u), C_{i}(v)] = ıℏd_{i}(A_{i}(u)D_{i}(v) − A_{i}(v)D_{i}(u)) ,  (2.15)
(u − v)[B_{i}(u), D_{i}(v)] = ıℏd_{i}(B_{i}(u)D_{i}(v) − B_{i}(v)D_{i}(u)) ,  (2.16)
(u − v)[C_{i}(u), D_{i}(v)] = ıℏd_{i}(D_{i}(u)C_{i}(v) − D_{i}(v)C_{i}(u)) ,  (2.17)
(u − v)[A_{i}(u), D_{i}(v)] = ıℏd_{i}(B_{i}(u)C_{i}(v) − B_{i}(v)C_{i}(u)) .  (2.18)

In addition, the following analog of the quantum determinant relation holds:

\begin{matrix}A_{i}(u)D_{i}(u − \frac{ıℏ}{2}(α_{i}, α_{i})) − B_{i}(u)C_{i}(u − \frac{ıℏ}{2}(α_{i}, α_{i})) = \\ ∏_{s≠i} ∏_{r=1}^{−a_{si}} A_{s}(u − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s})) . \end{matrix}  (2.19)

The described set of relations between A_{i}(u), B_{i}(u), C_{i}(u) and D_{i}(u) is a generalization of the Y (𝔰𝔩(ℓ + 1)) relations of [18] to the case of Y (𝔤) with an arbitrary simple Lie algebra 𝔤.

By (2.11), the coefficients of the generating series A_{i}(u), i = 1, …, ℓ define a commutative subalgebra of Y (𝔤). This property will be important in the construction of the representations of the Yangian in the next section.

3 Construction of the representations of Y (𝔤) and Y (𝔟)

In this section the explicit construction of the representations of the Yangian in terms of difference operators is given. The explicit description of the representation of the Yangian in terms of difference/differential operators acting in some functional space is based on the choice of a commutative subalgebra. The elements of this subalgebra act in this representation by multiplication on functions. The proposed construction uses the subalgebra generated by A_{i}(u) as a distinguished commutative subalgebra. However, we start with explicit description of the resulting representation and then we make some comments how it could be derived starting with the representation of commutative subalgebra generated by A_{i}(u) and using the commutation relations between A_{i}(u), B_{i}(u), and C_{i}(u) described in Section 2.

Let us introduce a set of variables {γ_{i,k} ; i ∈ Γ; k = 1, …, m_{i}}, where m_{i} are arbitrary positive integer numbers and let M be the space of meromorphic functions in these variables. Let us define the following difference operators acting on M:

β_{i,k} = e^{\frac{ıℏ}{2} (α_{i},α_{i})∂_{γ_{i,k}}} .  (3.1)

where ∂_{γ_{i,k}} := \frac{∂}{∂γ_{i,k}} is a differentiation over γ_{i,k}. It is useful to arrange the variables into the set of polynomials of the formal variable u of degrees m_{i}, i ∈ Γ

P_{i}(u) = ∏_{p=1}^{m_{i}}(u − γ_{i,p}) , i ∈ Γ.  (3.2)

Consider the operators

H_{i}(u) = R_{i}(u)\frac{∏_{s≠i} ∏_{r=1}^{−a_{si}} P_{s}(u − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}))}{P_{i}(u)P_{i}(u − \frac{ıℏ}{2}(α_{i}, α_{i}))} ,  (3.3)

E_{i}(u) = d_{i}^{−1/2} ∑_{k=1}^{m_{i}} \frac{∏_{s=i+1}^{ℓ} ∏_{r=1}^{−a_{si}} P_{s}(γ_{i,k} − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}))}{(u − γ_{i,k}) ∏_{p≠k}(γ_{i,k} − γ_{i,p})} β_{i,k}^{−1} ,  (3.4)

\begin{matrix}F_{i}(u) = −d_{i}^{−1/2} ∑_{k=1}^{m_{i}} R_{i}(γ_{i,k} + \frac{ıℏ}{2}(α_{i}, α_{i})) × \\ \frac{∏_{s=1}^{i−1} ∏_{r=1}^{−a_{si}} P_{s}(γ_{i,k} − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}) + \frac{ıℏ}{2}(α_{i}, α_{i}))}{(u − γ_{i,k} − \frac{ıℏ}{2}(α_{i}, α_{i})) ∏_{p≠k}(γ_{i,k} − γ_{i,p})} β_{i,k} , \\ i = 1, …, ℓ, \end{matrix}  (3.5)

where R_{i}(u) will be specified below.

