Running title: Representations for Toroidal Lie Algebras

PRINCIPAL VERTEX OPERATOR REPRESENTATIONS FOR TOROIDAL LIE ALGEBRAS

Yuly Billig *

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, N.B., E3B 5A3, Canada

We introduce the principal vertex operator representations for the toroidal Lie algebras generalizing the construction for the affine Kac-Moody algebras. We also represent the derivations of the toroidal algebras and introduce analogues of the Sugawara operators.

* E-mail address: ybillig@unb.ca I. INTRODUCTION.

Vertex operators discovered by physicists in string theory have turned out to be important objects in mathematics. One can use vertex operators to construct various realizations of the irreducible highest weight representations for affine Kac-Moody algebras. Two of these, the principal and homogeneous realizations, are of particular interest. The principal vertex operator construction for the affine algebra A_{1}^{(1)} allows one to construct soliton solutions of the Korteweg - de Vries hierarchy of partial differential equations. On the other hand, the homogeneous realization is linked to the fundamental nonlinear Schrödinger equation ^{1} .

S.Eswara Rao and R.V. Moody ^{2} studied the homogeneous vertex operator construction for toroidal Lie algebras. The present paper is devoted to the principal realization. Here we construct the principal vertex operator representation for the toroidal Lie algebra 𝔤̂ which is a universal central extension of 𝔤̃ = 𝔤̇ ⊗ ℂ[t_{0}^{±}, …, t_{n}^{±}], in which 𝔤̇ is a simplylaced simple finite-dimensional Lie algebra over ℂ. This generalizes the principal vertex operator realization of the basic representations of affine Lie algebras ^{3,4} .

We add the Lie algebra D of vector fields on a torus to the toroidal Lie algebra 𝔤̂ to form a larger algebra 𝔤. This is necessary in order to have a sufficiently large principal Heisenberg subalgebra. To construct a representation of 𝔤 we consider the standard module F for the principal Heisenberg subalgebra and then extend the action to all of 𝔤 by means of the vertex operators. The module F is irreducible as a module over 𝔤, and it is reducible over 𝔤̂. It will be interesting to see how this representation fits in the framework of the Verma modules over the toroidal Lie algebras ^{5} .

In a subsequent article ^{6} we construct an extension of the KdV hierarchy using the representations introduced in the present paper.

The vertex operator representations for the toroidal Lie algebras constructed here may also be useful for quantum field theories in space-time of more than two dimensions ^{7,8} .

This paper is organized as follows. In Section II we recall the construction of the toroidal Lie algebra and in Section III we sketch the principal vertex operator realization of the basic representations of affine Lie algebras. In Section IV we present the principal Heisenberg subalgebra of 𝔤 and define its standard module. We also introduce there vertex operators that will allow us to extend this representation to 𝔤. At the end of the section we state the main theorem of the paper. The proof of the main theorem occupies Section V. In the final section we construct the analogues of the Sugawara operators.

II. DEFINITIONS AND NOTATIONS.

Let 𝔤̇ be a simple finite-dimensional complex Lie algebra of type A_{ℓ}, D_{ℓ} or E_{ℓ} (i.e., simply-laced). The Killing form (·|·) is non-degenerate on the Cartan subalgebra 𝔥̇ of 𝔤̇ and induces the map ν : 𝔥̇ → 𝔥̇^{∗}. Let Δ̇ be the root system of 𝔤̇. Since 𝔤̇ is simply-laced then (α|α) = 2 for all nonzero roots α ∈ Δ̇.

To construct a toroidal algebra let us consider a tensor product of 𝔤̇ with the algebra of Laurent polynomials in n + 1 variable:

𝔤̃ = 𝔤̇ ⊗ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}].

We define the toroidal Lie algebra corresponding to 𝔤̇ as the universal central extension of 𝔤̃. The explicit construction of this extension ^{9,10} can be given as follows. Let K̇ be an (n + 1)-dimensional space with the basis {K_{0}, K_{1}, …, K_{n}}. Consider a derivation d_{p} = t_{p} \frac{d}{dt_{p}} of the algebra ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}] which can be naturally extended on the tensor product

K̃ = K̇ ⊗ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}].

The space of the universal central extension of 𝔤̃ is

K = K̃/dK̃,

where

dK̃ = {∑_{p=0}^{n} K_{p} ⊗ d_{p}(f)|f ∈ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}]} ⊂ K̃.

We will denote the image of K_{p} ⊗ t_{0}^{r_{0}}t^{r} in K by t_{0}^{r_{0}}t^{r}K_{p}, where r = (r_{1}, … r_{n}) and t^{r} = t_{1}^{r_{1}} … t_{n}^{r_{n}}. Note that K has the defining relations

r_{0}t_{0}^{r_{0}}t^{r}K_{0} + r_{1}t_{0}^{r_{0}}t^{r}K_{1} + … + r_{n}t_{0}^{r_{0}}t^{r}K_{n} = 0.

The toroidal algebra is the Lie algebra

𝔤̂ = 𝔤̇ ⊗ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}] ⊕ K

with the bracket

[g_{1} ⊗ f_{1}(t_{0} … t_{n}), g_{2} ⊗ f_{2}(t_{0} … t_{n})] = [g_{1}, g_{2}] ⊗ (f_{1}f_{2}) + (g_{1}|g_{2}) ∑_{p=0}^{n} d_{p}(f_{1})f_{2}K_{p} (2.1)

and

[𝔤̂, K] = 0.  (2.2)

Finally, we shall add certain derivations to 𝔤̂. Specifically, let D be the Lie algebra of derivations of ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}]:

D = {∑_{p=0}^{n} f_{p}(t_{0}, …, t_{n})d_{p}|f_{0}, …, f_{n} ∈ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}]} .

It is a general fact, that a derivation acting on a Lie algebra can be lifted in a unique way to a derivation of the universal central extension of this Lie algebra ^{11}. Thus the natural action of D on 𝔤̃

f_{1}(t_{0}, …, t_{n})d_{p} (g ⊗ f_{2}(t_{0}, …, t_{n})) = g ⊗ f_{1}d_{p}(f_{2})  (2.3)

has a unique extension to 𝔤̂. We shall denote the lift of f(t_{0}, …, t_{n})d_{p} by f(t_{0}, …, t_{n})D_{p}. Its action on the subspace 𝔤̃ is unchanged, while the action on K is given by the formula ^{2}:

f_{1}D_{a}(f_{2}K_{b}) = f_{1}d_{a}(f_{2})K_{b} + δ_{ab} ∑_{p=0}^{n} f_{2}d_{p}(f_{1})K_{p}.  (2.4)

Consider a subalgebra D_{+} in D:

D_{+} = {∑_{p=1}^{n} f_{p}(t_{0}, …, t_{n})D_{p}|f_{1}, …, f_{n} ∈ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}]} .

We will be working with the algebra 𝔤 which is a deformation of the semidirect product of 𝔤̂ with D_{+}:

𝔤 = 𝔤̇ ⊗ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}] ⊕ K ⊕ D_{+}.

Here multiplication in 𝔤̂ is given by (2.1) and (2.2), the action of D_{+} on 𝔤̃ is given by (2.3) and on K by (2.4). The Lie bracket in D_{+} is the following:

[t_{0}^{r_{0}}t^{r}D_{a}, t_{0}^{m_{0}}t^{m}D_{b}] = m_{a}t_{0}^{r_{0}+m_{0}}t^{r+m}D_{b}−r_{b}t_{0}^{r_{0}+m_{0}}t^{r+m}D_{a}−m_{a}r_{b}{∑_{p=0}^{n} r_{p}t_{0}^{r_{0}+m_{0}}t^{r+m}K_{p}}.

The last term in this formula is a 2-cocycle with values in a non-trivial D-module K. This abelian extension of the Lie algebra of vector fields on a torus is a generalization of the Virasoro algebra and was introduced by Rao and Moody ^{2} .

III. PRINCIPAL REALIZATION OF THE BASIC REPRESENTATION OF AFFINE LIE ALGEBRA.

When n = 0 the algebra 𝔤 constructed above yields the derived affine Kac-Moody algebra. We set n = 0 throughout this section. In this case K is one-dimensional and is spanned by K_{0}, while D_{+} is trivial, so

𝔤 = 𝔤̇ ⊗ ℂ[t_{0}, t_{0}^{−1}] ⊕ ℂK_{0}.

Lepowsky and Wilson ^{3} constructed the principal realization of the basic representation of affine Kac-Moody algebra A_{1}^{(1)}, while the construction for a general affine algebra was given by Kac, Kazhdan, Lepowsky and Wilson ^{4}. Let us recall this construction (see also Ref.1 for details). Its starting point is the principal Heisenberg subalgebra in 𝔤.

Let {α_{1}, …, α_{ℓ}} be simple roots and θ be the highest positive root in Δ̇. We define the height for a root β = ∑_{i=1}^{ℓ} k_{i}α_{i} as

ht(β) = ∑_{i=1}^{ℓ} k_{i}.

Note that ht(θ) = h − 1, where h is a Coxeter number of 𝔤̇ ^{12} .

Consider the principle gradation of 𝔤̃ defined by

deg(𝔤̇^{α_{1}} ⊗ 1) = … = deg(𝔤̇^{α_{ℓ}} ⊗ 1) = deg(𝔤̇^{−θ} ⊗ t_{0}) = 1,
deg(𝔤̇^{−α_{1}} ⊗ 1) = … = deg(𝔤̇^{−α_{ℓ}} ⊗ 1) = deg(𝔤̇^{θ} ⊗ t_{0}^{−1}) = −1.

We choose nonzero root vectors e_{0} ∈ 𝔤̇^{−θ} ⊗ t_{0}, e_{1} ∈ 𝔤̇^{α_{1}} ⊗ 1, …, e_{ℓ} ∈ 𝔤̇^{α_{ℓ}} and form the element \overline{e} = ∑_{i=0}^{ℓ} e_{i} ∈ 𝔤̃. Since \overline{e} is of degree 1 then its centralizer 𝔰̃ in g̃ is homogeneous with respect to the gradation.

Next, consider a projection

π : 𝔤̇ ⊗ ℂ[t_{0}, t_{0}^{−1}] → 𝔤̇ by t_{0} ↦ 1.

