Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus.

Yuly Billig^{1}, Alexander Molev^{2} and Ruibin Zhang^{2}

1Carleton University, School of Mathematics and Statistics, 1125 Colonel By Drive, Ottawa, K1S 5B6, Canada 2University of Sydney, School of Mathematics and Statistics, Sydney, NSW 2006, Australia

Abstract. We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.

Mathematics Subject Classification (2000): 17B66, 17B69.

Key words: Lie algebra of vector fields, vertex algebras, generalized Verma modules.

0. Introduction.

In recent years a new area in representation theory has emerged – the theory of bounded modules for infinite-dimensional Lie algebras with a dense ℤ^{n}-grading. The classical case n = 1 includes Kac-Moody algebras and the Virasoro algebra, and the main tool in the representation theory in this case is the concept of a Verma module. For n > 1, however, there is no natural way to split ℤ^{n}\{0} into a positive and a negative parts, and the Verma module technique does not yield interesting irreducible modules.

Nonetheless Berman and Billig [BB] showed that for a class of Lie algebras with a polynomial multiplication, the generalized Verma modules obtained by cutting ℤ^{n} with a hyperplane that intersects ℤ^{n} in a lattice of rank n − 1, have irreducible quotients with finite-dimensional weight spaces. Billig and Zhao [BZ] extended this result to Lie algebras and modules with exp-polynomial multiplication and gave many examples illustrating this theorem.

In the present paper we investigate representations of one of the most natural Lie algebras with a dense ℤ^{2}-grading – the Lie algebra of vector fields on a 2-dimensional torus:

D = Der ℂ[t_{0}^{±1}, t_{1}^{±1}].

This Lie algebra is ℤ^{2}-graded by the eigenvalues of the adjoint action of

d_{0} = t_{0} \frac{∂}{∂t_{0}} and d_{1} = t_{1} \frac{∂}{∂t_{1}} .

This grading is dense, i.e., its support is all of ℤ^{2}.

One class of representations of this Lie algebra with finite-dimensional weight spaces, the tensor modules, is well understood. These modules have a geometric realization as tensor fields on a torus.

Here we will study another family of simple modules for D – a class of bounded modules. Bounded modules are constructed using the general scheme given in [BB]. We consider a decomposition

D = D_{−} ⊕ D_{0} ⊕ D_{+}

given by the eigenvalues of d_{0}.

The degree zero part, D_{0}, is isomorphic to a semidirect product of the Lie algebra of vector fields on a circle with the abelian Lie algebra of functions on a circle. Tensor modules for D_{0} are parametrized by triples α, β, γ ∈ ℂ. The module T = T (α, β, γ) has a basis {v(n)|n ∈ γ + ℤ}, and the action of D_{0} is given by the formulas

\begin{aligned}(t_{1}^{m}d_{1})v(n) & = (n − αm)v(n + m), \\ (t_{1}^{m}d_{0})v(n) & = βv(n + m). \end{aligned}

We use these D_{0}-modules to construct the generalized Verma modules M(α, β, γ) for the Lie algebra D. Although M(α, β, γ) has infinite-dimensional weight spaces below the top T, the general result of [BB] implies that its irreducible quotient L(α, β, γ) has finite-dimensional weight spaces.

It is natural to ask for the dimensions of the weight spaces in L(α, β, γ). Here we give an explicit realization and a character formula for these modules when α ∉ ℚ and β = −\frac{α(α+1)}{2}. We show that in this case the picture is analogous to the situation with the basic module for affine Kac-Moody algebras.

Recall that an affine Kac-Moody algebra has an infinite-dimensional (principal) Heisenberg subalgebra, and its basic module remains irreducible when restricted to the Heisenberg subalgebra [LW], [K1]. This yields a realization of the basic module as a Fock space, and the action of the affine algebra is given by means of the vertex operators that are constructed as solutions of certain operator-valued differential equations.

For the class of modules for the Lie algebra of vector fields that we consider here, the role of the Heisenberg subalgebra is played by the loop subalgebra L = ℂ[t_{0}, t_{0}^{−1}]⊗sl_{2}(ℂ) ⊂ D.

Our main result, Theorem 1.2, states that for α ∉ ℚ, β = −\frac{α(α+1)}{2}, the module L(α, β, γ) remains irreducible when restricted to the loop subalgebra L. We shall show that L(α, β, γ) can be realized as an induced module for L:

L(α, β, γ) ≅ U(L_{−}) ⊗ T,  (0.1)

from which we immediately get a formula for the dimensions of the weight spaces.

To establish this result, we develop new methods that seem to be interesting in themselves. The main tool we use is the machinery of vertex algebras. We consider for the Lie algebra D the universal enveloping vertex algebra V_{D} and a certain quotient \overline{V}_{D} of it. In a straightforward way one can compute the elements of low degrees in the kernel of the projection

V_{D} → \overline{V}_{D}.

Applying the state-field correspondence map Y to the elements of the kernel, we obtain non-trivial relations on the action of the formal series

d_{0}(m, z) = ∑_{j∈ℤ} t_{0}^{j}t_{1}^{m}d_{0}z^{−j−2}

and

d_{1}(m, z) = ∑_{j∈ℤ} t_{0}^{j}t_{1}^{m}d_{1}z^{−j−1}.

In this way we derive the following relations in \overline{V}_{D}:

d_{0}(m, z) = \frac{1}{4} :d_{1}(1, z)d_{1}(m − 1, z): − \frac{1}{2} :d_{1}(0, z)d_{1}(m, z): + \frac{1}{4} :d_{1}(−1, z)d_{1}(m + 1, z):, (0.2)

\frac{∂}{∂z} d_{1}(m, z) = −\frac{m + 1}{2} :d_{1}(1, z)d_{1}(m − 1, z):

+m:d_{1}(0, z)d_{1}(m, z): − \frac{m − 1}{2} :d_{1}(−1, z)d_{1}(m + 1, z):.(0.3)

We prove that the same relations hold in the simple modules L(α, β, γ), provided that β = −\frac{α(α+1)}{2}. The relation (0.2) is a generalization of the Sugawara construction and implies that the action of d_{0}(m, z) may be expressed by means of the action of the subalgebra of the horizontal vector fields on the torus,

H = ℂ[t_{0}^{±1}, t_{1}^{±1}]d_{1}.

Hence, the module L(α, β, γ) with β = −\frac{α(α+1)}{2}, remains irreducible when restricted to the subalgebra H.

Our second step is to show that we can restrict to the yet smaller loop subalgebra L and still have an irreducible module. We achieve this with the help of the differential equation (0.3).

The idea of our proof may be seen from a well-known formula for the solution of an ordinary differential equation

\frac{∂}{∂z} x(z) = a(z)x(z),  (0.4)

which expresses x(z) as

x(z) = exp (∫_{0}^{z} a(z)dz) x(0).  (0.5)

This formula shows that the solution x(z) is “built” from a(z) and the initial value x(0).

We transform the differential equation (0.3) into an infinite system of differential equations for P (m, n, z) = d_{1}(m, z)v(n), m, n ∈ ℤ (see (5.1)). These differential equations are considerably more complicated than (0.4): it is an infinite system of equations; there is more than one term in the right hand side; the components of the operator-valued series in the right hand side do not commute; it involves normally ordered products; the series contains positive and negative powers of z.

These equations are highly non-commutative, even compared to the Fock space situation, where the commutator relations in the Heisenberg algebra are much simpler, and most of its components actually commute.

Because of this we need to redefine the exponential function, and this is done using iterative integration. To illustrate our method, we point out that the solution (0.5) may be alternatively written as a series

x(z) = ∑_{k=0}^{∞} b_{k}(z),

where b_{0}(z) = x(0) and b_{k+1}(z) = ∫_{0}^{z} a(z)b_{k}(z)dz.

In order to view an infinite system of equations as a single differential equation, we treat m and n as formal variables and work over the polynomial algebra ℂ[m, n].

The equation on P (m, n, z) that we get will be of the form (see (5.6)):

\frac{∂}{∂z} P (m, n, z) = (z^{−1}Ω + A_{+}(z)) P (m, n, z),  (0.6)

where the operator Ω acts in the following way:

ΩP (m, n, z) =

−\frac{1}{2}(m+1)(n−α)P (m−1, n+1, z)+mnP (m, n, z)− \frac{1}{2}(m−1)(n+α)P (m+1, n−1, z), (0.7)

and A_{+}(z) is a power series in non-negative powers of z with operator coefficients that correspond to the action of the loop subalgebra L.

Another difficulty that we have to overcome here is that the term z^{−1}Ω can’t be integrated in Laurent series. To handle this problem, we introduce a special integration formalism.

We find the solution of (0.6) as a series

P (m, n, z) = ∑_{k=0}^{∞} P_{k}(m, n, z),  (0.8)

where the terms in the right hand side are given by the recurrence relation

P_{k+1}(m, n, z) = z^{Ω} ∫ z^{−Ω}A_{+}(z)P_{k}(m, n, z)dz  (0.9)

with the integral defined in the following way:

z^{Ω} ∫ z^{−Ω} (∑_{j} R_{j}z^{j}) dz = ∑_{j} ((j + 1)I − Ω)^{−1} R_{j}z^{j+1}.

In order for this solution to be well-defined, we require the operators Ω − (j + 1)I to be invertible whenever R_{j} ≠ 0.

The formula (0.7) does not manifest invertibility of these operators on the space of polynomials ℂ[m, n]. In order to find the answer to this question, we give a representationtheoretic interpretation for the operator Ω. It turns out that Ω may be viewed as a Casimir operator on a tensor product of two Verma modules for sl_{2}(ℂ).

We fix a parameter α ∈ ℂ and consider the space of polynomials ℂ[x] as a module over the Witt algebra Der ℂ[t, t^{−1}] with the action given by

(t^{m}d)p(x) = (αm − x)p(x + m).

We denote this module by ℂ_{α}[x]. Restricting to the subalgebra sl_{2}(ℂ) = ℂt^{−1}d⊕ℂd⊕ℂtd, we get a representation of sl_{2}(ℂ) on ℂ_{α}[x]. In this framework, the operator Ω is just the bilinear Casimir operator on the tensor product ℂ_{α_{1}}[m] ⊗ ℂ_{α_{2}}[n] with α_{1} = 1, α_{2} = α. A related construction of the action of gl_{n}(ℂ) on the space of meromorphic functions in \frac{1}{2}n(n − 1) variables, has been given in [GKL].

In spite of the fact that the action of degree zero derivation d is not diagonalizable, dp(x) = −xp(x), the module ℂ_{α}[x] is nonetheless isomorphic to a Verma module for sl_{2}(ℂ) under a different choice of the Cartan element. This identification allows us to easily calculate the spectrum of Ω on the space ℂ[m, n] and determine that for α ∉ ℚ the operators Ω − jI are invertible for all j = 0, 1, 2, …

As a result, we conclude that the formulas (0.8) and (0.9) are valid, and the components of the series P (m, n, z) belong to the space U(L_{−})T. Using this fact, it is easy to establish the isomorphism (0.1).