Theorem 3.1 (i). For any set of integer numbers {m_{i}} satisfying the condition l_{i} := ∑_{j=1}^{ℓ} m_{j}a_{ji} ∈ ℤ_{+}, consider the polynomials

R_{i}(u) = ∏_{k=1}^{l_{i}}(u − ν_{i,k}),  (3.6)

where {ν_{i,k} , i ∈ Γ , k = 1, …, l_{i}} is a set of arbitrary complex parameters. Then the operators (3.3)-(3.5) considered as formal power series in u^{−1}, form a representation of Y (𝔤) in the space M. This representation is parameterized by a choice of {m_{i}} obeying the above restrictions (l_{i} ≥ 0) and by arbitrary complex parameters {ν_{i,k}}.

(ii). Let {m_{i}} be arbitrary integers and R_{i}(u) be rational functions of the following form

R_{i}(u) = \frac{∏_{k_{+}=1}^{l_{i}^{+}}(u − ν_{i,k_{+}}^{+})}{∏_{k_{−}=1}^{l_{i}^{−}}(u − ν_{i,k_{−}}^{−})},  (3.7)

where {ν_{i,k_{±}}^{±} , i ∈ Γ , k = 1, …, l_{i}^{±}} is a set of arbitrary complex parameters and l_{i}^{+} − l_{i}^{−} = ∑_{j=1}^{ℓ} m_{j}a_{ji}. Then the operators (3.3)-(3.4) considered as formal power series in u^{−1}, form a representation of Y (𝔟) in the space M. This representation is parameterized by the choice of {m_{i}} and by arbitrary complex parameters {ν_{i,k}^{±}}.

Proof. To prove the theorem introduce the following difference operators

κ_{i,k} = d_{i}^{−1/2} \frac{∏_{s=i+1}^{ℓ} ∏_{r=1}^{−a_{si}} P_{s}(γ_{i,k} − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}))}{∏_{p≠k}(γ_{i,k} − γ_{i,p})} β_{i,k}^{−1} ,  (3.8)

and

κ′_{i,k} = −d_{i}^{−1/2}R_{i}(γ_{ik} + _{\frac{ıℏ}{2}}(α_{i}, α_{i}))\frac{∏_{s=1}^{i−1} ∏_{r=1}^{−a_{si}} P_{s}(γ_{i,k} − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}) + \frac{ıℏ}{2}(α_{i}, α_{i}))}{∏_{p≠k}(γ_{i,k} − γ_{i,p})} β_{i,k} . (3.9)

where R_{i}(u) are rational functions of u. It is easy to show that the operators γ_{i,k}, κ_{i,k}, κ′_{i,k} satisfy the relations

\begin{matrix}κ_{i,k}γ_{j,l} − γ_{j,l}κ_{i,k} = −\frac{ıℏ}{2}(α_{i}, α_{i})δ_{i,j}δ_{k,l}κ_{i,k} , \\ κ′_{i,k}γ_{j,l} − γ_{jl}κ′_{i,k} = \frac{ıℏ}{2} (α_{i}, α_{i})δ_{ij}δ_{kl}κ′_{i,k} , \end{matrix}  (3.10)

\begin{matrix}(γ_{i,k} − γ_{j,l} − \frac{ıℏ}{2}(α_{i}, α_{j}))κ_{i,k}κ_{j,l} = (γ_{i,k} − γ_{j,l} + \frac{ıℏ}{2}(α_{i}, α_{j}))κ_{j,l}κ_{i,k} , \\ κ′_{i,k}κ′_{j,l}(γ_{i,k} − γ_{j,l} + \frac{ıℏ}{2} (α_{i}, α_{j})) = κ′_{j,l}κ′_{i,k}(γ_{i,k} − γ_{j,l} − \frac{ıℏ}{2} (α_{i}, α_{j})) , \end{matrix}  (3.11)