Note that deg(𝔤̇^{−θ} ⊗ t_{0}) = 1 and deg(𝔤̇^{−θ} ⊗ 1) = ht(−θ) = −h + 1, while π(𝔤̇^{−θ} ⊗ t_{0}) = π(𝔤̇^{−θ} ⊗ 1), and thus the principal ℤ-gradation of 𝔤̃ induces the ℤ_{h}-gradation of 𝔤̇:

𝔤̇ = ∑_{j∈ℤ_{h}} 𝔤̇_{j}.

The element π(\overline{e}) is regular in 𝔤̇, hence its centralizer 𝔰̇ = π(𝔰̃) in 𝔤̇ is a Cartan subalgebra, which implies that both 𝔰̇ and 𝔰̃ = 𝔰̇ ⊗ ℂ[t_{0}, t_{0}^{−1}] are abelian.

The preimage 𝔰 = 𝔰̇ ⊗ ℂ[t_{0}, t_{0}^{−1}] ⊕ ℂK_{0} of 𝔰̃ in 𝔤 is an infinite-dimensional Heisenberg algebra which is called the principal Heisenberg subalgebra of 𝔤. This subalgebra is the cornerstone of the construction of the vertex operator representation of 𝔤. The scheme of this construction is the following: one starts with the standard irreducible representation of 𝔰 and then magically one can lift this representation to the whole of 𝔤. Moreover the action of 𝔤 is prescribed and given by vertex operators. Finally one has only to check in one way or another that this construction works.

The irreducible representation of 𝔤 constructed in this way is of the highest weight Λ_{0}, where the linear functional Λ_{0} is given (we consider the simply-laced case) by

Λ_{0}(ν^{−1}(α_{i}) ⊗ t_{0}^{0}) = 0, Λ_{0}(K_{0}) = 1.

This highest weight representation is called the basic representation of 𝔤.

The Cartan subalgebra 𝔰̇ is homogeneous with respect to the principle ℤ_{h}-gradation:

𝔰̇ = ∑_{j∈ℤ_{h}} 𝔰̇_{j}.

Since (𝔤̇_{i}|𝔤̇_{j}) = 0 unless i + j = 0 (mod h) and (·|·) is nondegenerate on 𝔰̇ then one can choose a basis in 𝔰̇: {T_{1}, T_{2}, …, T_{ℓ}} such that T_{i} ∈ 𝔤̇_{m_{i}}, where 1 ≤ m_{1} ≤ m_{2} ≤ … ≤ m_{ℓ} < h and

(T_{i}|T_{ℓ+1−j}) = hδ_{ij}.

The numbers {m_{1}, …, m_{ℓ}} are called the exponents of 𝔤̇ ^{12} . Note that 𝔰̇ ∩ 𝔥̇ = (0), and therefore zero is not an exponent. Also, m_{ℓ+1−i} = h − m_{i}.

The principal Heisenberg subalgebra 𝔰 is spanned by T_{i} ⊗ t_{0}^{j}, j ∈ ℤ, i = 1, …, ℓ and K_{0}. We can now write the multiplication in 𝔰 explicitly:

[T_{i_{1}} ⊗ t_{0}^{j_{1}}, T_{ℓ+1−i_{2}} ⊗ t_{0}^{j_{2}}] = hj_{1}δ_{i_{1}i_{2}}δ_{j_{1},−j_{2}}.

It turns out that it is more convenient to work with a slightly different realization of 𝔤 based on the ℤ_{h}-gradation of 𝔤̇. Consider

𝔤_{s} = ∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j} ⊕ ℂK_{0},

with the Lie bracket

[g_{1} ⊗ s^{i}, g_{2} ⊗ s^{j}] = [g_{1}, g_{2}] ⊗ s^{i+j} + \frac{i}{h}(g_{1}|g_{2})δ_{i,−j}K_{0},

[K_{0}, 𝔤_{s}] = 0.

The proof of the following lemma is straightforward, so we omit it.

Lemma 1. The Lie algebras 𝔤 and 𝔤_{s} are isomorphic and the isomorphism

ψ : 𝔤̇ ⊗ ℂ[t_{0}, t_{0}^{−1}] ⊕ ℂK_{0} → ∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j} ⊕ ℂK_{0}

is given by

ψ(e_{α} ⊗ t^{i}) = e_{α} ⊗ s^{ht(α)+ih},

ψ(ν^{−1}(α) ⊗ t^{i}) = ν^{−1}(α) ⊗ s^{ih} + δ_{i,0} \frac{ht(α)}{h}K_{0},

ψ(K_{0}) = K_{0}.

We shall identify 𝔤 and 𝔤_{s} using this isomorphism.

In order to describe a basis in the principal Heisenberg subalgebra ψ(𝔰) we need to introduce the sequence {b_{i}}_{i∈ℤ} such that b_{i+jℓ} = m_{i} + jh for j ∈ ℤ, i = 1, …, ℓ. Then ψ(𝔰) is spanned by {T_{i} ⊗ s^{b_{i}}, K_{0}} , i ∈ ℤ. Note that b_{1−i} = −b_{i}.

Let Δ̇_{s} be the root system of 𝔤̇ with respect to the Cartan subalgebra 𝔰̇. For α ∈ Δ̇_{s} choose a root element

A^{α} = ∑_{j∈ℤ_{h}} A_{j}^{α}.

Since π(\overline{e}) is a regular element of the Cartan subalgebra then [π(\overline{e}), A^{α}] = λ^{α}A^{α} with λ^{α} ≠ 0. Hence A_{j}^{α} = (\frac{adπ(\overline{e})}{λ^{α}})^{j}A_{0}^{α} and all the components A_{j}^{α} are nonzero. We let the indices i, j in T_{i} and A_{j}^{α} run over ℤ by setting T_{i_{1}} = T_{i_{2}} if i_{1} ≡ i_{2}(mod ℓ) and A_{j_{1}}^{α} = A_{j_{2}}^{α} if j_{1} ≡ j_{2}(mod h).

Define constants λ_{i}^{α} = α(T_{i}). Then [T_{i}, A^{α}] = λ_{i}^{α}A^{α} and [T_{i}, A_{j}^{α}] = λ_{i}^{α}A_{j+m_{i}}^{α}.

Consider an automorphism σ of 𝔤̇ such that

σ = ζ^{j}Id on 𝔤̇_{j},

where ζ ∈ ℂ is a primitive h-root of 1. For a root element A^{α} = ∑_{j=1}^{h} A_{j}^{α} we have

[T_{i}, σ(A^{α})] = σ([σ^{−1}(T_{i}), A^{α}]) = σ([ζ^{−m_{i}}T_{i}, A^{α}]) = ζ^{−m_{i}}λ_{i}^{α}σ(A^{α}).

Thus σ(A^{α}) = ∑_{j=1}^{h} ζ^{j}A_{j}^{α} is also a root element and σ induces an automorphism of Δ̇_{s}.

Since all A_{j}^{α} are nonzero then the length of each orbit of σ in Δ̇_{s} is h.

We may choose ℓ roots β_{1}, …, β_{ℓ} ∈ Δ̇_{s} such that A_{0}^{β_{1}}, …, A_{0}^{β_{ℓ}} span 𝔤̇_{0}. Note that if two root elements belong to the same orbit under the action of σ then their zero componens are proportional. Therefore every orbit of σ contains exactly one of β_{1}, …, β_{ℓ}. Consequently, we may choose {σ^{j}(A^{β_{i}})} , i = 1, …, ℓ, j = 1, …, h as our family of the root elements. The set {A_{j}^{β_{i}} ⊗ s^{j}, T_{j} ⊗ s^{b_{j}}, K_{0}} , i = 1, …, ℓ, j ∈ ℤ forms a basis of 𝔤_{s}.

Remark. The properties of the automorphism σ were studied by Kostant ^{12} . He proved, in particular, that σ acts on the root system Δ̇_{s} as a Coxeter transformation. Remarkably, the complex coefficients λ_{i}^{α} could be interpreted as the orthogonal projections of the roots on a real plane invariant under the Coxeter transformation. These projections were introduced by Coxeter in order to visualize the regular polytopes of higher dimensions. In Chapters 12, 13 of Ref.13 and in section 12.5 of Ref.14 the case of the root system of type H_{4} is treated, which also gives the answer for E_{8}.

Now we shall construct the standard representation (φ, F) of 𝔰. The space of this representation (called the Fock space) is the polynomial algebra in the infinitely many variables:

F = ℂ[x_{1}, x_{2}, x_{3}, …].

For i ≥ 1, T_{1−i} ⊗ s^{−b_{i}} is represented by a multiplication operator:

φ(T_{1−i} ⊗ s^{−b_{i}}) = b_{i}x_{i},

T_{i} ⊗ s^{b_{i}} is represented by a differentiation operator:

φ(T_{i} ⊗ s^{b_{i}}) = \frac{∂}{∂x_{i}}

and φ(K_{0}) = Id.

Indeed, the only relation to be checked is

[φ(T_{i} ⊗ s^{b_{i}}), φ(T_{1−j} ⊗ s^{b_{1−j}}) = \frac{1}{h}(T_{i}|T_{1−j})b_{i}δ_{b_{i}b_{j}}φ(K_{0}).

But as it can be easily seen, both expressions are equal to b_{i}δ_{ij} Id.

One can lift this representation of 𝔰 to the whole 𝔤 using vertex operators. We consider the space 𝔤[[z, z^{−1}]] of formal Laurent series in a variable z with coefficients in 𝔤 and the space End(F)[[z, z^{−1}]] of series with coefficients in End(F) (see Chapter 2 in Ref.15 for details). The adjoint action of 𝔤 on 𝔤[[z, z^{−1}]] is well-defined.

Every representation φ : 𝔤 → End(F) defines a homomorphism φ : 𝔤[[z, z^{−1}]] → End(F)[[z, z^{−1}]].

The following theorem shows how to lift the standard representation of the principal Heisenberg subalgebra to 𝔤: Theorem 2 (Ref.1). There exists a representation

φ : 𝔤 → End(ℂ[x_{1}, x_{2}, …])

such that for i ∈ ℕ φ(T_{1−i} ⊗ s^{−b_{i}}) = b_{i}x_{i},

φ(T_{i} ⊗ s^{b_{i}}) = \frac{∂}{∂x_{i}} ,

φ(K_{0}) = Id,

∑_{j∈ℤ} φ(A_{j}^{α} ⊗ s^{j})z^{−j} = A^{α}(z), where

A^{α}(z) = Λ_{0}(A_{0}^{α} ⊗ s^{0})exp (∑_{i=1}^{∞} λ_{i}^{α}z^{b_{i}}x_{i}) exp (− ∑_{i=1}^{∞} λ_{1−i}^{α} \frac{z^{−b_{i}}}{b_{i}} \frac{∂}{∂x_{i}} ) .