The structure of the modules L(α, β, γ) for the values of the parameters not covered by Theorem 1.2, remains an open problem.

Acknowledgments. The first author thanks the University of Sydney for hospitality during his visit there. Financial support from the Natural Sciences and Engineering Research Council of Canada and the Australian Research Council is gratefully acknowledged.

1. Representations of Lie algebra of vector fields on a torus.

The algebra of Fourier polynomial functions on a 2-dimensional torus is isomorphic to the algebra of Laurent polynomials in two variables R = ℂ[t_{0}^{±1}, t_{1}^{±1}]. The Lie algebra of (polynomial) vector fields on a torus is the algebra of derivations of R:

D = Der (R) = R\frac{∂}{∂t_{0}} ⊕ R\frac{∂}{∂t_{1}} .

It is convenient to use slightly different generators of Der (R) as a free R-module:

d_{0} = t_{0} \frac{∂}{∂t_{0}} , d_{1} = t_{1} \frac{∂}{∂t_{1}} .

The Lie bracket is given by the formula:

[t_{0}^{m_{0}}t_{1}^{m_{1}}d_{a}, t_{0}^{r_{0}}t_{1}^{r_{1}}d_{b}] = r_{a}t_{0}^{m_{0}+r_{0}}t_{1}^{m_{1}+r_{1}}d_{b} − m_{b}t_{0}^{m_{0}+r_{0}}t_{1}^{m_{1}+r_{1}}d_{a}.  (1.1)

The degree operators d_{0}, d_{1} induce a ℤ^{2}-grading on D by their eigenvalues in the adjoint representation. We will also consider a ℤ-grading on D induced just by d_{0}.

We will need two subalgebras in D – the Lie algebra of “horizontal” vector fields

H = Rd_{1}

and the loop algebra

L = ℂ[t_{0}, t_{0}^{−1}] {ℂt_{1}^{−1}d_{1} ⊕ ℂd_{1} ⊕ ℂt_{1}d_{1}} .

It is easy to see that the latter subalgebra is isomorphic to the loop Lie algebra

L ≅ ℂ[t_{0}, t_{0}^{−1}] ⊗ sl_{2}(ℂ).

Next we will construct a class of bounded modules for these Lie algebras.

We take a decomposition of D into three parts,

D = D_{+} ⊕ D_{0} ⊕ D_{−},

corresponding to positive, zero and negative eigenvalues relative to d_{0}. In particular,

D_{0} = ℂ[t_{1}, t_{1}^{−1}]d_{0} ⊕ ℂ[t_{1}, t_{1}^{−1}]d_{1}.

In the same way we consider the decompositions for H and L, where H_{0} = H ∩ D_{0}, L_{0} = L ∩ D_{0}, etc. Note that H_{0} is the algebra of vector fields on a circle and L_{0} ≅ sl_{2}(ℂ).

Consider the family of modules T = T (α, β, γ), α, β, γ ∈ ℂ, for D_{0} with the basis {v(n)|n ∈ γ + ℤ} and the action defined by the formulas:

\begin{aligned}(t_{1}^{m}d_{1})v(n) & = (n − αm)v(n + m), \\ (t_{1}^{m}d_{0})v(n) & = βv(n + m). \end{aligned}

It is well-known that T (α, β, γ) is irreducible as a D_{0}-module unless α ∈ {−1, 0}, β = 0 and γ ∈ ℤ (see e.g. [E1]).

We let D_{+} act on T trivially, and construct the generalized Verma module

M_{D}(T) = M_{D}(α, β, γ) = Ind_{D_{0}⊕D_{+}}^{D}T (α, β, γ) ≅ U(D_{−}) ⊗ T (α, β, γ).

We call T the top of M_{D}(T).

In the same way, viewing T as a module for H_{0} and L_{0}, we construct the generalized Verma modules over H and L:

M_{H}(T) = M_{H}(α, γ) = U(H_{−}) ⊗ T, M_{L}(T) = M_{L}(α, γ) = U(L_{−}) ⊗ T.

In the notations above we dropped β since the actions of H_{0} and L_{0} on T (α, β, γ) are independent of β.

These modules have ℤ^{2}-gradings compatible with the gradings on the Lie algebras. In these gradings, the modules M_{D}(T) and M_{H}(T) have infinite-dimensional components below the top, whereas all components of M_{L}(T) are finite-dimensional.

The module M_{D}(T) has a unique maximal submodule. Indeed, any submodule in it is homogeneous with respect to ℤ^{2}-grading. Since T as a D_{0}-module has a unique maximal submodule (which is trivial in most cases), the intersection of any proper D-submodule with T should be in the maximal submodule of T. Hence the sum of proper submodules of M_{D}(T) is again a proper submodule, and the sum of all proper submodules is the unique maximal submodule.

The same argument shows that M_{H}(T) has a unique maximal homogeneous (with respect to ℤ^{2}-grading) submodule. We define L_{D}(T) = L_{D}(α, β, γ) (resp. L_{H}(T)) as a quotient of M_{D}(T) (resp. M_{H}(T)) by the maximal (resp. maximal homogeneous) submodule.

The following Theorem is a direct corollary of a general result of Berman and Billig ([BB], Theorem 1.12):

Theorem 1.1. The modules L_{D}(α, β, γ) and L_{H}(α, γ) have finite-dimensional components in ℤ^{2}-grading.

A natural question stems from Theorem 1.1:

Question. What are the characters of the modules L_{D}(α, β, γ) and L_{H}(α, γ)?

This question is also motivated by a conjecture which is due to Eswara Rao [E2], which in the case of a 2-torus states:

Conjecture. Every simple ℤ^{2}-graded module for the Lie algebra of vector fields on a 2-dimensional torus with finite-dimensional components of the grading is either a subquotient of a tensor module or is isomorphic to L_{D}(α, β, γ), possibly twisted with an automorphism of D.

This conjecture is an analogue of Kac’s conjecture, solved by Mathieu [M], which describes the case of the vector fields on a circle.

Whereas the tensor modules are completely understood, nothing was previously known about the modules L_{D}(α, β, γ). The goal of the present paper is to give an explicit description for some of these modules. We will prove the following

Theorem 1.2. Let α ∈ ℂ, α ∉ ℚ, β = −\frac{1}{2}α(α + 1), γ ∈ ℂ. Then

L_{D}(α, β, γ) ≅ L_{H}(α, γ) ≅ M_{L}(α, γ).

That is, the action of the loop algebra L on M_{L}(α, γ) can be extended to an irreducible action of the whole algebra of vector fields D turning M_{L}(α, γ) into L_{D}(α, β, γ).

Let V be a module over the Lie algebra D, graded by eigenvalues of d_{0} and d_{1}:

V = ⨁_{(s_{0},s_{1})∈ℂ×ℂ} V_{s_{0},s_{1}},

where

V_{s_{0},s_{1}} = {v ∈ V | d_{0}v = s_{0}v, d_{1}v = s_{1}v} ,

and define the character of V to be

char V = ∑_{(s_{0},s_{1})∈ℂ×ℂ} dim (V_{s_{0},s_{1}}) q_{0}^{s_{0}}q_{1}^{s_{1}}.

Corollary 1.3. Under the assumptions of Theorem 1.2 on α, β, γ,

char L_{D}(α, β, γ) = q_{0}^{β} ( ∑_{j∈γ+ℤ} q_{1}^{j}) ∏_{n≥1}(1 − q_{0}^{−n})^{−3}.  (1.2)

2. Vertex algebras and vertex Lie algebras.

In this section we are going to construct vertex algebras associated with the Lie algebras D, H and L. We briefly recall some properties of vertex algebras and refer to [K2] and [L] for details.

A vertex algebra V is a vector space with additional structures (Y, D, 1), where the state-field correspondence Y is a map

Y : V → End(V)[[z, z^{−1}]],

the infinitesimal translation D is an operator D : V → V, and 1 is a vacuum vector 1 ∈ V. For a ∈ V we write

Y (a, z) = ∑_{n∈ℤ} a_{(n)}z^{−n−1}, a_{(n)} ∈ End(V).

For all a, b ∈ V,

a_{(n)}b = 0 for n ≫ 0.

The infinitesimal translation map D satisfies the axiom

Y (Da, z) = \frac{∂}{∂z} Y (a, z).

Finally, we will need the commutator formula:

[Y (a, z_{1}), Y (b, z_{2})] = ∑_{n≥0} \frac{1}{n!}Y (a_{(n)}b, z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{n} δ (\frac{z_{2}}{z_{1}} )] .  (2.1)

The delta function that appears in (2.1) above is

δ(z) = ∑_{n∈ℤ} z^{n}.

There exists a uniform procedure of constructing vertex algebras from vertex Lie algebras, and the Lie algebras under consideration are all vertex Lie algebras. Let us recall a definition of a vertex Lie algebra [DLM].

Let S be a Lie algebra with the basis {u(n), c(−1)|u ∈ U, c ∈ C, n ∈ ℤ} (U, C are some index sets). Define the corresponding fields in S[[z, z^{−1}]]:

u(z) = ∑_{n∈ℤ} u(n)z^{−n−1}, c(z) = c(−1)z^{0}, u ∈ U, c ∈ C.

Let F be a subspace in S[[z, z^{−1}]] spanned by all the fields u(z), c(z) and their derivatives of all orders.

Definition. A Lie algebra S with the basis as above is called a vertex Lie algebra if the following two conditions hold:

(VL1) for all u_{1}, u_{2} ∈ U,

[u_{1}(z_{1}), u_{2}(z_{2})] = ∑_{j=0}^{n} f_{j}(z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{j} δ (\frac{z_{2}}{z_{1}} )] ,  (2.2)

where f_{j}(z) ∈ F, n ≥ 0 and both depend on u_{1}, u_{2}, (VL2) for all c ∈ C, the elements c(−1) are central in S.

Let S^{(+)} be a subspace in S with the basis {u(n)|u ∈ U, n ≥ 0} and let S^{(−)} be a subspace with the basis {u(n), c(−1)|u ∈ U, c ∈ C, n < 0}. Then S = S^{(+)} ⊕ S^{(−)} and S^{(+)}, S^{(−)} are in fact subalgebras in S.

The universal enveloping vertex algebra V_{S} of a vertex Lie algebra S is defined as an induced module

V_{S} = Ind_{S^{(+)}}^{S}(ℂ1) = U(S^{(−)}) ⊗ 1,

where ℂ1 is a trivial 1-dimensional S^{(+)} module.