[κ_{i,k}, κ′_{j,l}] = 0, i ≠ j, [κ_{i,k}, κ′_{i,l}] = 0, k ≠ l ,  (3.12)

\begin{matrix}κ_{i,k}κ′_{i,k} = −d_{i}^{−1} \frac{R_{i}(γ_{i,k}) ∏_{s≠i} ∏_{r=1}^{−a_{si}} P_{s}(γ_{i,k} − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}))}{∏_{p≠k}(γ_{i,k} − γ_{i,p}) ∏_{p≠k}(γ_{i,k} − γ_{i,p} − \frac{ıℏ}{2}(α_{i}, α_{i}))} , \\ κ′_{i,k}κ_{i,k} = −d_{i}^{−1} \frac{R_{i}(γ_{i,k} + \frac{ıℏ}{2}(α_{i}, α_{i})) ∏_{s≠i} ∏_{r=1}^{−a_{si}} P_{s}(γ_{i,k} − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}) + \frac{ıℏ}{2}(α_{i}, α_{i}))}{∏_{p≠k}(γ_{i,k} − γ_{i,p}) ∏_{p≠k}(γ_{i,k} − γ_{i,p} + \frac{ıℏ}{2}(α_{i}, α_{i}))} . \end{matrix} (3.13)

Now let us define the generators as

H_{i}(u) = R_{i}(u) \frac{∏_{s≠i} ∏_{r=1}^{−a_{si}} P_{s}(u − \frac{ıℏ}{2}(α_{i} + rα_{s}, α_{s}))}{P_{i}(u)P_{i}(u − \frac{ıℏ}{2}(α_{i}, α_{i}))} ,  (3.14)

E_{i}(u) = ∑_{k=1}^{m_{i}} \frac{1}{u − γ_{i,k}} κ_{i,k} , i ∈ Γ,  (3.15)

F_{i}(u) = ∑_{k=1}^{m_{i}} κ′_{i,k} \frac{1}{u − γ_{i,k}} , i ∈ Γ ,  (3.16)

and let R(u_{i}) be rational functions compatible with the expansion (2.1). Then the relations (2.2),(2.3), and (2.5) may be derived by straightforward calculations. If we further restrict R_{i}(u) to be polynomial functions then the additional relations (2.4) hold. To complete the proof of the theorem one should verify the relations (2.6), which forms in fact the only nontrivial part of the proof. One can see that such a verification is reduced to the following combinatorial lemma.

Lemma 3.1 Let γ_{ik}, κ_{ik}, and κ′_{ik} satisfy the relations (3.10), (3.11). Then for any n = 2, 3, 4 the following formulas holds:

\begin{matrix}∑_{σ∈𝔖_{n}}[κ_{i,k_{σ(1)}}, […, [κ_{i,k_{σ(n)}}, κ_{j,l}]]...] = \\ = (\frac{ıℏ(α_{i}, α_{i})}{2})^{n} ∏_{s=0}^{n−1}(a_{ij} + s) · ( ∑_{σ∈𝔖_{n}} κ̃_{i,k_{σ(1)}} · . . . · κ̃_{i,k_{σ(n)}}) · κ_{j,l} , \end{matrix}  (3.17)

\begin{matrix}∑_{σ∈𝔖_{n}}[κ′_{i,k_{σ(1)}}, […, [κ′_{i,k_{σ(n)}}, κ′_{j,l}]]...] = \\ = (\frac{ıℏ(α_{i}, α_{i})}{2})^{n} ∏_{s=0}^{n−1}(a_{ij} + s) · ( ∑_{σ∈𝔖_{n}} κ̃′_{i,k_{σ(1)}} · . . . · κ̃′_{i,k_{σ(n)}}) · κ′_{j,l} , \end{matrix}  (3.18)

where

\begin{matrix}κ̃_{i,k} := \frac{1}{γ_{i,k} − γ_{j,l} + \frac{ıℏ(α_{i},α_{j})}{2}} κ_{i,k} , \\ κ̃′_{i,k} := κ′_{i,k} \frac{1}{γ_{i,k} − γ_{j,l} + \frac{ıℏ(α_{i},α_{j})}{2}} . \end{matrix}  (3.19)

Proof. We outline the proof of the only non-trivial relations (3.17). Thus, we should calculate the following expression for n = 2, 3, 4

X_{n} = ∑_{σ∈𝔖_{n}}[κ_{i,k_{σ(1)}}, […, [κ_{i,k_{σ(n)}}, κ_{j,l}]]...]  (3.20)

We consider only the case of non-coincident indexes k_{i} ≠ k_{j} for any i ≠ j. The proof of the general case is quite similar. Let η_{ij} = \frac{ıℏ(α_{i},α_{j})}{2}. For n = 2, 3, 4, X_{n} may be represented as a sum of n! terms of the first type:

η_{ij}^{n} ∏_{k=1}^{n}(γ_{i,k} − γ_{j,m})^{−1}[κ_{i,k_{σ(1)}}, […, [κ_{i,k_{σ(n)}}, κ_{j,l}]_{+}]_{+}...]_{+}  (3.21)

and n^{n} − n! terms of the second type:

\begin{matrix}η_{ii}^{s}η_{ij}^{n−s} ∏_{k=1}^{s}(γ_{i,k} − γ_{j,m})^{−1} ∏_{α,β}(γ_{i,α} − γ_{i,β})^{−1} × \\ [κ_{i,k_{σ(1)}}, […, κ_{j,l}[κ_{i,k_{σ(α)}}, κ_{i,k_{σ(β)}}]]_{+}]_{+}...]_{+} . \end{matrix}  (3.22)

One can reduce the terms of the second type to \binom{n}{2} terms of the first type as follows. Given I_{n} := {1, …, n} and the set of variables {a_{i}, i ∈ I}, d the following iterative formula holds.

∏_{i∈I_{n}}(a_{i} − d)^{−1} = ∑_{r∈I_{n}}(a_{r} − d)^{−1} ∏_{i∈I_{n}\{r}}(a_{i} − a_{r})^{−1}.

The left hand side ∏_{s=1}^{n}(a_{s} − d)^{−1} of this formula is exactly of the first type for a_{s} := γ_{i,k_{s}} and

d := γ_{j,l}. The iterations on the right hand side coincide with all the terms of the second type (3.22) and thus we are left with only the terms of the first type. The simple transformations then lead to (3.17).

Finally, let us briefly explain how these representations naturally arise from the relations (2.11)-(2.19). Due to (2.11) A_{i}^{(s)}, i ∈ Γ, s = 0, 1, 2, … generate a commutative subalgebra of Y (𝔤). We would like to construct the representation in the space of functions of the finite collection of the variables {γ_{ik}} such that A_{i}^{(s)} act through the multiplication by certain functions of {γ_{ik}}. It is natural to look for the representation of A_{i}(u) in the form A_{i}(u) = X_{i}(u)P_{i}(u) where P_{i}(u) are given by (3.2) and X_{i}(u) = 1 + ∑_{s=0}^{∞} X_{i}^{(s)}u^{−1−s} are some γ_{ik}- independent series. From the commutation relations (2.11)-(2.19) one derives that B_{i}(γ_{i,k}) and (C_{i}(γ_{i,k}))^{−1} are proportional to the shift operator (3.1). Therefore, by (2.9) the residues of E_{i}(u) and F_{i}(u) are proportional to the B_{i}(γ_{i,k}) and C_{i}(γ_{i,k}) respectively. This explains the ansatz (3.15), and (3.16) for the generators E_{i}(u) and F_{i}(u).

4 Symplectic leaves of the Yangian and the monopole moduli spaces

In this section we describe the Poisson geometry relevant to the description of the Yangian representations proposed above. It appears that this leads to the direct connection with moduli spaces of G-monopoles such that Lie(G) = 𝔤.

Let Y_{cl}(𝔤) and Y_{cl}(𝔟) be the Poisson algebras corresponding to the classical limit of Y (𝔤) and Y (𝔟) in the sense of [20], [21]. The elements Y_{cl}(𝔤) may be described as functions on the formal loop group LG_{−} based at the trivial loop g(u) = e where e ∈ G is unity element. We use its parameterization in terms of the infinite series of the form F (u) = ∑_{s=0}^{∞} F^{(s)}u^{−1−s}.

The description of the Poisson algebra Y_{cl}(𝔤) in terms of the generators and relations can be obtained from (2.2)-(2.6) by taking the limit ℏ → 0:

{h_{i}(u), h_{j}(v)} = 0 ,  (4.1)

\begin{matrix}\{h_{i}(u), e_{j}(v)\} = − (α_{i}, α_{j}) \frac{h_{i}(u)(e_{j}(u) − e_{j}(v))}{u − v} , \\ \{h_{i}(u), f_{j}(v)\} = (α_{i}, α_{j}) \frac{h_{i}(u)(f_{j}(u) − f_{j}(v))}{u − v} , \end{matrix}  (4.2)