Since A_{j}^{α} together with T_{i} ⊗ s^{b_{i}} and K_{0} span 𝔤 then the above formulas completely determine this representation.

We will use extensively the following Laurent series:

δ(z) = ∑_{j∈ℤ} z^{j} and Dδ(z) = ∑_{j∈ℤ} jz^{j}.

The first of these is the formal analogue of the delta function.

For an element B = ∑_{j∈ℤ_{h}} B_{j} ∈ 𝔤̇ denote by B(z) the formal series ∑_{j∈ℤ} φ(B_{j} ⊗s^{j})z^{−j} ∈

End(F)[[z, z^{−1}]].

Corollary 3. Let A = ∑_{j=1}^{h} A_{j}, B = ∑_{j=1}^{h} B_{j} ∈ 𝔤̇ and let C^{j} = [A_{j}, B] ∈ 𝔤̇. Then

[A(z_{1}), B(z_{2})] = ∑_{j=1}^{h}(C^{j}(z_{2}) + \frac{j}{h}(A_{j}|B_{−j}))(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h}) + ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}Dδ((\frac{z_{2}}{z_{1}} )^{h}).

Proof of the Corollary.

[A(z_{1}), B(z_{2})] = ∑_{j∈ℤ} ∑_{i∈ℤ}[φ(A_{j} ⊗ s^{j}), φ(B_{i} ⊗ s^{i})]z_{1}^{−j}z_{2}^{−i} =

= ∑_{j∈ℤ} ∑_{i∈ℤ} \{φ([A_{j}, B_{i}] ⊗ s^{j+i}) + \frac{1}{h}(A_{j}|B_{i})jδ_{j,−i}φ(K_{0})\} z_{1}^{−j}z_{2}^{−i} =

= ∑_{j∈ℤ} ∑_{k=i+j∈ℤ} φ([A_{j}, B_{k−j}] ⊗ s^{k})z_{1}^{−j}z_{2}^{j−k} + \frac{1}{h} ∑_{j∈ℤ}(A_{j}|B_{−j})j(\frac{z_{2}}{z_{1}} )^{j} =

= ∑_{j∈ℤ} C^{j}(z_{2})(\frac{z_{2}}{z_{1}} )^{j} + \frac{1}{h} ∑_{j_{1}=1}^{h} ∑_{{j_{2}∈ℤ \atop j=j_{1}+hj_{2}}} (A_{j_{1}}|B_{−j_{1}})(j_{1} + hj_{2})(\frac{z_{2}}{z_{1}} )^{j_{1}+hj_{2}} =

= ∑_{j_{1}=1}^{h}(C^{j_{1}}(z_{2}) + \frac{j_{1}}{h} (A_{j_{1}}|B_{−j_{1}}))(\frac{z_{2}}{z_{1}} )^{j_{1}}δ((\frac{z_{2}}{z_{1}} )^{h}) + ∑_{j_{1}=1}^{h}(A_{j_{1}}|B_{−j_{1}})(\frac{z_{2}}{z_{1}} )^{j_{1}}Dδ((\frac{z_{2}}{z_{1}} )^{h}).

IV. CONSTRUCTION OF THE VERTEX OPERATOR REPRESENTA-TION FOR THE TOROIDAL LIE ALGEBRA.

In this section we construct a vertex operator representation for the toroidal Lie algebra that generalizes the principal realization of the basic representation of affine Lie algebra. Again it will be convenient to replace 𝔤 with

𝔤_{s} = ∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j}ℂ[t_{1}^{±}, …, t_{n}^{±}] ⊕ K ⊕ D_{+},

with the Lie bracket

[g_{1} ⊗f_{1}(s, t), g_{2} ⊗f_{2}(s, t)] = [g_{1}, g_{2}]⊗(f_{1}f_{2})+(g_{1}|g_{2}) \{\frac{1}{h}(s\frac{d}{ds}f_{1})f_{2}K_{0} + ∑_{p=1}^{n} d_{p}(f_{1})f_{2}K_{p}\} ,

[f_{1}(s, t)D_{p}, g ⊗ f_{2}(s, t)] = g ⊗ f_{1}(d_{p}f_{2})

while the multiplication in D_{+} and its action on K are the same as in 𝔤. Here and below we make identifications s^{hr_{0}}t^{r}K_{a} = t_{0}^{r_{0}}t^{r}K_{a} and s^{hr_{0}}t^{r}D_{b} = t_{0}^{r_{0}}t^{r}D_{b}, a = 0, 1, …, n, b = 1, …, n.

The following lemma is an immediate generalization of Lemma 1:

Lemma 4. The Lie algebras 𝔤 and 𝔤_{s} are isomorphic and the isomorphism is given by

ψ(e_{α} ⊗ t_{0}^{r_{0}}t^{r}) = e_{α} ⊗ s^{ht(α)+hr_{0}}t^{r},

ψ(ν^{−1}(α) ⊗ t_{0}^{r_{0}}t^{r}) = ν^{−1}(α) ⊗ s^{hr_{0}}t^{r} + \frac{ht(α)}{h}s^{hr_{0}}t^{r}K_{0},

ψ = Id on K ⊕ D_{+}.

We shall identify 𝔤 and 𝔤_{s} using this isomorphism.

The subalgebra 𝔰 with the basis {T_{i} ⊗ s^{b_{i}}, s^{ih}D_{p}, s^{ih}K_{p}, K_{0}} , i ∈ ℤ, p = 1, …, n is the principal (degenerate) Heisenberg subalgebra of 𝔤.

Indeed, K_{0} is its central element and the multiplication in 𝔰 is given by

[T_{i} ⊗ s^{b_{i}}, T_{1−j} ⊗ s^{−b_{j}}] = b_{i}δ_{ij}K_{0},
[s^{ih}D_{a}, s^{jh}K_{b}] = δ_{ab}is^{(i+j)h}K_{0} = iδ_{ab}δ_{i,−j}K_{0},
[s^{jh}D_{p}, T_{i} ⊗ s^{b_{i}}] = 0, [s^{ih}D_{a}, s^{jh}D_{b}] = 0,
[T_{i} ⊗ s^{b_{i}}, s^{jh}K_{p}] = 0, [s^{ih}K_{a}, s^{jh}K_{b}] = 0,

where i, j ∈ ℤ, a, b, p = 1, …, n. This subalgebra is degenerate as a Heisenberg algebra since [D_{p}, K_{p}] = 0.

The Heisenberg algebra 𝔰 can be represented on the space

F = ℂ[q_{p}^{±}, x_{i}, u_{pi}, v_{pi}]_{i∈ℕ}^{p=1,...,n}

by differentiation and multiplication operators:

φ(T_{i} ⊗ s^{b_{i}}) = \frac{∂}{∂x_{i}} , φ(T_{1−i} ⊗ s^{−b_{i}}) = b_{i}x_{i},

φ(s^{ih}D_{p}) = \frac{∂}{∂u_{pi}} , φ(s^{−ih}D_{p}) = iv_{pi},

φ(s^{ih}K_{p}) = \frac{∂}{∂v_{pi}} , φ(s^{−ih}K_{p}) = iu_{pi},

φ(D_{p}) = q_{p} \frac{∂}{∂q_{p}} , φ(K_{p}) = 0,

φ(K_{0}) = Id,

where i ≥ 1 and p = 1, …, n. Our goal is to extend this representation of 𝔰 to 𝔤. Consider the following elements of 𝔤[[z, z^{−1}]]:

∑_{j∈ℤ} s^{jh}t^{r}K_{0}z^{−jh} and ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{r}z^{−j}.

Note that

[T_{i} ⊗ s^{b_{i}}, ∑_{j∈ℤ} s^{jh}t^{r}K_{0}z^{−jh}] = 0,

[s^{ih}K_{p}, ∑_{j∈ℤ} s^{jh}t^{r}K_{0}z^{−jh}] = 0,

[s^{ih}D_{p}, ∑_{j∈ℤ} s^{jh}t^{r}K_{0}z^{−jh}] = ∑_{j∈ℤ} r_{p}s^{(i+j)h}t^{r}K_{0}z^{−jh} =

= ∑_{k=i+j∈ℤ} r_{p}s^{kh}t^{r}K_{0}z^{−kh+ih} = r_{p}z^{ih} ∑_{j∈ℤ} s^{jh}t^{r}K_{0}z^{−jh}.

The theory of vertex operators suggests (see Lemma 14.5 in Ref.1) that ∑_{j∈ℤ} s^{jh}t^{r}K_{0}z^{−jh} should be represented by

K_{0}(z, r) = q^{r}exp (∑_{p=1}^{n} r_{p} ∑_{j≥1} z^{jh}u_{pj}) exp (− ∑_{p=1}^{n} r_{p} ∑_{j≥1} \frac{z^{−jh}}{j} \frac{∂}{∂v_{pj}} ) .

Here q^{r} = q_{1}^{r_{1}} … q_{n}^{r_{n}}.

In a similar way the commutator relations

[T_{i} ⊗ s^{b_{i}}, ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{r}z^{−j}] = ∑_{j∈ℤ} λ_{i}^{α}A_{j+b_{i}}^{α} ⊗ s^{j+b_{i}}t^{r}z^{−j} =

= λ_{i}^{α} ∑_{k=j+b_{i}∈ℤ} A_{k}^{α} ⊗ s^{k}t^{r}z^{−k+b_{i}} = λ_{i}^{α}z^{b_{i}} ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{r}z^{−j},

[s^{ih}K_{p}, ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{r}z^{−j}] = 0,

[s^{ih}D_{p}, ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{r}z^{−j}] = ∑_{j∈ℤ} r_{p}A_{j}^{α} ⊗ s^{j+ih}t^{r}z^{−j} =

= r_{p} ∑_{k=j+ih∈ℤ} A_{j}^{α} ⊗ s^{k}t^{r}z^{−k+ih} = r_{p}z^{ih} ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{r}z^{−j}

suggest that ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{r}z^{−j} should be represented by

A^{α}(z, r) =

= q^{r}Λ_{0}(A_{0}^{α} ⊗ s^{0})exp (∑_{i=1}^{∞} λ_{i}^{α}z^{b_{i}}x_{i}) exp (− ∑_{i=1}^{∞} λ_{1−i}^{α} \frac{z^{−b_{i}}}{b_{i}} \frac{∂}{∂x_{i}} ) ×

×exp (∑_{p=1}^{n} r_{p} ∑_{j∈ℤ} z^{jh}u_{pj}) exp (− ∑_{p=1}^{n} r_{p} ∑_{j∈ℤ} \frac{z^{−jh}}{j} \frac{∂}{∂v_{pj}} ) =

= A^{α}(z)K_{0}(z, r).