Theorem 2.1. ([DLM], Theorem 4.8) Let S be a vertex Lie algebra. Then

(a) V_{S} has a structure of a vertex algebra with the vacuum vector 1, infinitesimal translation D being a natural extension of the derivation of S given by

D(u(n)) = −nu(n − 1), D(c(−1)) = 0, u ∈ U, c ∈ C,

and the state-field correspondence map Y defined by the formula:

Y (a_{1}(−1 − n_{1}) … a_{k−1}(−1 − n_{k−1})a_{k}(−1 − n_{k})1, z)

= : (\frac{1}{n_{1}!} (\frac{∂}{∂z} )^{n_{1}} a_{1}(z)) …: (\frac{1}{n_{k−1}!} (\frac{∂}{∂z} )^{n_{k−1}} a_{k−1}(z)) (\frac{1}{n_{k}!} (\frac{∂}{∂z} )^{n_{k}} a_{k}(z)): …: ,

(2.3)

where a_{j} ∈ U, n_{j} ≥ 0 or a_{j} ∈ C, n_{j} = 0.

(b) Any bounded S-module is a vertex algebra module for V_{S}.

In the formula (2.3) above, the normal ordering of two fields :a(z)b(z): is defined as

:a(z)b(z): = a_{+}(z)b(z) + b(z)a_{−}(z),

where a_{+}(z) (resp. a_{−}(z)) is the part of the power series a(z) involving non-negative (resp. negative) powers of z, a(z) = a_{+}(z) + a_{−}(z).

There is a natural relation between S-submodules of V_{S} and vertex algebra ideals of V_{S}.

Theorem 2.2. (see e.g. [B]) Any D-invariant S-submodule in V_{S} is a vertex algebra ideal in V_{S}. Conversely, every vertex algebra ideal in V_{S} is a D-invariant S-submodule.

Theorem 2.3. The Lie algebras D, H and L are vertex Lie algebras. The generating fields in D are

d_{1}(m, z) = ∑_{j∈ℤ} t_{0}^{j}t_{1}^{m}d_{1}z^{−j−1},  (2.4)

d_{0}(m, z) = ∑_{j∈ℤ} t_{0}^{j}t_{1}^{m}d_{0}z^{−j−2},  (2.5)

where m ∈ ℤ. The generating fields of H are (2.4) with m ∈ ℤ, and the generating fields of L are (2.4) with m = −1, 0, 1.

Note that the centers of these Lie algebras are trivial, and the set C is empty in all of these cases.

Proof. We need to show that the commutators between the fields (2.4) and (2.5) conform with (2.2).

[d_{1}(m, z_{1}), d_{1}(r, z_{2})] = ∑_{ij} [t_{0}^{i}t_{1}^{m}d_{1}, t_{0}^{j}t_{1}^{r}d_{1}] z_{1}^{−i−1}z_{2}^{−j−1}

= (r−m) ∑_{ij} (t_{0}^{i+j}t_{1}^{m+r}d_{1}z_{2}^{−i−j−1}) (z_{1}^{−i−1}z_{2}^{i}) = (r−m)d_{1}(m+r, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] . (2.6)

[d_{1}(m, z_{1}), d_{0}(r, z_{2})] = ∑_{ij} [t_{0}^{i}t_{1}^{m}d_{1}, t_{0}^{j}t_{1}^{r}d_{0}] z_{1}^{−i−1}z_{2}^{−j−2}

= r ∑_{ij} (t_{0}^{i+j}t_{1}^{m+r}d_{0}z_{2}^{−i−j−2}) (z_{1}^{−i−1}z_{2}^{i}) − ∑_{ij} (t_{0}^{i+j}t_{1}^{m+r}d_{1}z_{2}^{−i−j−1}) (iz_{1}^{−i−1}z_{2}^{i−1})

= rd_{0}(m + r, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] − d_{1}(m + r, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] .  (2.7)

[d_{0}(m, z_{1}), d_{0}(r, z_{2})] = ∑_{ij} [t_{0}^{i}t_{1}^{m}d_{0}, t_{0}^{j}t_{1}^{r}d_{0}] z_{1}^{−i−2}z_{2}^{−j−2}

= ∑_{ij}(j − i)t_{0}^{i+j}t_{1}^{m+r}d_{0}z_{1}^{−i−2}z_{2}^{−j−2}

= ∑_{ij} ((i + j + 2)t_{0}^{i+j}t_{1}^{m+r}d_{0}z_{2}^{−i−j−3}) (z_{1}^{−i−2}z_{2}^{i+1})

−2 ∑_{ij} (t_{0}^{i+j}t_{1}^{m+r}d_{0}z_{2}^{−i−j−2}) ((i + 1)z_{1}^{−i−2}z_{2}^{i})

= − (\frac{∂}{∂z_{2}} d_{0}(m + r, z_{2})) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] − 2d_{0}(m + r, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ (\frac{z_{2}}{z_{1}} )] .  (2.8)

This proves the claim of the theorem.

It follows from (2.4), (2.5) that

D^{(+)} = Span {t_{0}^{i}t_{1}^{m}d_{1}, t_{0}^{j}t_{1}^{m}d_{0} | i ≥ 0, j ≥ −1, m ∈ ℤ} ,

D^{(−)} = Span {t_{0}^{i}t_{1}^{m}d_{1}, t_{0}^{j}t_{1}^{m}d_{0} | i ≤ −1, j ≤ −2, m ∈ ℤ} ,

and H^{(±)} = H ∩ D^{(±)}, L^{(±)} = L ∩ D^{(±)}.

By applying Theorem 2.1, we obtain the vertex algebras V_{D} = U(D^{(−)}) ⊗ 1, V_{H} = U(H^{(−)})⊗1 and V_{L} = U(L^{(−)})⊗1. We have natural embeddings of these vertex algebras: V_{L} ⊂ V_{H} ⊂ V_{D}.

3. Relations in vertex algebras and modules.

The vertex algebras V_{D}, V_{H} and V_{L} are graded by ℤ^{2} with 1 having degree (0, 0). In the ℤ-grading by the degree in t_{0}, these algebras are trivial in positive degree. At the top, in degree zero relative to t_{0}, we have the only non-trivial component ℂ1. When we go to the next layer in degree −1, then in V_{D} and V_{H} we get an infinite direct sum of 1-dimensional spaces spanned by (t_{0}^{−1}t_{1}^{m}d_{1})1 in each degree (−1, m). Note the following relation in V_{D}:

(t_{0}^{−1}t_{1}^{m}d_{0})1 = 0.  (3.1)

The vertex algebras V_{D}, V_{H} and V_{L} are not simple. In fact, their maximal ideals correspond to the augmentation ideals in U(D^{(−)}), etc., and thus the quotients by the maximal ideals yield just a 1-dimensional trivial vertex algebra ℂ1. Nonetheless, it is possible to extract useful information for the representation theory of the corresponding Lie algebras by studying certain quotients of these vertex algebras.

Let V be one of the vertex algebras, V_{D}, V_{H} or V_{L}. As an alternative to the maximal ideal, we will define the subradical of V as an ideal V^{sr} in V, maximal among the homogeneous (in ℤ^{2}-grading) ideals trivially intersecting with the space V_{−1} of elements of degree −1 with respect to t_{0}.

Let \overline{V} be the quotient \overline{V} = V/V^{sr}. Theorem 3.1. The vertex algebras \overline{V}_{D} and \overline{V}_{H} are isomorphic. More precisely, there

exists an isomorphism \overline{V}_{H} → \overline{V}_{D}, making the following diagram commutative:

V_{H} ⟶ V_{D}
↓  ↓
\overline{V}_{H} ⟶ \overline{V}_{D}

This theorem implies that the H-module \overline{V}_{H} admits a natural structure of a module over the larger Lie algebra D.

In order to prove this theorem, we need to study the subradical V_{D}^{sr} at degree −2. The elements of the subradical yield relations in \overline{V}_{D} which will be crucial for our work.

Proposition 3.2. (a) The component of \overline{V}_{D} of degree (−2, m) has dimension 3, and is spanned by the elements (t_{0}^{−1}t_{1}^{k}d_{1})(t_{0}^{−1}t_{1}^{m−k}d_{1})1 with k = −1, 0, 1.

(b) The following relations in \overline{V}_{D} describe the projection V_{D} → \overline{V}_{D} in degree (−2, m):

(t_{0}^{−2}t_{1}^{m}d_{0})1 =

\frac{1}{4}(t_{0}^{−1}t_{1}d_{1})(t_{0}^{−1}t_{1}^{m−1}d_{1})1 − \frac{1}{2}(t_{0}^{−1}d_{1})(t_{0}^{−1}t_{1}^{m}d_{1})1 + \frac{1}{4}(t_{0}^{−1}t_{1}^{−1}d_{1})(t_{0}^{−1}t_{1}^{m+1}d_{1})1, (3.2)

(t_{0}^{−2}t_{1}^{m}d_{1})1 = −\frac{m + 1}{2}(t_{0}^{−1}t_{1}d_{1})(t_{0}^{−1}t_{1}^{m−1}d_{1})1

+m(t_{0}^{−1}d_{1})(t_{0}^{−1}t_{1}^{m}d_{1})1 − \frac{m − 1}{2}(t_{0}^{−1}t_{1}^{−1}d_{1})(t_{0}^{−1}t_{1}^{m+1}d_{1})1, (3.3)

(t_{0}^{−1}t_{1}^{r}d_{1})(t_{0}^{−1}t_{1}^{m−r}d_{1})1 = \frac{r(r + 1)}{2}(t_{0}^{−1}t_{1}d_{1})(t_{0}^{−1}t_{1}^{m−1}d_{1})1

−(r − 1)(r + 1)(t_{0}^{−1}d_{1})(t_{0}^{−1}t_{1}^{m}d_{1})1 + \frac{r(r − 1)}{2}(t_{0}^{−1}t_{1}^{−1}d_{1})(t_{0}^{−1}t_{1}^{m+1}d_{1})1. (3.4)

Proof. In order to prove this proposition we need to study the action of the raising operators t_{0}t_{1}^{s}d_{0} and t_{0}t_{1}^{s}d_{1} on the component of degree (−2, m) of V_{D}. The corresponding component of the subradical V_{D}^{sr} is the joint kernel of these raising operators.