\{e_{i}(u), f_{j}(v)\} = − δ_{ij} \frac{h_{i}(u) − h_{i}(v)}{u − v} ,  (4.3)

\begin{matrix}\{e_{i}(u), e_{i}(v)\} = −(α_{i}, α_{i})\frac{(e_{i}(u) − e_{i}(v))^{2}}{u − v} , \\ \{f_{i}(u), f_{i}(v)\} = (α_{i}, α_{i})\frac{(e_{i}(u) − e_{i}(v))^{2}}{u − v} , \\ \{e_{i}(u), e_{j}(v)\} = −(α_{i}, α_{j})\frac{e_{i}(u)(e_{j}(u) − e_{j}(v))}{u − v} − \frac{\{e_{i}^{(0)}, (e_{j}(u) − e_{j}(v))\}}{u − v}, \\ \{f_{i}(u), f_{j}(v)\} = (α_{i}, α_{j})\frac{f_{i}(u)(f_{j}(u) − f_{j}(v))}{u − v} − \frac{\{f_{i}^{(0)}, (f_{j}(u) − f_{j}(v))\}}{u − v}, \\ i ≠ j, a_{ij} ≠ 0; \end{matrix}  (4.4)

\begin{matrix}∑_{σ∈𝔖_{n}}\{e_{i}(u_{σ(1)}), \{e_{i}(u_{σ(2)}), …, \{e_{i}(u_{σ(n)}), e_{j}(v)\} …\}\} = 0 , \\ ∑_{σ∈𝔖_{n}}\{f_{i}(u_{σ(1)}), \{f_{i}(u_{σ(2)}), …, \{f_{i}(u_{σ(n)}), f_{j}(v)\} …\}\} = 0 , \\ n = 1 − a_{ij}, (i ≠ j) . \end{matrix}  (4.5)

There exists the following simple interpretation of the generators h_{i}(u), e_{i}(u), f_{i}(u). Let us fix the Gauss decomposition of an element g(u) ∈ LG_{−}:

g(u) = exp ( ∑_{α} f_{α}(u)F_{α}) · exp ( ∑_{i=1}^{ℓ} ϕ_{i}(u)H_{i}) · exp ( ∑_{α} e_{α}(u)E_{α}) ,  (4.6)

where H_{i}, F_{α}, E_{α} provide a basis of 𝔤 labeled by positive roots α, and ϕ_{i}(u), f_{α}(u), e_{α}(u) are the local exponential coordinates on the group LG_{−}. Note that we consider the functions on the formal loop group LG_{−} (i.e. we deal with functions on the formal neighbourhood of e ∈ LG_{−}) and thus the Gauss decomposition (4.6) is valid ”everywhere”. The functions e_{i}(u) := e_{α_{i}}(u), f_{i}(u) := f_{α_{i}}(u) corresponding to the simple roots α_{i} together with h_{i}(u) = exp{− ∑_{j=1}^{ℓ} a_{ji}ϕ_{j}(u)} give us a set of the generators satisfying the relations (4.1)- (4.5). The local coordinates ϕ_{i}(u), e_{i}(u), f_{i}(u) may be expressed explicitly in terms of the matrix elements of the fundamental representations of U(𝔤). Let {π_{i}} be a set of fundamental representations corresponding to the fundamental weights {ω_{i}} of 𝔤 and v_{+/−}^{(i)} be the highest/lowest vectors in these representation. Denote by a_{i}(u), b_{i}(u), c_{i}(u), d_{i}(u) the following formal series

\begin{matrix}a_{i}(u) = ⟨v_{−}^{(i)}|π_{i}(g(u))|v_{+}^{(i)}⟩ , \\ b_{i}(u) = ⟨v_{−}^{(i)}|π_{i}(g(u))π_{i}(F_{i})|v_{+}^{(i)}⟩ , \\ c_{i}(u) = ⟨v_{−}^{(i)}|π_{i}(E_{i})π_{i}(g(u))|v_{+}^{(i)}⟩ , \\ d_{i}(u) = ⟨v_{−}^{(i)}|π_{i}(E_{i})π_{i}(g(u))π_{i}(F_{i})|v_{+}^{(i)}⟩ . \end{matrix}  (4.7)