The last formula hints that for B ∈ 𝔤̇ we may try to represent ∑_{j∈ℤ} B_{j} ⊗ s^{j}t^{r}z^{−j} by

B(z, r) = B(z)K_{0}(z, r).

Note that B(z) is known from the affine case (Theorem 2).

In fact, we shall see that the same approach is valid even for K and D_{+}, but for these we should first discuss the concept of the normal ordering.

Let X(z) = ∑_{i∈ℤ} X_{i}z^{i}, Y (z) = ∑_{j∈ℤ} Y_{j}z^{j} ∈ End(F)[[z, z^{−1}]]. We say that the product

X(z)Y (z) exists if for every k ∈ ℤ and for every v ∈ F the sum

∑_{i∈ℤ} X_{i}Y_{k−i}v

has finitely many nonzero terms. If this is the case then

X(z)Y (z) = ∑_{k∈ℤ}(∑_{i∈ℤ} X_{i}Y_{k−i})z^{k}.

It is possible that X(z)Y (z) exists while Y (z)X(z) does not. For example, this happens for

X(z) = ∑_{i≥1} x_{i}z^{i} and Y (z) = ∑_{i≥1} \frac{∂}{∂x_{i}} z^{−i}.

To improve the situation we define the normal ordering for the product of operators on F. For a differential operator P ∈ End(F) the normal ordering :x_{i}P: is defined as x_{i}P, while the normal ordering of the product :\frac{∂}{∂x_{i}} P: is defined as P \frac{∂}{∂x_{i}}. Note that for X(z) and Y (z) from the above example :X(z)Y (z): = :Y (z)X(z): = X(z)Y (z).

From the action of the principle Heisenberg subalgebra on F we see that the series

∑_{j∈ℤ} s^{jh}K_{p}z^{−jh} and ∑_{j∈ℤ} s^{jh}D_{p}z^{−jh} are represented by

K_{p}(z) = ∑_{i≥1} iu_{pi}z^{ih} + ∑_{i≥1} \frac{∂}{∂v_{pi}} z^{−ih}

and

D_{p}(z) = ∑_{i≥1} iv_{pi}z^{ih} + q_{p} \frac{∂}{∂q_{p}} + ∑_{i≥1} \frac{∂}{∂u_{pi}} z^{−ih}.

Using the analogy with A^{α}(z, r) we shall represent ∑_{j∈ℤ} s^{jh}t^{r}K_{p}z^{−jh} by

K_{p}(z, r) = K_{p}(z)K_{0}(z, r)

and ∑_{j∈ℤ} s^{jh}t^{r}D_{p}z^{−jh} by

D_{p}(z, r) = :D_{p}(z)K_{0}(z, r): .

Note that we use the normal ordering in the case of D_{p}(z, r) in order for the product to exist.

We summarize this discussion in our main theorem:

Theorem 5. There exists a representation φ of the toroidal Lie algebra 𝔤_{s} on the space

F = ℂ[q_{p}^{±}, x_{i}, u_{pi}, v_{pi}]_{i∈ℕ}^{p=1,...,n}

such that for i = 1, …, ℓ, p = 1, …, n, α ∈ Δ̇_{s}, r ∈ ℤ^{n}

∑_{j∈ℤ} φ(s^{jh}t^{r}K_{0})z^{−jh} = K_{0}(z, r) =

= q^{r}exp (∑_{p=1}^{n} r_{p} ∑_{j≥1} z^{jh}u_{pj}) exp (− ∑_{p=1}^{n} r_{p} ∑_{j≥1} \frac{z^{−jh}}{j} \frac{∂}{∂v_{pj}} ) ,  (4.1)

∑_{j∈ℤ} φ(T_{i} ⊗ s^{m_{i}+jh}t^{r})z^{−m_{i}−jh} = T_{i}(z, r) =

= \{∑_{j≥1}(jh − m_{i})z^{jh−m_{i}}x_{jℓ+1−i} + ∑_{j≥0} z^{−jh−m_{i}} \frac{∂}{∂x_{jℓ+i}} \} K_{0}(z, r),  (4.2)

∑_{j∈ℤ} φ(A_{j}^{α} ⊗ s^{j}t^{r})z^{−j} = A^{α}(z, r) =

Λ_{0}(A_{0}^{α} ⊗ s^{0})exp (∑_{i=1}^{∞} λ_{i}^{α}z^{b_{i}}x_{i}) exp (− ∑_{i=1}^{∞} λ_{1−i}^{α} \frac{z^{−b_{i}}}{b_{i}} \frac{∂}{∂x_{i}} ) K_{0}(z, r),  (4.3)

∑_{j∈ℤ} φ(s^{jh}t^{r}K_{p})z^{−jh} = K_{p}(z, r) = \{∑_{i≥1} iu_{pi}z^{ih} + ∑_{i≥1} \frac{∂}{∂v_{pi}} z^{−ih}\} K_{0}(z, r),  (4.4)

∑_{j∈ℤ} φ(s^{jh}t^{r}D_{p})z^{−jh} = D_{p}(z, r) = : \{∑_{i≥1} iv_{pi}z^{ih} + q_{p} \frac{∂}{∂q_{p}} + ∑_{i≥1} \frac{∂}{∂u_{pi}} z^{−ih}\} K_{0}(z, r): .

(4.5)

Remark. By choosing a specialization q_{p} ∈ ℂ\{0}, p = 1, …, n, we obtain a representation of the toroidal algebra 𝔤̂ on the Fock space

F̂ = ℂ[x_{i}, u_{pi}, v_{pi}]_{i∈ℕ}^{p=1,...,n}.

V. PROOF OF THE MAIN THEOREM.

First of all, let us check that formulas (4.1) - (4.5) define a linear map φ : 𝔤 → End(F). The linear dependencies between the momenta at the left hand side are of the form

φ(A_{j}^{σ^{k}(α)} ⊗ s^{j}t^{r}) = ζ^{kj}φ(A_{j}^{α} ⊗ s^{j}t^{r})

and

jφ(s^{jh}t^{r}K_{0}) + ∑_{p=1}^{n} r_{p}φ(s^{jh}t^{r}K_{p}) = 0.

These dependencies extend to the following relations for the corresponding series:

∑_{j∈ℤ} φ(A_{j}^{σ^{k}(α)} ⊗ s^{j}t^{r})z^{−j} = ∑_{j∈ℤ} φ(A_{j}^{α} ⊗ s^{j}t^{r})(ζ^{−k}z)^{−j}

and

−\frac{1}{h}D_{z} ∑_{j∈ℤ} φ(s^{jh}t^{r}K_{0})z^{−jh} + ∑_{p=1}^{n} r_{p} ∑_{j∈ℤ} φ(s^{jh}t^{r}K_{p})z^{−jh} = 0,

where D_{z} = z \frac{∂}{∂z}. We have to show that the same relations hold for the right hand sides of (4.1) - (4.5). Indeed, noting that λ_{i}^{σ^{k}(α)} = ζ^{−km_{i}}λ_{i}^{α} we obtain

A^{σ^{k}(α)}(z, r) = Λ_{0}(A_{0}^{σ^{k}(α)})exp(∑_{i=1}^{∞} λ_{i}^{α}ζ^{−km_{i}}z^{b_{i}}x_{i})exp(− ∑_{i=1}^{∞} λ_{1−i}^{α}ζ^{km_{i}} \frac{z^{−b_{i}}}{b_{i}} \frac{∂}{∂x_{i}} )K_{0}(z, r) =

= Λ_{0}(A_{0}^{α})exp(∑_{i=1}^{∞} λ_{i}^{α}(ζ^{−k}z)^{b_{i}}x_{i})exp(− ∑_{i=1}^{∞} λ_{1−i}^{α} \frac{(ζ^{−k}z)^{−b_{i}}}{b_{i}} \frac{∂}{∂x_{i}} )K_{0}(ζ^{−k}z, r) = A^{α}(ζ^{−k}z, r).

The verification of the second relation is also straightforward:

D_{z}K_{0}(z, r) = D_{z}q^{r}exp (∑_{p=1}^{n} r_{p} ∑_{j≥1} z^{jh}u_{pj}) exp (− ∑_{p=1}^{n} r_{p} ∑_{j≥1} \frac{z^{−jh}}{j} \frac{∂}{∂v_{pj}} ) =

∑_{p=1}^{n} r_{p} \{∑_{j≥1} z^{jh}jhu_{pj} + ∑_{j≥1} z^{−jh}h\frac{∂}{∂v_{pj}} \} K_{0}(z, r) = h ∑_{p=1}^{n} r_{p}K_{p}(z, r).

Observing that the momenta of the series in the left hand sides of (4.1) - (4.5) span 𝔤 we conclude that the linear map φ : 𝔤 → End(F) is well-defined. We need to show now that φ is a homomorphism of Lie algebras.

The following Lemma and its Corollary which we state without proof are very useful for the computations with formal series.

Lemma 6 (cf. Proposition 2.2.2 in Ref.15). Let

δ(z) = ∑_{k∈ℤ} z^{k} and Dδ(z) = D_{z}δ(z) = ∑_{k∈ℤ} kz^{k}.