The following calculations are straightforward and use (3.1):

\begin{aligned}(t_{0}t_{1}^{s}d_{0})(t_{0}^{−2}t_{1}^{m}d_{0})1 & = 0, \\ (t_{0}t_{1}^{s}d_{1})(t_{0}^{−2}t_{1}^{m}d_{0})1 & = −(t_{0}^{−1}t_{1}^{m+s}d_{1})1, \end{aligned}

\begin{aligned}(t_{0}t_{1}^{s}d_{0})(t_{0}^{−2}t_{1}^{m}d_{1})1 & = −2(t_{0}^{−1}t_{1}^{m+s}d_{1})1, \\ (t_{0}t_{1}^{s}d_{1})(t_{0}^{−2}t_{1}^{m}d_{1})1 & = (m − s)(t_{0}^{−1}t_{1}^{m+s}d_{1})1, \end{aligned}

(t_{0}t_{1}^{s}d_{0})(t_{0}^{−1}t_{1}^{r}d_{1})(t_{0}^{−1}t_{1}^{m−r}d_{1})1 = (2s + 2r − m)(t_{0}^{−1}t_{1}^{m+s}d_{1})1,

(t_{0}t_{1}^{s}d_{1})(t_{0}^{−1}t_{1}^{r}d_{1})(t_{0}^{−1}t_{1}^{m−r}d_{1})1 = (r − s)(m − 2r − s)(t_{0}^{−1}t_{1}^{m+s}d_{1})1. (3.5)

Since s is an arbitrary integer, we will treat the coefficients in the right hand sides above as polynomials in s. We consider a map ψ from the (−2, m)-component of V_{D} to ℂ[s] ⊕ ℂ[s] where the two components of ψ(x) are the coefficients at (t_{0}^{−1}t_{1}^{m+s}d_{1})1 in (t_{0}t_{1}^{s}d_{0})x and (t_{0}t_{1}^{s}d_{1})x. Then the kernel of ψ consists of the elements in the subradical, and the dimension of the quotient is equal to the dimension of the image of ψ.

By inspection, we can see that the image of ψ is spanned by {(0, 1), (2, s), (2s+m, s^{2})}, thus the quotient by the subradical in degree (−2, m) has dimension 3. It is easy to verify that ψ((t_{0}^{−1}t_{1}^{r}d_{1})(t_{0}^{−1}t_{1}^{m−r}d_{1})1) with r = −1, 0, 1 span the image of ψ. Using the formulas above it is easy to check that the differences between left hand sides and right hand sides in (3.2)-(3.4) are in the subradical.

Remark. In fact (3.3) can be derived from (3.4) by using the relation t_{0}^{−2}t_{1}^{m}d_{1} = (m − 2r)^{−1}[t_{0}^{−1}t_{1}^{r}d_{1}, t_{0}^{−1}t_{1}^{m−r}d_{1}] with m − 2r ≠ 0.

Corollary 3.3. The following relations hold in the vertex algebra \overline{V}_{D}:

d_{0}(m, z) = \frac{1}{4} :d_{1}(1, z)d_{1}(m − 1, z): − \frac{1}{2} :d_{1}(0, z)d_{1}(m, z): + \frac{1}{4} :d_{1}(−1, z)d_{1}(m + 1, z):, (3.6)

\frac{∂}{∂z} d_{1}(m, z) = −\frac{m + 1}{2} :d_{1}(1, z)d_{1}(m − 1, z):

+m:d_{1}(0, z)d_{1}(m, z): − \frac{m − 1}{2} :d_{1}(−1, z)d_{1}(m + 1, z):, (3.7)

:d_{1}(r, z)d_{1}(m − r, z): = \frac{r(r + 1)}{2} :d_{1}(1, z)d_{1}(m − 1, z):

−(r − 1)(r + 1):d_{1}(0, z)d_{1}(m, z): + \frac{r(r − 1)}{2} :d_{1}(−1, z)d_{1}(m + 1, z):. (3.8)

Proof. The relations (3.6)-(3.8) are derived from (3.2)-(3.4) by applying the state-field correspondence map Y, and noting that from the construction of the universal enveloping vertex algebra we have Y ((t_{0}^{−2}t_{1}^{m}d_{0})1, z) = d_{0}(m, z), Y ((t_{0}^{−1}t_{1}^{m}d_{1})1, z) = d_{1}(m, z) and Y ((t_{0}^{−2}t_{1}^{m}d_{1})1, z) = \frac{∂}{∂z} d_{1}(m, z) both in V_{D} and \overline{V}_{D}.

Now we are ready to give a proof of Theorem 3.1. It follows from (3.6) that for any v ∈ \overline{V}_{D} we have (t_{0}^{j}t_{1}^{m}d_{0})v ∈ U(H)v for all j, m ∈ ℤ. This implies that the map

V_{H} → \overline{V}_{D},

obtained as a composition of embedding V_{H} → V_{D} and projection V_{D} → \overline{V}_{D}, is surjective. It remains to show that the kernel of this map is the subradical of V_{H}, which follows from Lemma 3.4. V_{D}^{sr} ∩ V_{H} = V_{H}^{sr}. Proof. Indeed, let v be a homogeneous element in V_{H} ⊂ V_{D}. This element belongs to

V_{D}^{sr} if and only if U(D)v trivially intersects with the degree −1 layer of V_{D}. Note that the degree −1 layers in V_{D} and V_{H} coincide. Due to (3.6), U(D)v = U(H)v mod V_{D}^{sr}. Since V_{D}^{sr} trivially intersects with the degree −1 layer of V_{D}, we get that v ∈ V_{D}^{sr} if and only if v ∈ V_{H}^{sr}. This completes the proof of the Lemma.

Thus the map V_{H} → \overline{V}_{D} factors through \overline{V}_{H} and the resulting map
\overline{V}_{H} → \overline{V}_{D}

is an isomorphism of H-modules. Theorem 3.1 is now proved.

We will prove below an analogue of Theorem 3.1 for the modules over D and H. We will show that irreducible H-modules L_{H}(α, γ) admit the action of the bigger algebra D, with the action extended by (3.6). In order to prove this result, we will need a description of the layer of degree −3 in \overline{V}_{H}. Since the subalgebra H^{(−)} is generated by its elements of degree −1, the component of degree (−3, m) in V_{H} is spanned by the elements (t_{0}^{−1}t_{1}^{r}d_{1})(t_{0}^{−1}t_{1}^{k}d_{1})(t_{0}^{−1}t_{1}^{m−r−k}d_{1})1, and hence we will need to describe how such elements reduce in \overline{V}_{H}.

Proposition 3.5. (a) The component of \overline{V}_{H} of degree (−3, m) has dimension 8, and is spanned by the elements (t_{0}^{−1}t_{1}^{i}d_{1})(t_{0}^{−1}t_{1}^{j}d_{1})(t_{0}^{−1}t_{1}^{m−i−j}d_{1})1 with i, j = −1, 0, 1, which satisfy the following linear relation in \overline{V}_{H}:

∑_{i,j=−1,0,1} A_{i+2,j+2}(t_{0}^{−1}t_{1}^{i}d_{1})(t_{0}^{−1}t_{1}^{j}d_{1})(t_{0}^{−1}t_{1}^{m−i−j}d_{1})1 = 0,  (3.9)

with

A = ( \begin{matrix}m(m − 1)(m − 2) & −2m(m − 1)(m − 2) & (m − 3)(m − 2)(m + 2) \\ −2(m − 1)(m − 2)(m + 3) & 4m(m − 2)(m + 2) & −2(m − 3)(m + 1)(m + 2) \\ (m − 2)(m + 2)(m + 3) & −2m(m + 1)(m + 2) & m(m + 1)(m + 2) \end{matrix} ) .

(b) The following relation holds in \overline{V}_{D} and describes the projection V_{H} → \overline{V}_{H} in degree (−3, m):

(t_{0}^{−1}t_{1}^{r}d_{1})(t_{0}^{−1}t_{1}^{k}d_{1})(t_{0}^{−1}t_{1}^{m−r−k}d_{1})1 = ∑_{i,j=−1,0,1} B_{i+2,j+2}(t_{0}^{−1}t_{1}^{i}d_{1})(t_{0}^{−1}t_{1}^{j}d_{1})(t_{0}^{−1}t_{1}^{m−i−j}d_{1})1

(3.10)

with

B = \frac{1}{12} ( \begin{matrix}r(r − 1) & 0 & 0 \\ 0 & (r − 1)(r + 1) & 0 \\ 0 & 0 & r(r + 1) \end{matrix} ) ×

× ( \begin{matrix}c + r − k & −2(c + 4r + 2k) & c + 7r + 5k + 6 \\ −2(c − 3r − 3k) & 4(c − 3) & −2(c + 3r + 3k) \\ c − 7r − 5k + 6 & −2(c − 4r − 2k) & c − r + k \end{matrix} ) ,

where c = r^{2} + 2rk + 3k^{2}.

This proposition is proved in the same way as Proposition 3.2, and we will omit the details of this straightforward calculation.

Let R be the vertex algebra ideal in V_{H} generated by the elements of V_{H}^{sr} corresponding to the relations (3.4), (3.9) and (3.10). Define Ṽ_{H} to be the quotient V_{H}/R. The reason for considering such a quotient is that we need relations corresponding to the elements of degrees −2 and −3 in V_{H}^{sr}, and yet when we study modules for V_{H}, it is easier to check that the relations corresponding to (3.4), (3.9) and (3.10) hold in the given module, rather than checking that all relations corresponding to the subradical V_{H}^{sr} hold.

Theorem 3.6. There is an epimorphism of vertex algebras

V_{D} → Ṽ_{H}

which is defined by (3.2) on the generators (t_{0}^{−2}t_{1}^{m}d_{0})1 and by the identity map on (t_{0}^{−1}t_{1}^{m}d_{1})1.

Proof. Denote the right hand side of (3.2) by d̃_{0}(m). We need to show that the fields Y (d̃_{0}(m), z) and Y ((t_{0}^{−1}t_{1}^{m}d_{1})1, z) in Ṽ_{H} yield a representation of the Lie algebra D, i.e., they satisfy relations (2.6)-(2.8). We can use the commutator formula (2.1) to express these relations via n-th products:

(t_{0}^{−1}t_{1}^{m}d_{1}1)_{(0)}(t_{0}^{−1}t_{1}^{r}d_{1}1) = (r − m)(t_{0}^{−1}t_{1}^{r+m}d_{1}1),  (3.11)
(t_{0}^{−1}t_{1}^{m}d_{1}1)_{(n)}(t_{0}^{−1}t_{1}^{r}d_{1}1) = 0 for n ≥ 1,  (3.12)
(t_{0}^{−1}t_{1}^{m}d_{1}1)_{(0)}d̃_{0}(r) = rd̃_{0}(r + m),  (3.13)
(t_{0}^{−1}t_{1}^{m}d_{1}1)_{(1)}d̃_{0}(r) = −(t_{0}^{−1}t_{1}^{r+m}d_{1}1),  (3.14)
(t_{0}^{−1}t_{1}^{m}d_{1}1)_{(n)}d̃_{0}(r) = 0 for n ≥ 2,  (3.15)
d̃_{0}(m)_{(0)}d̃_{0}(r) = −Dd̃_{0}(r + m),  (3.16)
d̃_{0}(m)_{(1)}d̃_{0}(r) = −2d̃_{0}(r + m),  (3.17)
d̃_{0}(m)_{(n)}d̃_{0}(r) = 0 for n ≥ 2.  (3.18)

Relations (3.11) and (3.12) obviously hold. To verify (3.16), we will work in the vertex algebra V_{D}. We have

d̃_{0}(m)_{(0)}d̃_{0}(r) = (t_{0}^{−2}t_{1}^{m}d_{0}1)_{(0)}(t_{0}^{−2}t_{1}^{r}d_{0}1) mod V_{D}^{sr}
= −D(t_{0}^{−2}t_{1}^{r+m}d_{0}1) = −Dd̃_{0}(r + m) mod V_{D}^{sr}.