Lemma 4.1 The coordinates corresponding to the simple roots and Cartan elements entering the Gauss decomposition (4.6) can be expressed through the matrix elements (4.7) as follows

\begin{matrix}e^{ϕ_{i}(u)} = a_{i}(u) , \\ e_{i}(u) = \frac{b_{i}(u)}{a_{i}(u)} , \\ f_{i}(u) = \frac{c_{i}(u)}{a_{i}(u)} . \end{matrix}  (4.8)

The variables a_{i}(u), b_{i}(u), c_{i}(u), d_{i}(u) are the classical counterparts of the variables A_{i}(u), B_{i}(u), C_{i}(u), D_{i}(u) defined in (2.7),(2.9),(2.10). A similar description of Y (𝔤𝔩(ℓ + 1)) was given in [3] (see also [19], [22]). In this case the functions a_{i}(u), b_{i}(u), c_{i}(u), d_{i}(u) are given by the minors of the matrix π_{∗}(g(u)) where π_{∗} is the tautological representation π_{∗} : 𝔤𝔩(ℓ + 1) → End(ℂ^{ℓ+1}), which agrees with (4.7). Let us remark that (4.7) may be considered as a classical counterpart of the universal R-matrix map of the dual Hopf algebra with the opposite comultiplication A^{0} to the Hopf algebra A (see [2] for details). Together with the explicit Gauss form of the universal R-matrix this should provide the quantum version of Lemma 4.1. The simplest example of Y (𝔰𝔩(2)) may be extracted from [17].

Now consider the classical counterpart of the Yangian representations constructed in the Section 3. These representations have the following property: the images of E_{i}(u) are rational operator-valued functions in u with simple poles. Given the Poisson brackets (4.1)-(4.5) on LG_{−}, the representations of the Yangian correspond to the symplectic leaves in LG_{−}. A similar description holds for Y_{cl}(𝔟) in terms of the symplectic leaves in LB_{−}. We define the symplectic leaf O to be rational if the restriction of the generators e_{i}(u) is a rational function over u and let O^{(0)} ⊂ O be an open part corresponding to e_{i}(u) having only simple poles. Thus the symplectic leaves corresponding to the representations constructed in Section 3 are rational. One can describe the symplectic leaves as symplectic manifolds as follows. Open parts O^{(0)} of the rational symplectic leaves in LB_{−} corresponding to the representations constructed in Theorem 3.1 are isomorphic (as abstract manifolds) to the open subsets of the space of M̃_{b}(m) of the based rational maps

e = (e_{1}, ⋯, e_{ℓ}) : (ℙ^{1}, ∞) ⟶^{e} (\underbrace{ℙ^{1} × ⋯ × ℙ^{1}}_{ℓ} , 0 × ⋯ × 0),  (4.9)

of the fixed multi-degree m = (m_{1}, ⋯, m_{ℓ}) where e_{i}(u) are the generators of Y_{cl}(𝔟) corresponding to simple roots. Analogously open parts O^{(0)} of the rational symplectic leaves of LG_{−} corresponding to the representations constructed in Theorem 3.1 are isomorphic to the open subsets of M̃_{b}(m) with additional restrictions ∑_{j=1}^{ℓ} m_{j}a_{ji} = l_{i} ∈ ℤ_{+}.

Taking into account the results of [15] one can reformulate the description of the symplectic leaves as follows. Consider the space M(m) of the holomorphic maps ℙ^{1} → G/B of multi-degree m = (m_{1}, …, m_{ℓ}) ∈ Λ_{W}^{∨}, where Λ_{W}^{∨} = H_{2}(G/B, ℤ) is the co-weight lattice of 𝔤. It will be useful to consider G/B as a manifold parameterizing the Borel subgroups in G. Choose some Borel subgroup B_{+} and let b_{+} ⊂ G/B be the corresponding point in the flag manifold. Let us fix the local coordinate on ℙ^{1} and consider the evaluation map ev_{∞} : M(m) ⟶ G/B defined as ev_{∞} : f → f(∞). Thus M(m) is supplied with the structure of the fibred space over G/B and the fibre is naturally identified with the moduli space M_{b}(m) of the based holomorphic maps f : (ℙ^{1}, ∞) → (G/B, b_{+}) of the multi-degree m. It appears that the open part of M_{b}(m) can be naturally identified with the moduli space M̃_{b}(m) introduced above. Actually this follows from the results of Drinfeld in the form presented in [15]. Thus it was shown in [15] that M_{b}(m) is a smooth manifold of dimension

dim M_{b}(m) = 2|m| = 2(m_{1} + … + m_{ℓ}).