If the products in the left hand sides exist then the following equalities hold:

(i)  X(z_{1})δ(\frac{z_{2}}{z_{1}} ) = X(z_{2})δ(\frac{z_{2}}{z_{1}} ),

(ii)  X(z_{1})Dδ(\frac{z_{2}}{z_{1}} ) = X(z_{2})Dδ(\frac{z_{2}}{z_{1}} ) + D_{z_{2}}(X(z_{2}))δ(\frac{z_{2}}{z_{1}} ),

Corollary 7. Let X(z) = Y (z^{h}). The following equalities hold provided the products in the left hand sides exist:

(i)  X(z_{1})δ ((\frac{z_{2}}{z_{1}} )^{h}) = X(z_{2})δ ((\frac{z_{2}}{z_{1}} )^{h}) ,

(ii)  X(z_{1})Dδ ((\frac{z_{2}}{z_{1}} )^{h}) = X(z_{2})Dδ ((\frac{z_{2}}{z_{1}} )^{h}) + \frac{1}{h}D_{z_{2}}(X(z_{2}))δ ((\frac{z_{2}}{z_{1}} )^{h}) ,

Note that A^{α}(z, r) = A^{α}(z)K_{0}(z, r) and T_{i}(z, r) = T_{i}(z)K_{0}(z, r) , hence for every B ∈ 𝔤̇ we have B(z, r) = B(z)K_{0}(z, r).

In order to verify the commutator relations between the elements of

∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j}ℂ[t_{1}^{±}, …, t_{n}^{±}] we have to show that for every A, B ∈ 𝔤̇

∑_{j,i∈ℤ} φ([A_{j} ⊗ s^{j}t^{r}, B_{i} ⊗ s^{i}t^{m}])z_{1}^{−j}z_{2}^{−i} = ∑_{j,i∈ℤ} [φ(A_{j} ⊗ s^{j}t^{r}), φ(B_{i} ⊗ s^{i}t^{m})] z_{1}^{−j}z_{2}^{−i}.

(5.1)

Indeed, let C^{j} = [A_{j}, B]. Then

∑_{j,i∈ℤ} φ([A_{j} ⊗ s^{j}t^{r}, B_{i} ⊗ s^{i}t^{m}])z_{1}^{−j}z_{2}^{−i} =

= ∑_{j,i∈ℤ} φ([A_{j}, B_{i}] ⊗ s^{j+i}t^{r+m})z_{1}^{−j}z_{2}^{−i}

+ ∑_{j,i∈ℤ}(A_{j}|B_{i})φ(s^{j+i}t^{r+m}(\frac{j}{h}K_{0} + ∑_{p=1}^{n} r_{p}K_{p})) =

= ∑_{j∈ℤ} ∑_{k=j+i∈ℤ} φ([A_{j}, B_{k−j}] ⊗ s^{k}t^{r+m})z_{1}^{−j}z_{2}^{j−k}

+ ∑_{j_{1}=1}^{h} ∑_{{j_{2}∈ℤ \atop j=j_{1}+hj_{2}}} ∑_{{i_{2}∈ℤ \atop i=−j_{1}+hi_{2}}} (A_{j_{1}}|B_{−j_{1}})\frac{j_{1} + hj_{2}}{h} φ(s^{h(j_{2}+i_{2})}t^{r+m}K_{0})z_{1}^{−j_{1}−hj_{2}}z_{2}^{j_{1}−hi_{2}}

+ ∑_{j_{1}=1}^{h} ∑_{{j_{2}∈ℤ \atop j=j_{1}+hj_{2}}} ∑_{{i_{2}∈ℤ \atop i=−j_{1}+hi_{2}}} (A_{j_{1}}|B_{−j_{1}}) ∑_{p=1}^{n} r_{p}φ(s^{h(j_{2}+i_{2})}t^{r+m}K_{p})z_{1}^{−j_{1}−hj_{2}}z_{2}^{j_{1}−hi_{2}} =

= ∑_{j_{1}=1}^{h} ∑_{j_{2}∈ℤ} ∑_{k∈ℤ} φ(C_{k}^{j_{1}} ⊗ s^{k}t^{r+m})z_{2}^{−k}(\frac{z_{2}}{z_{1}} )^{j_{1}}(\frac{z_{2}}{z_{1}} )^{hj_{2}}

+ ∑_{j_{1}=1}^{h} ∑_{j_{2}∈ℤ} ∑_{k=j_{2}+i_{2}∈ℤ}(A_{j_{1}}|B_{−j_{1}})\frac{j_{1} + hj_{2}}{h} φ(s^{hk}t^{r+m}K_{0})z_{1}^{−j_{1}−hj_{2}}z_{2}^{j_{1}+hj_{2}−hk}

+ ∑_{j_{1}=1}^{h} ∑_{j_{2}∈ℤ} ∑_{k=j_{2}+i_{2}∈ℤ}(A_{j_{1}}|B_{−j_{1}}) ∑_{p=1}^{n} r_{p}φ(s^{hk}t^{r+m}K_{p})z_{1}^{−j_{1}−hj_{2}}z_{2}^{j_{1}+hj_{2}−hk} =

= ∑_{j_{1}=1}^{h} C^{j_{1}}(z_{2}, r + m)(\frac{z_{2}}{z_{1}} )^{j_{1}}δ((\frac{z_{2}}{z_{1}} )^{h})

+ \frac{1}{h} ∑_{j_{1}=1}^{h} j_{1}(A_{j_{1}}|B_{−j_{1}})K_{0}(z_{2}, r + m)(\frac{z_{2}}{z_{1}} )^{j_{1}}δ((\frac{z_{2}}{z_{1}} )^{h})

+ ∑_{p=1}^{n} r_{p} ∑_{j_{1}=1}^{h}(A_{j_{1}}|B_{−j_{1}})K_{p}(z_{2}, r + m)(\frac{z_{2}}{z_{1}} )^{j_{1}}δ((\frac{z_{2}}{z_{1}} )^{h})

+ ∑_{j_{1}=1}^{h}(A_{j_{1}}|B_{−j_{1}})K_{0}(z_{2}, r + m)(\frac{z_{2}}{z_{1}} )^{j_{1}}Dδ((\frac{z_{2}}{z_{1}} )^{h}).

For the right hand side of (5.1) we use Corollary 3, Corollary 7 and the fact that K_{0}(z, r)K_{0}(z, m) = K_{0}(z, r + m):

∑_{j,i∈ℤ} [φ(A_{j} ⊗ s^{j}t^{r}), φ(B_{i} ⊗ s^{i}t^{m})] z_{1}^{−j}z_{2}^{−i} = [A(z_{1}, r), B(z_{2}, m)] =

= [A(z_{1})K_{0}(z_{1}, r), B(z_{2})K_{0}(z_{2}, m)] = [A(z_{1}), B(z_{2})] K_{0}(z_{1}, r)K_{0}(z_{2}, m) =

= ∑_{j=1}^{h} C^{j}(z_{2})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{1}, r)K_{0}(z_{2}, m)

+ \frac{1}{h} ∑_{j=1}^{h} j(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{1}, r)K_{0}(z_{2}, m)

+ ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}Dδ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{1}, r)K_{0}(z_{2}, m) =

= ∑_{j=1}^{h} C^{j}(z_{2})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r)K_{0}(z_{2}, m)

+ \frac{1}{h} ∑_{j=1}^{h} j(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r)K_{0}(z_{2}, m)

+ ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}Dδ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r)K_{0}(z_{2}, m)

+ ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})\frac{1}{h}D_{z_{2}}(K_{0}(z_{2}, r))K_{0}(z_{2}, m) =

= ∑_{j=1}^{h} C^{j}(z_{2})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r + m)

+ \frac{1}{h} ∑_{j=1}^{h} j(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r + m)

+ ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}Dδ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r + m)

+ ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h}) ∑_{p=1}^{n} r_{p}K_{p}(z_{2})K_{0}(z_{2}, r)K_{0}(z_{2}, m) =

= ∑_{j=1}^{h} C^{j}(z_{2})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r + m)

+ \frac{1}{h} ∑_{j=1}^{h} j(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r + m)

+ ∑_{p=1}^{n} r_{p} ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}δ((\frac{z_{2}}{z_{1}} )^{h})K_{p}(z_{2}, r + m)

+ ∑_{j=1}^{h}(A_{j}|B_{−j})(\frac{z_{2}}{z_{1}} )^{j}Dδ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, r + m).

Thus (5.1) holds.

It is easy to see that operators φ(s^{jh}t^{r}K_{p}), p = 0, 1, …, n commute with φ( ∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j}ℂ[t_{1}^{±}, …, t_{n}^{±}] ⊕ K). This follows from the obvious equalities:

[K_{p}(z_{1}, r), A^{α}(z_{2}, m)] = [K_{p}(z_{1}, r), T_{i}(z_{2}, m)] = 0,
[K_{a}(z_{1}, r), K_{b}(z_{2}, m)] = 0, p, a, b = 0, 1, …, n.

In order to check the action of D_{+} on ∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j}ℂ[t_{1}^{±}, …, t_{n}^{±}] it is sufficient to verify

the following equality for the generating series:

∑_{j,i∈ℤ} φ([s^{jh}t^{r}D_{p}, s^{i}t^{m}A_{i}^{α}])z_{1}^{−jh}z_{2}^{−i} = ∑_{j,i∈ℤ} [φ(s^{jh}t^{r}D_{p}), φ(A_{i}^{α} ⊗ s^{i}t^{m})] z_{1}^{−jh}z_{2}^{−i}. (5.2)

The corresponding equalities for T_{i} ⊗ s^{b_{i}}t^{m} and s^{ih}t^{m}K_{p} will follow since s^{jh}t^{r}D_{p} acts as

a derivation and s^{i}t^{m}A_{i}^{α} generate ∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j}ℂ[t_{1}^{±}, …, t_{n}^{±}] ⊕ K.

Let us prove that (5.2) holds.

∑_{j,i∈ℤ} φ([s^{jh}t^{r}D_{p}, s^{i}t^{m}A_{i}^{α}])z_{1}^{−jh}z_{2}^{−i} = ∑_{j,i∈ℤ} m_{p}φ(s^{i+jh}t^{r+m}A_{i}^{α})z_{1}^{−jh}z_{2}^{−i} =

= ∑_{j∈ℤ} ∑_{k=i+jh∈ℤ} m_{p}φ(s^{k}t^{r+m}A_{k}^{α})z_{1}^{−jh}z_{2}^{−k+jh} = m_{p}A^{α}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h}).