Thus the difference between the right hand side and the left hand side in (3.16) belongs to V_{D}^{sr}. By Lemma 3.4 it is actually in V_{H}^{sr}. Note also that these elements are of degree −3, and the ideal R contains all elements of V_{H}^{sr} of degree −3 by its construction. Hence the two sides of (3.16) are equal in Ṽ_{H}. The verification of other relations is completely analogous.

Corollary 3.7. Every module for the vertex algebra Ṽ_{H} admits the action of the Lie algebra D defined by (3.6).

Theorem 3.8. Let β = −\frac{α(α+1)}{2}. Then the modules L_{H}(α, γ) and L_{D}(α, β, γ) are isomorphic. The action of H on L_{H}(α, γ) extends to the action of D by (3.6).

The idea of the proof is to show that L_{H}(α, γ) is a module for the vertex algebra Ṽ_{H} and then apply Corollary 3.7. To show that L_{H}(α, γ) admits the action of Ṽ_{H}, we need to prove that for every generator u of the ideal R in V_{H}, we have

Y (u, z)L_{H}(α, γ) = 0.

Expand

Y (u, z) = ∑_{j∈ℤ} u_{(j)}z^{−j−1}, u_{(j)} ∈ End(L_{H}(α, γ)).

Consider the subspace S in End(L_{H}(α, γ)) spanned by u_{(j)}, j ∈ ℤ, where u runs over the elements of V_{H}^{sr} of degrees −2 and −3.

Lemma 3.9. The space S is invariant with respect to the adjoint action of H.

Proof. The adjoint action of H may be expressed using n-th products:

[t_{0}^{i}t_{1}^{m}d_{1}, u_{(j)}] = [(t_{0}^{−1}t_{1}^{m}d_{1}1)_{(i)}, u_{(j)}].

To deal with the last expression we recall the Borcherds commutator formula:

[a_{(i)}, b_{(j)}] = ∑_{k≥0} ( \begin{matrix}i \\ k \end{matrix} ) (a_{(k)}b)_{(i+j−k)}.

Note that the operators (t_{0}^{−1}t_{1}^{m}d_{1}1)_{(k)} with k ≥ 0 increase the degree by k. Since the subradical V_{H}^{sr} is a vertex algebra ideal, which is trivial in degree greater or equal to −1, we get that the span of elements of V_{H}^{sr} of degrees −2 and −3 is stable under the action of (t_{0}^{−1}t_{1}^{m}d_{1}1)_{(k)} with k ≥ 0. Thus for k ≥ 0, ((t_{0}^{−1}t_{1}^{m}d_{1}1)_{(k)}u)_{(i+j−k)} belongs to S and the Lemma is proved.

Lemma 3.10. The operators in S of degree 0 act trivially on the top T of the module L_{H}(α, γ).

Proof. The space S is spanned by the moments of the following generating series:

:d_{1}(r, z)d_{1}(m − r): − \frac{r(r + 1)}{2} :d_{1}(1, z)d_{1}(m − 1, z):+

+(r − 1)(r + 1):d_{1}(0, z)d_{1}(m, z): − \frac{r(r − 1)}{2} :d_{1}(−1, z)d_{1}(m + 1, z):,  (3.19)

:d_{1}(r, z):d_{1}(k, z)d_{1}(m − r − k, z):: − ∑_{i,j=−1,0,1} B_{i+2,j+2}:d_{1}(i, z):d_{1}(j, z)d_{1}(m − i − j, z)::,

(3.20)

and

∑_{i,j=−1,0,1} A_{i+2,j+2}:d_{1}(i, z):d_{1}(j, z)d_{1}(m − i − j, z)::.  (3.21)

The degree zero component of :d_{1}(p, z)d_{1}(s, z): is the z^{−2} moment of this formal series, and it acts on v(n) as

(t_{1}^{s}d_{1})(t_{1}^{p}d_{1})v(n) = (n − αp)(n + p − αs)v(n + p + s).

Verifying that the zero moments of (3.19), (3.20) and (3.21) vanish on T, amounts to checking that the corresponding sums of polynomials are zero. This can be easily done with Maple or any other software for symbolic computation, or directly by hand if the reader favours the traditional method. This completes the proof of the Lemma.

Proof of Theorem 3.8. Let u be a homogeneous element of the subradical V_{H}^{sr} of degree −2 or −3. We need to show that the operators u_{(n)} vanish in L_{H}(α, γ). Let ∑_{k} w_{k}v(k) be a homogeneous element in L_{H}(α, γ), w_{k} ∈ U(H_{−}). To prove that u_{(n)} ∑_{k} w_{k}v(k) = 0, we need to show that for every element w′ ∈ U(H_{+}) we get w′u_{(n)} ∑_{k} w_{k}v(k) = 0 whenever the left hand side belongs to the top T.

Using the Poincaré-Birkhoff-Witt argument, we can move all raising operators in the product w′u_{(n)}w_{k} to the right, and all lowering operators to the left, keeping degree zero operators in the middle. By Lemma 3.9, each term will contain a factor from S. Applying such an expression to v(k) we see that all the summands where the raising operators are present, vanish. The terms that remain will be products of degree zero operators. But Lemma 3.10 implies that every degree zero operator from S vanishes on T. Thus w′u_{(n)} ∑_{k} w_{k}v(k) = 0 in L_{H}(α, γ), and so u_{(n)} vanishes in this module. This proves that L_{H}(α, γ) is a module for Ṽ_{H}, and hence by Corollary 3.7, the action of H on L_{H}(α, γ) extends to the action of D on the same space using (3.6).

In this way we get an irreducible module over D. In order to identify it with one of the modules L_{D}(α, β, γ), we need to compute the action of t_{1}^{m}d_{0} on T:

(t_{1}^{m}d_{0})v(n) = \frac{1}{4}(t_{1}^{m−1}d_{1})(t_{1}d_{1})v(n) − \frac{1}{2}(t_{1}^{m}d_{1})d_{1}v(n) + \frac{1}{4}(t_{1}^{m+1}d_{1})(t_{1}^{−1}d_{1})v(n)

= (\frac{1}{4}(n − α)(n + 1 − α(m − 1)) − \frac{1}{2}n(n − αm) + \frac{1}{4}(n + α)(n − 1 − α(m + 1))) v(n + m)

= −\frac{α(α + 1)}{2}v(n + m).

Thus L_{H}(α, γ) is isomorphic to L_{D}(α, β, γ) with β = −\frac{α(α+1)}{2}. This completes the proof of the theorem.

Corollary 3.11. The relations (3.7) hold in L_{H}(α, γ).

Proof. In the proof of Theorem 3.8, we actually showed that all the relations corresponding to the elements of the subradical V_{H}^{sr} of degree −2 and −3, hold in the modules L_{H}(α, γ).

We will view (3.7) as a system of differential equations. Solving these differential equations will allow us to prove that for α ∉ ℚ the module L_{H}(α, γ) is generated by T as a module over the loop subalgebra L.

4. Representations of sl_{2}(ℂ).

In order to solve the differential equations (3.7), we need to study a family of representations for the Lie algebra sl_{2}(ℂ). Fix α ∈ ℂ, and consider the following action of the Lie algebra of vector fields on a circle Der ℂ[t, t^{−1}] on the space of polynomials ℂ[x]:

(t^{m}d)p(x) = (αm − x)p(x + m).  (4.1)

Let us verify that this is indeed a representation of the Lie algebra:

(t^{m}d)(t^{r}d)p(x) − (t^{r}d)(t^{m}d)p(x)
= (αm − x)(αr − x − m)p(x + m + r) − (αr − x)(αm − x − r)p(x + m + r)
= (r − m)(α(m + r) − x)p(x + m + r) = [t^{m}d, t^{r}d]p(x).

We will denote this module by ℂ_{α}[x].

Let us restrict the module ℂ_{α}[x] to the subalgebra ℂt^{−1}d ⊕ ℂd ⊕ ℂtd, which is isomorphic to sl_{2}(ℂ). Note that the Cartan subalgebra element d is not diagonalizable on ℂ_{α}[x], as it acts by dp(x) = −xp(x). In particular, this shows that ℂ_{α}[x] is a cyclic module, generated by vector 1 under the action of d. Even though d is not diagonalizable on ℂ_{α}[x], still this module is isomorphic to a Verma module over sl_{2}(ℂ), but with respect to a non-standard Cartan subalgebra.

Theorem 4.1. The module ℂ_{α}[x] is isomorphic to the Verma module for sl_{2}(ℂ) with the highest weight 2α and the highest weight vector 1 with respect to the following Cartan decomposition of sl_{2}(ℂ):

h = td − t^{−1}d, e = \frac{1}{2} (td − 2d + t^{−1}d) , f = −\frac{1}{2} (td + 2d + t^{−1}d) .

Proof. Verification of the usual relations [h, e] = 2e, [h, f] = −2f, [e, f] = h is straightforward. Let us now check that 1 is the highest weight vector:

h1 = (td)1 − (t^{−1}d)1 = (α − x) − (−α − x) = (2α)1,

e1 = \frac{1}{2} ((td)1 − 2d1 + (t^{−1}d)1) = \frac{1}{2} ((α − x) + 2x + (−α − x)) = 0.

Finally, we have noted above that ℂ_{α}[x] is generated by its highest weight vector 1. Thus it is isomorphic to the Verma module over sl_{2}(ℂ).

Corollary 4.2. The action of sl_{2}(ℂ) on its Verma module M_{λ} can be extended to the action of the Lie algebra of vector fields on a circle, which is irreducible except when λ = 0.

Proof. The Verma module M_{λ} over sl_{2} can be realized as ℂ_{α}[x] with α = \frac{λ}{2}. It follows from (4.1) that the action of sl_{2} may be extended to the Lie algebra Der ℂ[t, t^{−1}]. Let us show that the module ℂ_{α}[x] is irreducible over the Lie algebra of vector fields on a circle, except when α = 0. It is well-known that when λ is a non-negative integer, the Verma module over sl_{2} has a single proper submodule of codimension λ + 1, and for all other λ the Verma module is irreducible. To prove that ℂ_{α}[x] is irreducible as a module over Der ℂ[t, t^{−1}], all we need to show is that the proper sl_{2}-submodule is not invariant under the action of the Lie algebra of vector fields on a circle. Assume the contrary. Then we would get a finite-dimensional quotient module. But the Lie algebra of vector fields on a circle, being a simple infinite-dimensional algebra, can’t have non-trivial finite-dimensional representations. Thus ℂ_{α}[x] is irreducible as a module over this Lie algebra, unless α = 0. In the case when α = 0, the action (4.1) on ℂ_{0}[x] is

(t^{m}d)p(x) = −xp(x + m)

and we can immediately see that the space of polynomials vanishing at 0 forms a submodule in ℂ_{0}[x].