The explicit description of the manifold M_{b}(m) can be obtained by generalizing the classical Plücker embedding of G/B into the product ∏_{i⊂Γ} ℙ(V_{ω_{i}}) of the projectivisations of the fundamental representations V_{ω_{i}} as follows. Let π_{λ} : U(𝔤) → End(V_{λ}) be the irreducible representation of the universal enveloping algebra U(𝔤) with the highest weights λ and V_{λ} = V_{λ} ⊗ O_{ℙ^{1}} be the corresponding trivial vector bundles on ℙ^{1}. Using the local coordinate we identify Γ(𝔸^{1}, V_{λ}) = V_{λ} ⊗ ℂ[u]. Denote by v_{+}^{λ} the highest weight vectors in V_{λ} with respect to the Borel subgroup B_{+}. Similarly let v_{−}^{λ} be the lowest weight vector in the dual representation V_{λ}^{∨} normalized by the condition ⟨v_{−}^{λ}|v_{+}^{λ}⟩ = 1. We also introduce the additional set of vectors π_{λ}(F_{i})v_{+}^{λ}, π_{λ}(E_{i})v_{−}^{λ} where F_{i} and E_{i} are the generators corresponding to the simple roots α_{i}. Thus given a section v^{λ} ∈ Γ(𝔸^{1}, V_{λ}) we have the following decomposition v^{λ}(u) = a_{λ}(u) · v_{+}^{λ} + ∑_{i∈Γ} b_{λ}^{i}(u)π_{λ}(F_{i})v_{+}^{λ} + ϕ^{λ}(u) with ϕ^{λ} ⊂ Γ(𝔸^{1}, V_{λ}) satisfying ⟨v_{−}^{λ}|ϕ^{λ}⟩ = ⟨v_{−}^{λ}|π_{λ}(E_{i})|ϕ^{λ}⟩ = 0.

According to Drinfeld (see [15] for the details) the moduli space M_{b}(m) is isomorphic to the space Z_{m} of the collections of sections v^{λ}(u) for each λ ∈ Λ_{W}^{+} satisfying the conditions

1. The polynomial a_{λ}(u) is monic of degree ⟨m, λ⟩;

2. The degree of (v^{λ} − a_{λ}(u)v_{+}^{λ}) is strictly less then ⟨m, λ⟩;

3. For any G-equivariant morphism ϕ : V_{λ} ⊗ V_{μ} → V_{ν} such that ν = μ + λ and the conjugated morphism satisfies ϕ^{∗}(v_{−}^{ν}) = v_{−}^{λ} ⊗ v_{−}^{μ} we have ϕ(v^{λ} ⊗ v^{μ}) = v^{ν};

4. For any G-equivariant morphism ϕ : V_{λ} ⊗ V_{μ} → V_{ν} such that ν < μ + λ we have ϕ(v^{λ} ⊗ v^{μ}) = 0.

It is easy to see that the set {v^{λ}, λ ∈ Λ_{W}^{+} } satisfying these conditions is determined by its subset {v^{ω_{i}}} corresponding to fundamental representations ω_{i}. Moreover, given arbitrary polynomials a_{ω_{i}}(u) and b_{ω_{i}}^{i}(u) satisfying the conditions (1) and (2) above (i.e. a_{ω_{i}}(u) are monic and deg(a_{ω_{i}}) = deg(b_{ω_{i}}^{i}) + 1 = m_{i}) there exist such ϕ^{ω_{i}}(u) that for v^{ω_{i}}(u) = a_{ω_{i}}(u) · v_{+}^{ω_{i}} + b_{ω_{i}}^{i}(u)π_{ω_{i}}(F_{i})v_{+}^{ω_{i}} + ϕ^{ω_{i}}(u) the conditions (3) and (4) hold (see [15], [10] for details). Let us consider the subset of the polynomials a_{ω_{i}} and b_{ω_{i}} such that the roots γ_{i,k} of a_{ω_{i}}(u) do not coincide γ_{i,k} ≠ γ_{j,l} for (i, k) ≠ (j, l). The space of such polynomials a_{i}(u) ≡ a_{ω_{i}}(u), b_{i}(u) ≡ b_{ω_{i}}^{i}(u) is 2|m|-dimensional and thus is isomorphic to the open subspace in the moduli space M_{b}(m). Note that the polynomials a_{i}(u) and b_{i}(u) define the map e ∈ M̃_{b}(m)

(ℙ^{1}, ∞) ⟶^{e} (\underbrace{ℙ^{1} × … × ℙ^{1}}_{ℓ} , 0 × … × 0),

given by

e(u) = (b_{1}(u)/a_{1}(u)) × … × (b_{ℓ}(u)/a_{ℓ}(u)).