Next, we will show that the right hand side of (5.2) reduces to the same expression. In the following computations it will be convenient to denote φ(s^{jh}D_{p}) = \frac{∂}{∂u_{pj}} by γ_{p}(j) , φ(s^{−jh}D_{p}) = jv_{pj} by γ_{p}(−j) and φ(D_{p}) = q_{p} \frac{∂}{∂q_{p}} by γ_{p}(0). Note that [γ_{p}(j), K_{0}(z, m)] = m_{p}z^{jh}K_{0}(z, m) for j ∈ ℤ.

∑_{j,i∈ℤ} [φ(s^{jh}t^{r}D_{p}), φ(A_{i}^{α} ⊗ s^{i}t^{m})] z_{1}^{−jh}z_{2}^{−i} = [D_{p}(z_{1}, r), A^{α}(z_{2}, m)] =

= [: {∑_{j∈ℤ} γ_{p}(j)z_{1}^{−jh}} K_{0}(z_{1}, r): , A^{α}(z_{2})K_{0}(z_{2}, m)] =

= A^{α}(z_{2}) [: {∑_{j∈ℤ} γ_{p}(j)z_{1}^{−jh}} K_{0}(z_{1}, r): , K_{0}(z_{2}, m)] =

= A^{α}(z_{2}) [{∑_{j<0} γ_{p}(j)z_{1}^{−jh}} K_{0}(z_{1}, r) , K_{0}(z_{2}, m)]

+A^{α}(z_{2}) [K_{0}(z_{1}, r) {∑_{j≥0} γ_{p}(j)z_{1}^{−jh}} , K_{0}(z_{2}, m)] =

= A^{α}(z_{2}) [∑_{j<0} γ_{p}(j)z_{1}^{−jh} , K_{0}(z_{2}, m)] K_{0}(z_{1}, r)

+A^{α}(z_{2})K_{0}(z_{1}, r) [∑_{j≥0} γ_{p}(j)z_{1}^{−jh} , K_{0}(z_{2}, m)] =

= A^{α}(z_{2}) ∑_{j<0} z_{1}^{−jh}z_{2}^{jh}m_{p}K_{0}(z_{2}, m)K_{0}(z_{1}, r)

+A^{α}(z_{2})K_{0}(z_{1}, r) ∑_{j≥0} z_{1}^{−jh}z_{2}^{jh}m_{p}K_{0}(z_{2}, m) =

= m_{p}A^{α}(z_{2})K_{0}(z_{1}, r)K_{0}(z_{2}, m)δ((\frac{z_{2}}{z_{1}} )^{h}) =

= m_{p}A^{α}(z_{2})K_{0}(z_{2}, r)K_{0}(z_{2}, m)δ((\frac{z_{2}}{z_{1}} )^{h}) =

= m_{p}A^{α}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h}).

From the above computation we can see that

[D_{p}(z_{1}, r), K_{0}(z_{2}, m)] = m_{p}K_{0}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h}).

To complete the proof of Theorem 5 we need to compute the commutators [φ(s^{jh}t^{r}D_{a}), φ(s^{ih}t^{m}D_{b})]. We will check the following equality for the generating series:

∑_{j,i∈ℤ} φ([s^{jh}t^{r}D_{a} , s^{ih}t^{m}D_{b}])z_{1}^{−jh}z_{2}^{−ih} = ∑_{j,i∈ℤ} [φ(s^{jh}t^{r}D_{a}) , φ(s^{ih}t^{m}D_{b})] z_{1}^{−jh}z_{2}^{−ih}.

(5.3)

To compute the left hand side we use multiplication in 𝔤:

∑_{j,i∈ℤ} φ([s^{jh}t^{r}D_{a}, s^{ih}t^{m}D_{b}])z_{1}^{−jh}z_{2}^{−ih} =

= ∑_{j,i∈ℤ} φ(m_{a}s^{(j+i)h}t^{r+m}D_{b} − r_{b}s^{(j+i)h}t^{r+m}D_{a})z_{1}^{−jh}z_{2}^{−ih}

− m_{a}r_{b}φ(s^{(j+i)h}t^{r+m} {jK_{0} + ∑_{p=1}^{n} r_{p}K_{p}})z_{1}^{−jh}z_{2}^{−ih} =

= ∑_{j∈ℤ} ∑_{k=j+i∈ℤ} (m_{a}φ(s^{kh}t^{r+m}D_{b}) − r_{b}φ(s^{kh}t^{r+m}D_{a}))z_{1}^{−jh}z_{2}^{−kh+jh}

− ∑_{j∈ℤ} ∑_{k=j+i∈ℤ} m_{a}r_{b}φ(s^{kh}t^{r+m} {jK_{0} + ∑_{p=1}^{n} r_{p}K_{p}}))z_{1}^{−jh}z_{2}^{−kh+jh} =

= (m_{a}D_{b}(z_{2}, r + m) − r_{b}D_{a}(z_{2}, r + m))δ((\frac{z_{2}}{z_{1}} )^{h})

− m_{a}r_{b} ∑_{p=1}^{n} r_{p}K_{p}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h}) − m_{a}r_{b}K_{0}(z_{2}, r + m)Dδ((\frac{z_{2}}{z_{1}} )^{h}).

Observe that

[D_{a}(z, r) , γ_{b}(i)] = [: {∑_{j∈ℤ} γ_{a}(j)z^{−jh}} K_{0}(z, r): , γ_{b}(i)] =

= [∑_{j<0} γ_{a}(j)z^{−jh}K_{0}(z, r) , γ_{b}(i)] + [K_{0}(z, r) ∑_{j≥0} γ_{a}(j)z^{−jh} , γ_{b}(i)] =

= ∑_{j<0} γ_{a}(j)z^{−jh} [K_{0}(z, r) , γ_{b}(i)] + [K_{0}(z, r) , γ_{b}(i)] ∑_{j≥0} γ_{a}(j)z^{−jh} =

= −r_{b}z^{ih} ∑_{j<0} γ_{a}(j)z^{−jh}K_{0}(z, r) − r_{b}z^{ih}K_{0}(z, r) ∑_{j≥0} γ_{a}(j)z^{−jh} =

= −r_{b}z^{ih}D_{a}(z, r).

We use the last equality to compute the right hand side of (5.3):

∑_{j,i∈ℤ} [φ(s^{jh}t^{r}D_{a}), φ(s^{ih}t^{m}D_{b})] z_{1}^{−jh}z_{2}^{−ih} = [D_{a}(z_{1}, r), D_{b}(z_{2}, m)] =

= [D_{a}(z_{1}, r), ∑_{i<0} γ_{b}(i)z_{2}^{−ih}K_{0}(z_{2}, m)] + [D_{a}(z_{1}, r), K_{0}(z_{2}, m) ∑_{i≥0} γ_{b}(i)z_{2}^{−ih}] =

= ∑_{i<0} [D_{a}(z_{1}, r), γ_{b}(i)] z_{2}^{−ih}K_{0}(z_{2}, m) + ∑_{i<0} γ_{b}(i)z_{2}^{−ih} [D_{a}(z_{1}, r), K_{0}(z_{2}, m)]

+ [D_{a}(z_{1}, r), K_{0}(z_{2}, m)] ∑_{i≥0} γ_{b}(i)z_{2}^{−ih} + K_{0}(z_{2}, m) ∑_{i≥0} [D_{a}(z_{1}, r), γ_{b}(i)] z_{2}^{−ih} =

= −r_{b} ∑_{i<0} D_{a}(z_{1}, r)z_{1}^{ih}z_{2}^{−ih}K_{0}(z_{2}, m) + ∑_{i<0} γ_{b}(i)z_{2}^{−ih}m_{a}K_{0}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})

+ m_{a}K_{0}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h}) ∑_{i≥0} γ_{b}(i)z_{2}^{−ih} − r_{b}K_{0}(z_{2}, m) ∑_{i≥0} D_{a}(z_{1}, r)z_{1}^{ih}z_{2}^{−ih} =

= m_{a}D_{b}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})

− r_{b} ∑_{i<0}(\frac{z_{1}}{z_{2}} )^{ih} ∑_{j<0} γ_{a}(j)z_{1}^{−jh}K_{0}(z_{1}, r)K_{0}(z_{2}, m)

− r_{b} ∑_{i<0}(\frac{z_{1}}{z_{2}} )^{ih}K_{0}(z_{1}, r) ∑_{j≥0} γ_{a}(j)z_{1}^{−jh}K_{0}(z_{2}, m)

− r_{b}K_{0}(z_{2}, m) ∑_{i≥0} ∑_{j<0} γ_{a}(j)z_{1}^{−jh}K_{0}(z_{1}, r)(\frac{z_{1}}{z_{2}} )^{ih}

− r_{b}K_{0}(z_{2}, m) ∑_{i≥0} K_{0}(z_{1}, r)(\frac{z_{1}}{z_{2}} )^{ih} ∑_{j≥0} γ_{a}(j)z_{1}^{−jh} =

= m_{a}D_{b}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})

− r_{b} ∑_{i<0} ∑_{j<0}(\frac{z_{1}}{z_{2}} )^{ih}z_{1}^{−jh}γ_{a}(j)K_{0}(z_{1}, r)K_{0}(z_{2}, m)

− r_{b} ∑_{i<0} ∑_{j≥0}(\frac{z_{1}}{z_{2}} )^{ih}z_{1}^{−jh}K_{0}(z_{1}, r) [γ_{a}(j), K_{0}(z_{2}, m)]

− r_{b} ∑_{i<0} ∑_{j≥0}(\frac{z_{1}}{z_{2}} )^{ih}z_{1}^{−jh}K_{0}(z_{1}, r)K_{0}(z_{2}, m)γ_{a}(j)

− r_{b} ∑_{i≥0} ∑_{j<0} z_{1}^{−jh}(\frac{z_{1}}{z_{2}} )^{ih} [K_{0}(z_{2}, m), γ_{a}(j)] K_{0}(z_{1}, r)

− r_{b} ∑_{i≥0} ∑_{j<0} z_{1}^{−jh}(\frac{z_{1}}{z_{2}} )^{ih}γ_{a}(j)K_{0}(z_{2}, m)K_{0}(z_{1}, r)

− r_{b} ∑_{i≥0} ∑_{j≥0}(\frac{z_{1}}{z_{2}} )^{ih}z_{1}^{−jh}K_{0}(z_{2}, m)K_{0}(z_{1}, r)γ_{a}(j) =