Let us now consider the action of the Casimir operator

Ω = −\frac{1}{2}(td) ⊗ (t^{−1}d) + d ⊗ d − \frac{1}{2}(td) ⊗ (t^{−1}d) = \frac{1}{2}e ⊗ f + \frac{1}{4}h ⊗ h + \frac{1}{2}f ⊗ e

on the tensor product of modules ℂ_{α_{1}}[x] ⊗ ℂ_{α_{2}}[y] ≅ M_{2α_{1}} ⊗ M_{2α_{2}}. It is well-known that Ω commutes with the action of sl_{2}(ℂ) on the tensor product of these modules.

Let us write down the action of Ω on ℂ_{α_{1}}[x] ⊗ ℂ_{α_{2}}[y] explicitly:

Ωp(x, y) = −\frac{1}{2}(x − α_{1})(y + α_{2})p(x + 1, y − 1) + xyp(x, y) − \frac{1}{2}(x + α_{1})(y − α_{2})p(x − 1, y + 1).

Although each term in the right hand side increases the degree of p(x, y), it can be seen that the Casimir operator Ω actually preserves the space of polynomials of total degree in x and y at most n.

The Casimir operator Ω plays an important role in the differential equations that we study in the next section. In particular we would need to know the spectrum of Ω on ℂ_{α_{1}}[x] ⊗ ℂ_{α_{2}}[y].

Proposition 4.3. The space ℂ_{α_{1}}[x] ⊗ ℂ_{α_{2}}[y] decomposes into a direct sum of finitedimensional Ω-invariant subspaces. The eigenvalues of Ω are

\{ω_{n} = \frac{n^{2}}{2} − n(α_{1} + α_{2} + \frac{1}{2}) + α_{1}α_{2}|n = 0, 1, 2, …\}

(each occurring with infinite multiplicity).

Proof. Since the Casimir operator has weight 0, it preserves the subspaces V_{n} = ⊕_{k=0}^{n} ℂf^{k}1_{α_{1}} ⊗ f^{n−k}1_{α_{2}}. This establishes the first claim of the proposition. Next, the

operator f ⊗ id + id ⊗ f is an injection of V_{n−1} into V_{n}. Since Ω commutes with this operator (it is the action of f on the tensor product of two modules), we conclude that V_{n} contains an Ω-invariant subspace of codimension 1, on which the spectrum of Ω coincides with its spectrum on V_{n−1}.

Hence there exists a sequence {ω_{n}|n = 0, 1, 2, …} such that the spectrum of Ω on V_{n} is {ω_{0}, …, ω_{n}}. The sum ω_{0} + … + ω_{n} equals the trace of Ω on V_{n}, which is easy to compute, since on these subspaces tr(Ω) = \frac{1}{4}tr(h ⊗ h) (the other two terms are traceless). We have

ω_{0} + … + ω_{n} = \frac{1}{4} ∑_{k=0}^{n}(−2k + 2α_{1})(−2n + 2k + 2α_{2}),

and thus for n ≥ 1,

ω_{n} = ∑_{k=0}^{n}(k − α_{1})(n − k − α_{2}) − ∑_{k=0}^{n−1}(k − α_{1})(n − 1 − k − α_{2}) = −(n − α_{1})α_{2} + ∑_{k=0}^{n−1}(k − α_{1})

= \frac{n^{2}}{2} − n(α_{1} + α_{2} + \frac{1}{2}) + α_{1}α_{2}.  (4.2)

Note that this formula also gives the correct value of ω_{0} = α_{1}α_{2}.

Remark. If all ω_{n} are distinct (which happens when the sum of the highest weights 2(α_{1} + α_{2}) is not a non-negative integer), then Ω is diagonalizable on the tensor product of these Verma modules, and the tensor product decomposes into a direct sum of Verma modules over sl_{2}(ℂ).

5. Differential equations in modules for vertex algebras.

Our goal is to show that the modules L_{H}(α, γ) with α ∉ ℚ, are generated by T as the modules over the loop subalgebra L. This will give a complete description of these modules. We will do this by studying a differential equation in L_{H}(α, γ) that arises from (3.7).

The first step towards this result is the following

Lemma 5.1. Let us suppose that for all j, (t_{0}^{−j}t_{1}^{m}d_{1})v(n) belongs to the subspace U(L_{−})T in the module L_{H}(α, γ). Then L_{H}(α, γ) = U(L_{−})T.

Proof. The proof is based on a simple Poincaré-Birkhoff-Witt argument. The module L_{H}(α, γ) is spanned by the vectors

(t_{0}^{−j_{1}}t_{1}^{m_{1}}d_{1}) … (t_{0}^{−j_{k}}t_{1}^{m_{k}}d_{1})v(n), j_{1}, …, j_{k} > 0.

We are going to show that these vectors belong to the space U(L_{−})T using a double induction on total degree −j = −(j_{1} +…+j_{k}) and on length k. If the total degree is zero, then there is nothing to prove. Suppose now that the claim of the lemma is true for all degrees above −j. We have that (t_{0}^{−j_{2}}t_{1}^{m_{2}}d_{1}) … (t_{0}^{−j_{k}}t_{1}^{m_{k}}d_{1})v(n) has degree greater than −j. By induction assumption, it belongs to the space U(L_{−})T. Thus without the loss of generality we may assume that |m_{2}|, …, |m_{k}| ≤ 1.

Let us prove by induction on length k that for the claim of the lemma holds for the vectors in L_{H}(α, γ) of degree −j. The basis of induction, k = 1, is precisely the assumption of the lemma. Let k ≥ 2. Then

(t_{0}^{−j_{1}}t_{1}^{m_{1}}d_{1})(t_{0}^{−j_{2}}t_{1}^{m_{2}}d_{1}) … (t_{0}^{−j_{k}}t_{1}^{m_{k}}d_{1})v(n)
= (m_{2} − m_{1})(t_{0}^{−j_{1}−j_{2}}t_{1}^{m_{1}+m_{2}}d_{1}) … (t_{0}^{−j_{k}}t_{1}^{m_{k}}d_{1})v(n)
+(t_{0}^{−j_{2}}t_{1}^{m_{2}}d_{1})(t_{0}^{−j_{1}}t_{1}^{m_{1}}d_{1}) … (t_{0}^{−j_{k}}t_{1}^{m_{k}}d_{1})v(n).

The first summand in the right hand side belongs to the space U(L_{−})T by induction assumption on length, while for the second summand we get that

(t_{0}^{−j_{1}}t_{1}^{m_{1}}d_{1})(t_{0}^{−j_{3}}t_{1}^{m_{3}}d_{1}) … (t_{0}^{−j_{k}}t_{1}^{m_{k}}d_{1})v(n) ∈ U(L_{−})T

by the induction assumption on degree. Noting that t_{0}^{−j_{2}}t_{1}^{m_{2}}d_{1} ∈ L_{−} since |m_{2}| ≤ 1, we obtain the desired claim. The lemma is now proved.

Thus it remains to prove that (t_{0}^{−j}t_{1}^{m}d_{1})v(n) ∈ U(L_{−})T for all j. We will achieve this by studying the generating series

P (m, n, z) = d_{1}(m, z)v(n) = z^{−1}(t_{1}^{m}d_{1})v(n) + (t_{0}^{−1}t_{1}^{m}d_{1})v(n) + z(t_{0}^{−2}t_{1}^{m}d_{1})v(n) + …

Since d_{1}(m, z) satisfies the differential equation (3.7), we conclude that P (m, n, z) satisfies an analogous differential equation

\frac{∂}{∂z} P (m, n, z) =

(−\frac{m + 1}{2} :d_{1}(1, z)d_{1}(m − 1, z): + m:d_{1}(0, z)d_{1}(m, z):

−\frac{m − 1}{2} :d_{1}(−1, z)d_{1}(m + 1, z):) v(n)

= (−\frac{m + 1}{2}d_{1}(1, z)_{+}d_{1}(m − 1, z) + md_{1}(0, z)_{+}d_{1}(m, z)

−\frac{m − 1}{2}d_{1}(−1, z)_{+}d_{1}(m + 1, z)) v(n)

+z^{−1} (−\frac{m + 1}{2}d_{1}(m − 1, z)(t_{1}d_{1}) + mz^{−1}d_{1}(m, z)d_{1}

−\frac{m − 1}{2}z^{−1}d_{1}(m + 1, z)(t_{1}^{−1}d_{1})) v(n)

= −\frac{m + 1}{2}d_{1}(1, z)_{+}P (m−1, n, z)+md_{1}(0, z)_{+}P (m, n, z)− \frac{m − 1}{2}d_{1}(−1, z)_{+}P (m+1, n, z)

+z^{−1}ΩP (m, n, z), (5.1)

where

d_{1}(k, z)_{+} = ∑_{j<0}(t_{0}^{j}t_{1}^{k}d_{1})z^{−j−1}  (5.2)

and

ΩP (m, n, z) =

−\frac{1}{2}(m+1)(n−α)P (m−1, n+1, z)+mnP (m, n, z)− \frac{1}{2}(m−1)(n+α)P (m+1, n−1, z).(5.3)

In order to solve this differential equation, we will treat m and n as formal variables. From this point of view, the transformation Ω defined in (5.3), is essentially the Casimir operator on the space ℂ_{α_{1}}[m] ⊗ ℂ_{α_{2}}[n] with α_{1} = 1, α_{2} = α.

Let us give a precise description of the spaces which we will be working with. Consider the algebra ℂ[m, n] of polynomials in two formal variables m, n. Let Γ be the space of affine functions in m with integer coefficients: Γ = {am + b|a, b ∈ ℤ}. We are going to replace in the definitions of Lie algebras and modules which we have previously discussed, the algebra of Laurent polynomials ℂ[t_{0}^{±1}, t_{1}^{±1}], which is a group algebra of ℤ ⊕ ℤ, with the group algebra of ℤ ⊕ Γ, and extend the scalars to ℂ[m, n]:

R̂ = ℂ[m, n] ⊗ ℂ[ℤ ⊕ Γ],
Ĥ = R̂d_{1},

with obvious modifications of the formulas (1.1) for the Lie bracket:

[t_{0}^{i}t_{1}^{μ}d_{1}, t_{0}^{j}t_{1}^{ν}d_{1}] = (ν − μ)t_{0}^{i+j}t_{1}^{μ+ν}d_{1}, i, j ∈ ℤ, μ, ν ∈ Γ.

In the same way we will replace the H_{0}-module T (α, γ) with the Ĥ_{0}-module T̂(α, γ), α, γ ∈ ℂ:

T̂ = ⨁_{ν∈Γ} ℂ[m, n]v(n + ν).