Therefore we have established the isomorphism of the open parts of the moduli spaces

ϕ : M_{b}(m) ⟶ M̃_{b}(m).

One can summarize this in the following

Proposition 4.1 (i) The open parts O^{(0)} of the rational symplectic leaves of Y_{cl}(𝔟) corresponding to the representations constructed in Theorem 3.1 are isomorphic to the open parts of the spaces of the based maps

(ℙ^{1}, ∞) → (G/B, b_{+})  (4.10)

of the fixed multi-degree m = (m_{1}, ⋯, m_{ℓ}) ∈ H_{2}(G/B, ℤ).

(ii) The open parts O^{(0)} of the rational symplectic leaves of Y_{cl}(𝔤) corresponding to the representations constructed in Theorem 3.1 are isomorphic to the spaces of the based maps (4.10) with additional restrictions ∑_{j=1}^{ℓ} m_{j}a_{ji} = l_{i} ∈ ℤ_{+}.

The connection with the explicit parameterization used in the previous sections is as follows. Let us parameterize the open subset U ⊂ M̃_{b}(m) by the following étale coordinates

(x_{i,k}, y_{i,k}), i = 1, …, ℓ, k = 1, …, m_{i}

defined by the conditions

a_{i}(x_{i,k}) = 0, y_{i,k} = b_{i}(x_{i,k}).

Then the coordinates (x_{ik}, y_{ik}) are related to the coordinates (γ_{i,k}, κ_{i,k}) by the simple redefinition

x_{i,k} = γ_{i,k} ,  (4.11)
y_{i,k} = κ_{i,k} ∏_{s≠k}(γ_{i,k} − γ_{i,s}) .  (4.12)

Note that the classical limit of the relations (3.10),(3.11) provides the open part of the space M_{b}(m) with the holomorphic symplectic structure

{γ_{i,k}, γ_{j,l}} = 0,  (4.13)
\{γ_{i,k}, κ_{j,l}\} = \frac{1}{2}(α_{i}, α_{i})δ_{i,j}δ_{k,l}κ_{j,l} ,  (4.14)
\{κ_{i,k}, κ_{j,l}\} = (α_{i}, α_{j})\frac{κ_{i,k}κ_{j,l}}{γ_{i,k} − γ_{j,l}} , (i, k) ≠ (j, l),  (4.15)

or equivalently in the coordinates (x_{i,k}, y_{i,k})

{x_{i,k}, x_{j,l}} = 0,  (4.16)
{y_{i,k}, y_{i,l}} = 0,  (4.17)
\{x_{i,k}, y_{j,l}\} = \frac{1}{2}(α_{i}, α_{i})δ_{i,j}δ_{k,l}y_{j,l},  (4.18)
\{y_{i,k}, y_{j,l}\} = (α_{i}, α_{j})\frac{y_{i,k}y_{j,l}}{x_{i,k} − x_{j,l}} , i ≠ j.  (4.19)

In general the symplectic leaves of the Poisson-Lie group G are connected components of the intersections of the double cosets of the Poisson-Lie dual G^{∗} in G × G with the diagonal G ⊂ G × G [21]. We are going to discuss the connection of this description with the algebrogeometric description considered above in the separate publication.

It appears that description of the symplectic leaves of the Poisson-Lie groups associated with the Yangian given in this section provides a direct connection with the moduli spaces of the G-monopoles with the maximal symmetry breaking. These moduli spaces are also given by the spaces of the based maps ℙ^{1} → G/B [23], [24], [25]. Our construction provides the holomorphic symplectic structure on these spaces. The explicit description of the holomorphic symplectic structure on the moduli space of the monopoles was given in the case of G = SU(N) in [9] (generalizing the results for SU(2) of [8]) and for general case in [10]. It turns out that our description of the coordinates on the moduli space and the expression for the Poisson structure in coordinates (x_{i,k}, y_{i,k}) exactly matches the description given in [10]. Thus the representation constructed in Section 2 can be considered as a quantization of the moduli space of the monopoles.

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