= m_{a}D_{b}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})

− r_{b}δ((\frac{z_{2}}{z_{1}} )^{h}): \{∑_{j∈ℤ} γ_{a}(j)z_{2}^{−jh}\} K_{0}(z_{2}, r)K_{0}(z_{2}, m):

− m_{a}r_{b} ∑_{i>0} ∑_{j≥0}(\frac{z_{2}}{z_{1}} )^{(i+j)h}K_{0}(z_{1}, r)K_{0}(z_{2}, m)

+ m_{a}r_{b} ∑_{i≤0} ∑_{j<0}(\frac{z_{2}}{z_{1}} )^{(i+j)h}K_{0}(z_{1}, r)K_{0}(z_{2}, m) =

= m_{a}D_{b}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h}) − r_{b}D_{a}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})

− m_{a}r_{b} ∑_{k=i+j>0} ∑_{j=0}^{k−1}(\frac{z_{2}}{z_{1}} )^{kh}K_{0}(z_{1}, r)K_{0}(z_{2}, m)

+ m_{a}r_{b} ∑_{k=i+j<0} ∑_{j=k}^{−1}(\frac{z_{2}}{z_{1}} )^{kh}K_{0}(z_{1}, r)K_{0}(z_{2}, m) =

= (m_{a}D_{b}(z_{2}, r + m) − r_{b}D_{a}(z_{2}, r + m))δ((\frac{z_{2}}{z_{1}} )^{h})

− m_{a}r_{b} ∑_{k∈ℤ} k(\frac{z_{2}}{z_{1}} )^{kh}K_{0}(z_{1}, r)K_{0}(z_{2}, m) =

= (m_{a}D_{b}(z_{2}, r + m) − r_{b}D_{a}(z_{2}, r + m))δ((\frac{z_{2}}{z_{1}} )^{h})

− m_{a}r_{b}K_{0}(z_{1}, r)K_{0}(z_{2}, m)Dδ((\frac{z_{2}}{z_{1}} )^{h}) =

= (m_{a}D_{b}(z_{2}, r + m) − r_{b}D_{a}(z_{2}, r + m))δ((\frac{z_{2}}{z_{1}} )^{h})

− \frac{1}{h}m_{a}r_{b}D_{z_{2}}(K_{0}(z_{2}, r))K_{0}(z_{2}, m)δ((\frac{z_{2}}{z_{1}} )^{h})

− m_{a}r_{b}K_{0}(z_{2}, r)K_{0}(z_{2}, m)Dδ((\frac{z_{2}}{z_{1}} )^{h}) =

= (m_{a}D_{b}(z_{2}, r + m) − r_{b}D_{a}(z_{2}, r + m))δ((\frac{z_{2}}{z_{1}} )^{h})

− m_{a}r_{b} ∑_{p=1}^{n} r_{p}K_{p}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})

− m_{a}r_{b}K_{0}(z_{2}, r + m)Dδ((\frac{z_{2}}{z_{1}} )^{h}).

This establishes (5.3) and completes the proof of Theorem 5.

VI. SUGAWARA OPERATORS.

In the representation we constructed the operators from D_{+} act as derivations on the algebra 𝔤̂ = 𝔤̇ ⊗ ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}] ⊕ K. It is possible to extend this action on D, that is to represent t_{0}^{j}t^{r}D_{0} by operators on F. For the affine case these are the Sugawara operators.

We will work with the toroidal algebra realized as ∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j}ℂ[t_{1}^{±}, …, t_{n}^{±}] ⊕ K. For

this realization

ψ(t_{0}^{j}t^{r}D_{0}) = \frac{1}{h}s^{jh}t^{r}D_{s} − \frac{1}{h}ν^{−1}(ρ) ⊗ s^{jh}t^{r},  (6.1)

where ρ ∈ 𝔥̇ with (ρ|α_{i}) = 1 for every simple root α_{i} ∈ Δ̇ and D_{s} = s\frac{d}{ds}.

Since for α ∈ Δ the elements A^{α} ⊗ t_{0}^{i}t^{m} generate 𝔤̂ then we have to check that

ψ([t_{0}^{j}t^{r}D_{0}, A^{α} ⊗ t_{0}^{i}t^{m}]) = [ψ(t_{0}^{j}t^{r}D_{0}), ψ(A^{α} ⊗ t_{0}^{i}t^{m})] .

Note that A^{α} ∈ 𝔤̇_{ht(α)} and (ρ|α) = ht(α).

We have

ψ([t_{0}^{j}t^{r}D_{0}, A^{α} ⊗ t_{0}^{i}t^{m}]) = iψ(A^{α} ⊗ t_{0}^{i+j}t^{r+m}) = iA^{α} ⊗ s^{ht(α)+h(i+j)}t^{r+m},

while

[ψ(t_{0}^{j}t^{r}D_{0}), ψ(A^{α} ⊗ t_{0}^{i}t^{m})] =

= [\frac{1}{h}s^{jh}t^{r}D_{s} − \frac{1}{h}ν^{−1}(ρ) ⊗ s^{jh}t^{r}, A^{α} ⊗ s^{ht(α)+ih}t^{m}] =

= \frac{1}{h}(ht(α) + ih − (ρ|α))A^{α} ⊗ s^{ht(α)+h(i+j)}t^{r+m} =

= iA^{α} ⊗ s^{ht(α)+h(i+j)}t^{r+m}.

This establishes (6.1). Tt is then sufficient to construct operators on F corresponding to s^{jh}t^{r}D_{s}.

It will be convenient to denote in this section φ(T_{i} ⊗ s^{b_{i}}) by τ(i) , φ(s^{jh}D_{p}) by γ_{p}(j) and φ(s^{jh}K_{p}) by κ_{p}(j).

In these notations

K_{0}(z, r) = q^{r}exp (∑_{p=1}^{n} r_{p} ∑_{j≥1} \frac{κ_{p}(−j)}{j}z^{jh}) exp (− ∑_{p=1}^{n} r_{p} ∑_{j≥1} \frac{κ_{p}(j)}{j}z^{−jh}) and

A^{α}(z) = Λ_{0}(A_{0}^{α} ⊗ s^{0})exp (∑_{i≥1} λ_{i}^{α} \frac{τ(1 − i)}{b_{i}}z^{b_{i}} ) exp (− ∑_{i≥1} λ_{1−i}^{α} \frac{τ(i)}{b_{i}}z^{−b_{i}} ) .

Nontrivial commutators in the principal Heisenberg subalgebra are

[τ(i), τ(1 − i)] = b_{i} and [γ_{p}(j), κ_{p}(−j)] = j.

The action of the Heisenberg subalgebra on the vertex operators is determined by

[τ(i), A^{α}(z)] = λ_{i}^{α}z^{b_{i}}A^{α}(z),
[γ_{p}(j), K_{0}(z, r)] = r_{p}z^{jh}K_{0}(z, r).

Consider the following operators on the space F:

L_{τ} (j) = \frac{1}{2} ∑_{i∈ℤ}:τ(1 − i)τ(i + jℓ): ,

L_{γκ}(j) = ∑_{p=1}^{n} ∑_{i∈ℤ}:γ_{p}(−i)κ_{p}(i + j): .

These are the analogues of the Sugawara operators.

Construct the formal generating series for these:

D_{s}(z) = − ∑_{j∈ℤ}(L_{τ} (j) + hL_{γκ}(j))z^{−jh}.

Proposition 8. Formula (4.5) together with

∑_{i∈ℤ} φ(s^{ih}t^{r}D_{s})z^{−ih} = D_{s}(z, r) = :D_{s}(z)K_{0}(z, r):

defines the representation of the Lie algebra D on the space

φ(𝔤̂) = φ(∑_{j∈ℤ} 𝔤̇_{j} ⊗ s^{j}ℂ[t_{1}^{±}, …, t_{n}^{±}] ⊕ K).

Proof. We have an action of D on 𝔤̂ which is a unique extension of its natural action on 𝔤̃. Thus we need to prove that φ([D, B]) = [φ(D), φ(B)] for every D ∈ D and B ∈ 𝔤̂. For subalgebra D_{+} this was proved in the course of Theorem 4, hence we need to consider only D = s^{ih}t^{r}D_{s}. Since s^{ih}t^{r}D_{s} act on 𝔤̂ as derivations and 𝔤̂ is generated as an algebra by elements A_{j}^{α} ⊗ s^{j}t^{m} then it is sufficient to prove that

φ([s^{ih}t^{r}D_{s}, A_{j}^{α} ⊗ s^{j}t^{m}]) = [φ(s^{ih}t^{r}D_{s}), φ(A_{j}^{α} ⊗ s^{j}t^{m})] .

Again we shall replace this with an equivalent identity for the corresponding series:

φ([∑_{i∈ℤ} s^{ih}t^{r}D_{s}z_{1}^{−ih}, ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{m}z_{2}^{−j}]) = [∑_{i∈ℤ} φ(s^{ih}t^{r}D_{s})z_{1}^{−ih}, ∑_{j∈ℤ} φ(A_{j}^{α} ⊗ s^{j}t^{m})z_{2}^{−j}].

(6.2)

We have

φ([∑_{i∈ℤ} s^{ih}t^{r}D_{s}z_{1}^{−ih}, ∑_{j∈ℤ} A_{j}^{α} ⊗ s^{j}t^{m}z_{2}^{−j}]) =

= ∑_{i∈ℤ} ∑_{j∈ℤ} jφ(A_{j}^{α} ⊗ s^{j+ih}t^{r+m})z_{1}^{−ih}z_{2}^{−j} =

= ∑_{i∈ℤ} ∑_{k=j+ih∈ℤ}(k − ih)φ(A_{k}^{α} ⊗ s^{k}t^{r+m})z_{1}^{−ih}z_{2}^{−k+ih} =

= ∑_{k∈ℤ} kφ(A_{k}^{α} ⊗ s^{k}t^{r+m})z_{2}^{−k}δ((\frac{z_{2}}{z_{1}} )^{h})

− h ∑_{k∈ℤ} φ(A_{k}^{α} ⊗ s^{k}t^{r+m})z_{2}^{−k}Dδ((\frac{z_{2}}{z_{1}} )^{h}) =

= −(D_{z_{2}}A^{α}(z_{2}, r + m))δ((\frac{z_{2}}{z_{1}} )^{h})

− A^{α}(z_{2}, r + m)(D_{z_{2}}δ((\frac{z_{2}}{z_{1}} )^{h})) =

= −D_{z_{2}}(A^{α}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})),

where D_{z_{2}} = z_{2} \frac{∂}{∂z_{2}} .