As before, we form the generalized Verma modules M_{H}(T̂) = U(Ĥ_{−}) ⊗_{ℂ[m,n]} T̂ and

M_{H}(T̂) = U(H_{−}) ⊗ℂ T̂. We define L_{H}(T̂) (resp. L_{H}(T̂)) to be the quotient of M_{H}(T̂) (resp. M_{H}(T̂)) by the maximal ℤ ⊕ Γ-graded submodule trivially intersecting T̂.

The following lemma is quite obvious:

Lemma 5.2. (a) A specialization m ↦ m_{0} ∈ ℤ, n ↦ n_{0} ∈ ℤ extends to a homomorphism of Lie algebras Ĥ → H and a homomorphism of modules

M_{H}(T̂) → M_{H}(T).

(b) The above map factors through to the following:

L_{H}(T̂) → L_{H}(T).

Our strategy will be to solve the differential equation (5.1) in the module L_{H}(T̂), where we treat m and n as formal variables, and then use part (b) of the Lemma above to obtain a solution in L_{H}(T).

We also need to establish a relation between M_{H}(T̂) and L_{H}(T̂):

Lemma 5.3. The composition

φ : M_{H}(T̂) → L_{H}(T̂)

of an embedding M_{H}(T̂) → M_{H}(T̂) with a projection M_{H}(T̂) → L_{H}(T̂) is surjective.

Proof. We need to show that for every x ∈ M_{H}(T̂) there exists a y ∈ M_{H}(T̂), such that the images of x and y in L_{H}(T̂) coincide. It is sufficient to consider the case when x is a monomial:

x = (t_{0}^{−j_{1}}t_{1}^{μ_{1}}d_{1}) … (t_{0}^{−j_{s}}t_{1}^{μ_{s}}d_{s})v(n + ν), μ_{1}, …, μ_{s}, ν ∈ Γ, j_{1}, … j_{s} > 0.

The kernel of the projection M_{H}(T̂) → L_{H}(T̂) is invariant under the shifts n ↦ n + μ, μ ∈ Γ. Thus we may assume without the loss of generality that μ_{1} + … + μ_{s} + ν = 0. A homogeneous element of M_{H}(T̂) is non-zero in L_{H}(T̂) if and only if a submodule generated by this element in M_{H}(T̂) intersects with T̂ non-trivially. Thus the image of x in L_{H}(T̂) is determined by the values of ux, where u are elements in U(Ĥ_{+}) of degree k = j_{1} + … + j_{s} in t_{0}. Moreover, it is sufficient to take u to be elements of degree k in U(H_{+}). This follows from the fact that a non-zero polynomial in m assumes a non-zero value at some integer. We may also assume that u is also homogeneous in t_{1} of degree i. Then

ux = f(m, n)v(n + i),

where f is a polynomial of degree at most s + k in m. However, a polynomial of degree s + k may be extrapolated through its s + k + 1 distinct integer values. Hence we can construct y to be an extrapolation in m through the specializations of x to any s + k + 1 distinct integer values of m. Note that such an extrapolation of x depends only on the degree s + k, but not on a particular raising element u.

Example. The same technique may be applied to the vertex algebras V_{H} and \overline{V}_{H}. Consider the element x = (t_{0}^{−1}t_{1}^{m}d_{1})(t_{0}^{−1}t_{1}^{n}d_{1})1 ∈ V_{H}. We would like to get a reduction rule for the image of x in \overline{V}_{H}. First, we make a shift n ↦ n − m, and get x′ = (t_{0}^{−1}t_{1}^{m}d_{1})(t_{0}^{−1}t_{1}^{n−m}d_{1})1. The image of x′ in \overline{V}_{H} is determined by the values of the raising operators of degree one, u = t_{0}t_{1}^{i}d_{1}, i ∈ ℤ, applied to x′. It can be easiy seen that the coefficients that occur is ux′ are polynomials in m of degree two (cf., last formula in (3.5)). Thus x′ may be interpolated in \overline{V}_{H} with a quadratic polynomial in m using any three integer specializations of m in x′. Choosing −1, 0, 1 as the interpolation points, we get (cf., (3.4)):

(t_{0}^{−1}t_{1}^{m}d_{1})(t_{0}^{−1}t_{1}^{n−m}d_{1})1 = \frac{m(m + 1)}{2}(t_{0}^{−1}t_{1}d_{1})(t_{0}^{−1}t_{1}^{n−1}d_{1})1

−(m − 1)(m + 1)(t_{0}^{−1}d_{1})(t_{0}^{−1}t_{1}^{n}d_{1})1 + \frac{m(m − 1)}{2}(t_{0}^{−1}t_{1}^{−1}d_{1})(t_{0}^{−1}t_{1}^{n+1}d_{1})1.

Taking the shift back, n ↦ n + m, we obtain analogous relation for x.

Let us consider a subspace W in L_{H}(T̂), spanned as a ℂ[m, n]-module by the elements (t_{0}^{−j_{1}}t_{1}^{μ_{1}}d_{1}) … (t_{0}^{−j_{s}}t_{1}^{μ_{s}}d_{s})v(n + ν), with μ_{1} + … + μ_{s} + ν = m + i, i ∈ ℤ, and the corresponding subspace W̃ in M_{H}(T̂):

W̃ = U(H_{−}) ⊗ (⨁_{i∈ℤ} ℂ[m, n]v(n + m + i)) .

We note that φ(W̃) = W.

The coefficients at all powers of z in P (m, n, z) belong to the space W, and the same is true for all summands in the right hand side of the differential equation (5.1). This allows us to restrict to subspace W when solving (5.1). The reason for considering such a restriction is that in space W̃ the Casimir operator Ω acts just on the coefficients:

Ω_{W} (f(m, n)uv(m + n + i)) = Ω (f(m, n)) uv(m + n + i), u ∈ U(H_{−}), i ∈ ℤ. (5.4)

Since the kernel of φ is invariant under the shift operators m ↦ m + j, n ↦ n + i, i, j ∈ ℤ, it is also invariant under Ω. Thus we may calculate the action of Ω in W by lifting elements to W̃:

Ω_{W} = φΩ_{W} φ^{−1}.  (5.5)

As a consequence, we get that the action of Ω_{W} is locally finite, i.e., every vector in W is contained in a finite-dimensional Ω-invariant ℂ-subspace, and that the spectrum of Ω_{W} is a subset of (4.2) with α_{1} = 1, α_{2} = α.

Denote by S_{i} the shift operator m ↦ m + i, i ∈ ℤ, on the space W. Then the differential equation (5.1) may be rewritten as follows:

\frac{∂}{∂z} P (m, n, z) = (z^{−1}Ω + A_{+}(z)) P (m, n, z),  (5.6)

where Ω is the Casimir operator (5.3) and

A_{+}(z) = −\frac{m + 1}{2}d_{1}(1, z)_{+}S_{−1} + md_{1}(0, z)_{+} − \frac{m − 1}{2}d_{1}(−1, z)_{+}S_{1}.

6. Solving non-commutative differential equations.

Consider a differential equation in a vector space W:

\frac{∂}{∂z} w(z) = A(z)w(z),  (6.1)

where

w(z) = ∑_{j≥r} w_{j}z^{j}, with w_{j} ∈ W,  (6.2)

and

A(z) = Ωz^{−1} + ∑_{i≥0} A_{i}z^{i}, with Ω, A_{i} ∈ End(W).  (6.3)

We would like to solve this differential equation for w(z) given the ”initial data” w_{r} ∈ W. We will denote by A_{+}(z) the tail of A(z):

A_{+}(z) = ∑_{i≥0} A_{i}z^{i}.

In order to solve this differential equation, we will introduce a special type of integra-

tion. An expression b(z) = ∑_{j=−∞}^{∞} b_{j}z^{j}, involving z^{−1}, can’t be integrated in a usual way,

since z^{−1} does not have an antiderivative in power series. To avoid this difficulty we define

z^{Ω} ∫ z^{−Ω}b(z)dz = ∑_{j=−∞}^{∞} ((j + 1)I − Ω)^{−1} b_{j}z^{j+1}.  (6.4)

For this integral to be defined we require that the operator (j + 1)I − Ω is invertible whenever b_{j} ≠ 0.

The notation for this integral is motivated by the formula

z^{ω} ∫ z^{−ω}b(z)dz = ∑_{j=−∞}^{∞} \frac{1}{j + 1 − ω} b_{j}z^{j+1},

for ω ∈ ℂ, ω ∉ ℤ.

Lemma 6.1. Assume that the integral (6.4) is well-defined. Then

\frac{∂}{∂z} z^{Ω} ∫ z^{−Ω}b(z)dz = z^{−1}Ω (z^{Ω} ∫ z^{−Ω}b(z)dz) + b(z).

Proof.

\frac{∂}{∂z} z^{Ω} ∫ z^{−Ω}b(z)dz

= ∑_{j=−∞}^{∞}(j + 1) ((j + 1)I − Ω)^{−1} b_{j}z^{j}

= ∑_{j=−∞}^{∞} ((j + 1)I − Ω) ((j + 1)I − Ω)^{−1} b_{j}z^{j}

+ ∑_{j=−∞}^{∞} Ω ((j + 1)I − Ω)^{−1} b_{j}z^{j}

= z^{−1}Ω (z^{Ω} ∫ z^{−Ω}b(z)dz) + b(z).

The following theorem gives a formula for the solution of the differential equation (6.1).

Theorem 6.2. Fix r ∈ ℤ. Consider a non-commutative differential equation

\frac{∂}{∂z} w(z) = A(z)w(z),

with w(z) and A(z) as in (6.2) and (6.3). Assume that the operators Ω − jI are invertible for all j > r. Given w_{r} ∈ W satisfying the consistency condition

(Ω − rI) w_{r} = 0,  (6.5)

there exists a unique solution of the above differential equation of the form

w(z) = w_{r}z^{r} + ∑_{j>r} w_{j}z^{j}, w_{j} ∈ W.

This solution is given by the formula

w(z) = ∑_{k=0}^{∞} P_{k}(z)  (6.6)

with P_{0}(z) = w_{r}z^{r} and

P_{k+1}(z) = z^{Ω} ∫ z^{−Ω}A_{+}(z)P_{k}(z)dz for all k ≥ 0.  (6.7)

The series P_{k}(z) has no powers of z below r + k, which makes the infinite sum (6.6) well-defined.

Proof. Let us expand the differential equation (6.1) in powers of z. We can see that it does not have any terms in powers of z below r − 1. The z^{r−1} term of this equation is

rw_{r} = Ωw_{r},

which holds by (6.5). The term at z^{j−1} with j > r is

jw_{j} = Ωw_{j} + ∑_{i=r}^{j−1} A_{j−i−1}w_{i}.