In order to compute the right hand side of (6.2), we will need two lemmas. Lemma 9.

(i)  [L_{τ} (j), A^{α}(z)] = z^{jh}(D_{z} + jh)A^{α}(z),
(ii)  [L_{τ} (z_{1}), A^{α}(z_{2})] = D_{z_{2}}(A^{α}(z_{2})δ((\frac{z_{2}}{z_{1}} )^{h})).

Proof. We will prove (i) assuming that j ≥ 0. The case j < 0 is analogous.

[L_{τ} (j), A^{α}(z)] = \frac{1}{2} [∑_{i∈ℤ}:τ(1 − i)τ(i + jℓ):, A^{α}(z)] =

= \frac{1}{2} [∑_{i>0} τ(1 − i)τ(i + jℓ), A^{α}(z)] + \frac{1}{2} [∑_{i≤0} τ(i + jℓ)τ(1 − i), A^{α}(z)] =

= \frac{1}{2} ∑_{i>0} τ(1 − i) [τ(i + jℓ), A^{α}(z)] + \frac{1}{2} ∑_{i>0} [τ(1 − i), A^{α}(z)] τ(i + jℓ)

+\frac{1}{2} ∑_{i≤0} τ(i + jℓ) [τ(1 − i), A^{α}(z)] + \frac{1}{2} ∑_{i≤0} [τ(i + jℓ), A^{α}(z)] τ(1 − i) =

= \frac{1}{2} ∑_{i>0} τ(1 − i)λ_{i}^{α}z^{b_{i}+jh}A^{α}(z) + \frac{1}{2} ∑_{i>0} λ_{1−i}^{α}z^{−b_{i}}A^{α}(z)τ(i + jℓ)

+\frac{1}{2} ∑_{i≤0} τ(i + jℓ)λ_{1−i}^{α}z^{−b_{i}}A^{α}(z) + \frac{1}{2} ∑_{i≤0} λ_{i}^{α}z^{b_{i}+jh}A^{α}(z)τ(1 − i) =

= \frac{1}{2} ∑_{i>0} τ(1 − i)λ_{i}^{α}z^{b_{i}+jh}A^{α}(z) + \frac{1}{2} ∑_{i>0} λ_{1−i}^{α}z^{−b_{i}}A^{α}(z)τ(i + jℓ)

+\frac{1}{2} ∑_{i≤−jℓ} τ(i + jℓ)λ_{1−i}^{α}z^{−b_{i}}A^{α}(z) + \frac{1}{2} ∑_{−jℓ<i≤0} λ_{1−i}^{α}z^{−b_{i}} [τ(i + jℓ), A^{α}(z)]

+\frac{1}{2} ∑_{−jℓ<i≤0} λ_{1−i}^{α}z^{−b_{i}}A^{α}(z)τ(i + jℓ) + \frac{1}{2} ∑_{i≤0} λ_{i}^{α}z^{b_{i}+jh}A^{α}(z)τ(1 − i) =

= \frac{1}{2} ∑_{k=i>0} τ(1 − k)λ_{k}^{α}z^{b_{k}+jh}A^{α}(z) + \frac{1}{2} ∑_{k=i+jℓ>jℓ} λ_{1−k}^{α}z^{−b_{k}+jh}A^{α}(z)τ(k)

+\frac{1}{2} ∑_{k=1−i−jℓ>0} λ_{k}^{α}z^{b_{k}+jh}τ(1 − k)A^{α}(z) + \frac{1}{2} ∑_{−jℓ<i≤0} λ_{1−i}^{α}z^{−b_{i}}λ_{i}^{α}z^{b_{i}+jh}A^{α}(z)

+\frac{1}{2} ∑_{0<k=i+jℓ≤jℓ} λ_{1−k}^{α}z^{−b_{k}+jh}A^{α}(z)τ(k) + \frac{1}{2} ∑_{k=1−i>0} λ_{1−k}^{α}z^{−b_{k}+jh}A^{α}(z)τ(k) =

= z^{jh}D_{z}A^{α}(z) + \frac{j}{2}z^{jh}(∑_{k=1}^{ℓ} λ_{k}^{α}λ_{1−k}^{α})A^{α}(z) =

= z^{jh}(D_{z} + hj)A^{α}(z).

At the last step we used the equality

\frac{1}{2} ∑_{k=1}^{ℓ} λ_{k}^{α}λ_{1−k}^{α} = \frac{1}{2} ∑_{k=1}^{ℓ}(T_{k}|ν^{−1}(α))(T_{1−k}|ν^{−1}(α)) =

\frac{h}{2}(ν^{−1}(α)|ν^{−1}(α)) = \frac{h}{2}(α|α) = h,

which follows from the fact that \{\frac{1}{h}T_{1−k}\} is the dual basis for \{T_{k}\}.

Part (ii) is an immediate consequence of (i):

[L_{τ} (z_{1}), A^{α}(z_{2})] = ∑_{j∈ℤ} z_{1}^{−jh} [L_{τ} (j), A^{α}(z_{2})] =

= ∑_{j∈ℤ} z_{1}^{−jh}z_{2}^{jh}(D_{z_{2}} + hj)A^{α}(z_{2}) = (D_{z_{2}}A^{α}(z_{2}))δ((\frac{z_{2}}{z_{1}} )^{h}) + A^{α}(z_{2})(D_{z_{2}}δ((\frac{z_{2}}{z_{1}} )^{h})) =

= D_{z_{2}}(A^{α}(z_{2})δ((\frac{z_{2}}{z_{1}} )^{h})).

Lemma 10.

[:L_{γκ}(z_{1})K_{0}(z_{1}, r):, K_{0}(z_{2}, m)] = \frac{1}{h}δ((\frac{z_{2}}{z_{1}} )^{h})D_{z_{2}}(K_{0}(z_{2}, m))K_{0}(z_{1}, r).

Proof.

[:L_{γκ}(z_{1})K_{0}(z_{1}, r):, K_{0}(z_{2}, m)] =

= ∑_{j∈ℤ} z_{1}^{−jh} ∑_{p=1}^{n} ∑_{i>0} [γ_{p}(−i)κ_{p}(i + j)K_{0}(z_{1}, r), K_{0}(z_{2}, m)]

+ ∑_{j∈ℤ} z_{1}^{−jh} ∑_{p=1}^{n} ∑_{i≤0} [κ_{p}(i + j)K_{0}(z_{1}, r)γ_{p}(−i), K_{0}(z_{2}, m)] =

= ∑_{j∈ℤ} z_{1}^{−jh} ∑_{p=1}^{n} ∑_{i>0} [γ_{p}(−i), K_{0}(z_{2}, m)] κ_{p}(i + j)K_{0}(z_{1}, r)

+ ∑_{j∈ℤ} z_{1}^{−jh} ∑_{p=1}^{n} ∑_{i≤0} κ_{p}(i + j)K_{0}(z_{1}, r) [γ_{p}(−i), K_{0}(z_{2}, m)] =

= ∑_{j∈ℤ} z_{1}^{−jh} ∑_{p=1}^{n} m_{p} ∑_{i>0} z_{2}^{−ih}K_{0}(z_{2}, m)κ_{p}(i + j)K_{0}(z_{1}, r)

+ ∑_{j∈ℤ} z_{1}^{−jh} ∑_{p=1}^{n} m_{p} ∑_{i≤0} κ_{p}(i + j)K_{0}(z_{1}, r)z_{2}^{−ih}K_{0}(z_{2}, m) =

= ∑_{j∈ℤ} ∑_{k=i+j∈ℤ} ∑_{p=1}^{n} m_{p}z_{1}^{−jh}z_{2}^{−kh+jh}κ_{p}(k)K_{0}(z_{2}, m)K_{0}(z_{1}, r) =

= \frac{1}{h}δ((\frac{z_{2}}{z_{1}} )^{h})D_{z_{2}}(K_{0}(z_{2}, m))K_{0}(z_{1}, r).

Using these two lemmas we can handle the right hand side of (6.2).

[∑_{i∈ℤ} φ(s^{ih}t^{r}D_{s})z_{1}^{−ih} , ∑_{j∈ℤ} φ(A_{j}^{α} ⊗ s^{j}t^{m})z_{2}^{−j}] =

= [D_{s}(z_{1}, r) , A^{α}(z_{2}, m)] =

= [:D_{s}(z_{1})K_{0}(z_{1}, r):, A^{α}(z_{2})K_{0}(z_{2}, m)] = = − [L_{τ} (z_{1})K_{0}(z_{1}, r), A^{α}(z_{2})K_{0}(z_{2}, m)]

− h [:L_{γκ}(z_{1})K_{0}(z_{1}, r):, A^{α}(z_{2})K_{0}(z_{2}, m)] = − [L_{τ} (z_{1}), A^{α}(z_{2})] K_{0}(z_{1}, r)K_{0}(z_{2}, m) − hA^{α}(z_{2}) [:L_{γκ}(z_{1})K_{0}(z_{1}, r):, K_{0}(z_{2}, m)] =

= −D_{z_{2}}(A^{α}(z_{2})δ((\frac{z_{2}}{z_{1}} )^{h}))K_{0}(z_{2}, m)K_{0}(z_{1}, r)

− A^{α}(z_{2})δ((\frac{z_{2}}{z_{1}} )^{h})D_{z_{2}}(K_{0}(z_{2}, m))K_{0}(z_{1}, r) =

= −D_{z_{2}}(A^{α}(z_{2})δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, m)K_{0}(z_{1}, r)) =

= −D_{z_{2}}(A^{α}(z_{2})δ((\frac{z_{2}}{z_{1}} )^{h})K_{0}(z_{2}, m)K_{0}(z_{2}, r)) =

= −D_{z_{2}}(A^{α}(z_{2}, r + m)δ((\frac{z_{2}}{z_{1}} )^{h})).

ACKNOWLEDGMENT. This work is supported by the Natural Sciences and Engineering Research Council of Canada.

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