By the assumption of the theorem, the operators Ω − jI with j > r are invertible, which gives the recursive formula for the (unique) solution of (6.1):

w_{j} = (jI − Ω)^{−1} ∑_{i=r}^{j−1} A_{j−i−1}w_{i}.  (6.8)

Finally, let us prove that the same solution is given by the formula (6.6). To do this, we need to show that the series defined by (6.6), (6.7) also satisfies (6.1).

Since A_{+}(z) contains only non-negative powers of z, we see that integration (6.4) increases the powers of z in (6.7) by 1. Thus P_{k}(z) has no powers of z below r + k, which makes the infinite sum (6.6) well-defined.

By Lemma 6.1,

\frac{∂}{∂z} P_{k+1}(z) = z^{−1}ΩP_{k+1}(z) + A_{+}(z)P_{k}(z).

Also by (6.5) we get that

\frac{∂}{∂z} P_{0}(z) = z^{−1}ΩP_{0}(z).

Thus

\frac{∂}{∂z} ∑_{k=0}^{∞} P_{k}(z) = z^{−1}Ω ∑_{k=0}^{∞} P_{k}(z) + A_{+}(z) ∑_{k=1}^{∞} P_{k−1}(z),

and we see that ∑_{k=0}^{∞} P_{k}(z) satisfies (6.1). Since the solution of (6.1) is unique, we conclude that the solutions (6.8) and (6.6)-(6.7) coincide.

We are now going to establish our main result, Theorem 1.2, by solving the differential equation (5.6) in the space W ⊂ L_{H}(T̂). Let us first verify that (5.6) satisfies the assumption of Theorem 6.2. By (5.4), (5.5) and Proposition 4.3, the action of the Casimir operator Ω on W is locally finite, and it has the spectrum

\{ω_{n} = \frac{1}{2} (n^{2} − (2α + 3)n + 2α) |n = 0, 1, 2, …\} .

It can be easily seen that ω_{1} = −1 for all α, whereas the spectrum of Ω contains a nonnegative integer if and only if α ∈ ℚ. Hence for all α ∉ ℚ the operators Ω−jI are invertible for all j = 0, 1, 2, …

The first term in the series P (m, n, z) is

P_{0}(m, n, z) = z^{−1}(t^{m}d)v(n) = z^{−1}(n − αm)v(n + m).

Thus the value of r that appears in the statement of Theorem 6.2 is r = −1. One can verify that the consistency condition

(Ω + I)(n − αm)v(n + m) = 0

holds – and actually this follows from the fact that (5.1) does have a solution in W. Hence, by Theorem 6.2, the solution of (5.1) is given by the formula

P (m, n, z) = ∑_{k=0}^{∞} P_{k}(m, n, z),

with

P_{k+1}(m, n, z) = z^{Ω} ∫ z^{−Ω}A_{+}(z)P_{k}(m, n, z)dz

= z^{Ω} ∫ z^{−Ω} (−\frac{m + 1}{2}d_{1}(1, z)_{+}P_{k}(m − 1, n, z) + md_{1}(0, z)_{+}P_{k}(m, n, z)

−\frac{m − 1}{2}d_{1}(−1, z)_{+}P_{k}(m + 1, n, z)) dz.

Since P_{0}(m, n, z) belongs to T̂ and the moments of d_{1}(1, z)_{+}, d_{1}(0, z)_{+}, d_{1}(−1, z)_{+} are in the loop algebra L, we conclude that the moments of P (m, n, z) = d_{1}(m, z)v(n), viewed as elements of L_{H}(T̂), belong in fact to the subspace U(L_{−})T̂. Applying Lemma 5.2 (b), we obtain that for all m, n, j ∈ ℤ,

(t_{0}^{j}t_{1}^{m}d_{1})v(n) ∈ U(L_{−})T

in the module L_{H}(T). Thus by Lemma 5.1, we get that L_{H}(T) is generated by T as an L-module,

L_{H}(T) = U(L_{−})T.

Hence we have a surjective map of L-modules

M_{L}(T) → L_{H}(T).

In the Appendix we use results of [F] and [KM] to show that the generalized Verma module M_{L}(α, γ) over the loop algebra L has no non-trivial submodules trivially intersecting with T whenever α ∉ \frac{1}{2}ℤ. Since α ∉ \{−1, 0\}, T is an irreducible H_{0}-module, and thus for α ∉ ℚ the kernel of the map M_{L}(T) → L_{H}(T) is trivial, and we obtain an isomorphism of L-modules:

L_{H}(T) ≅ M_{L}(T).

Finally, by Theorem 3.8, for β = −\frac{α(α+1)}{2}, the modules L_{D}(α, β, γ) and L_{H}(α, γ) are isomorphic as H-modules, and hence as L-modules. This gives us

L_{D}(α, β, γ) ≅ L_{H}(α, γ) ≅ M_{L}(α, γ).

We conclude that for α ∉ ℚ, β = −\frac{α(α+1)}{2}, the module L_{D}(α, β, γ) remains irreducible as a module over the loop subalgebra L, and its action on M_{L}(α, γ) can be extended to the larger algebra D of vector fields on a 2-dimensional torus.

Appendix. Generalized Verma modules for the loop Lie algebra.

Futorny [F] defined the Shapovalov form on the generalized Verma module M_{L}(α, γ), and used it to establish a criterion for irreducibility of M_{L}(α, γ) (in fact [F] treats a more general case of modules for the affine Kac-Moody algebra ŝl_{2}, however we restrict our attention here to the level zero case of the loop algebra L). A complete formula for the Shapovalov determinant was obtained by Khomenko and Mazorchuk in [KM]. Here we would like to address a related question of existence of non-trivial submodules in M_{L}(α, γ) having a trivial intersection with the top T (α, γ). We will do this with the help of the Shapovalov determinant formula given in [KM].

Consider three related partition functions,

p_{+}(s, n) = dim U(L_{+})_{(s,n)},
p(s, n) = dim U(ℂ(t_{1}d_{1}) ⊕ L_{+})_{(s,n)},
p_{3}(s) = dim U(L_{+})_{s}.

Here a ℤ^{2}-grading refers to the grading by degrees in t_{0} and t_{1}, while a ℤ-grading is taken just by degrees in t_{0}.

From the structure of the root system of L we see that p_{+}(s, n) = 0 when |n| > s and we have the symmetry p_{+}(s, n) = p_{+}(s, −n). The Poincaré-Birkhoff-Witt theorem implies that

p(s, n) = ∑_{k≥0} p_{+}(s, n − k),  (A.1)

while

p_{3}(s) = ∑_{n∈ℤ} p_{+}(s, n).  (A.2)

We also note that the partition function p_{3} appears in the expansion of (1.2):

∏_{n≥1}(1 − q^{n})^{−3} = ∑_{s=0}^{∞} p_{3}(s)q^{s}.

Let us define two determinants associated with the module M_{L}(α, γ). As before, we consider ℤ^{2}-grading of M_{L}(α, γ) by degrees in t_{0} and t_{1}. We will assume that the top T (α, γ) has degree zero in t_{0}. Note that any submodule in M_{L}(α, γ) is homogeneous with respect to ℤ^{2}-grading since d_{1} ∈ L, while the action of d_{0} can be recovered via the Sugawara construction (cf., (3.6)).

Fix homogeneous bases {u_{i}^{+}|i = 1, … p_{3}(s)} in U(L_{+})_{s} and {u_{i}^{−}} in U(L_{−})_{−s}. Let r_{i} be the degree in t_{1} of u_{i}^{+}. We may assume that the degree in t_{1} of u_{i}^{−} is −r_{i}. Define ϵ_{i} = 1 if r_{i} > 0 and ϵ_{i} = −1 if r_{i} < 0. The value of ϵ_{i} when r_{i} = 0 will not be relevant.

In this set-up, the set {u_{i}^{−}v(n + r_{i})} forms a basis of M_{L}(α, γ)_{(−s,n)}, where s ≥ 1, n ∈ γ + ℤ.

We introduce two square matrices. Define G_{ij} = G_{ij}^{s,n} to be the coefficient at v(n + r_{i}) in u_{i}^{+}u_{j}^{−}v(n + r_{j}). Similarly, define F_{ij} = F_{ij}^{s,n} to be the coefficient at v(n) in (t_{1}^{−ϵ_{i}}d_{1})^{|r_{i}|}u_{i}^{+}u_{j}^{−}(t_{1}^{ϵ_{j}}d_{1})^{|r_{j}|}v(n). We will consider the determinants of these matrices as polynomials in n and α.

It is clear that M_{L}(α, γ) has a non-trivial submodule that trivially intersects the top T (α, γ) if and only if det G^{s,n} = 0 for some s ≥ 1, n ∈ γ + ℤ. The second determinant, det F, is the determinant of the Shapovalov form, calculated in [KM]. Since there is a simple connection between the two determinants, we will get a formula for det G.

Indeed, since

(t_{1}^{ϵ_{j}}d_{1})^{|r_{j}|}v(n) = ∏_{k=1}^{|r_{j}|}(n + (k − 1)ϵ_{j} − αϵ_{j})v(n + r_{j}),

and similarly

(t_{1}^{−ϵ_{i}}d_{1})^{|r_{i}|}v(n + r_{i}) = ∏_{k=1}^{|r_{i}|}(n + kϵ_{i} + αϵ_{i})v(n),

we get that

det F = det G × ∏_{j=1}^{p_{3}(s)} ∏_{k=1}^{|r_{j}|}(n + (k − 1)ϵ_{j} − αϵ_{j}) × ∏_{i=1}^{p_{3}(s)} ∏_{k=1}^{|r_{i}|}(n + kϵ_{i} + αϵ_{i}).  (A.3)

Combining the factors in (A.3) that correspond to the basis elements of equal positive or negative degrees, we obtain

det F = det G × ∏_{m=1}^{s} ∏_{k=1}^{m} ((n + k − 1 − α)(n − k + 1 + α)(n + k + α)(n − k − α))^{p_{+}(s,−m)}.

Finally, interchanging the order of the products and taking (A.1) into account, we get

det F = det G × ∏_{k=1}^{s} ((n + k − 1 − α)(n − k + 1 + α)(n + k + α)(n − k − α))^{p(s,−k)}.

The Shapovalov determinant formula given in [KM], applied to our situation yields (up to a non-zero constant factor):

det F = ∏_{k=1}^{s} ((n + k − 1 − α)(n − k + 1 + α)(n + k + α)(n − k − α))^{p(s,−k)}

× ∏_{m=1}^{s} \begin{matrix}[ & \frac{s}{m} & ] \\ ∏ \\ k=1 \end{matrix} ((2m − k + 2α + 1)(2m − k − 2α − 1))^{p_{3}(s−mk)}.

Comparing the last two equalities, we obtain a formula for det G (up to a non-zero constant factor):

det G^{s,n} = ∏_{m=1}^{s} \begin{matrix}[ & \frac{s}{m} & ] \\ ∏ \\ k=1 \end{matrix} ((2m − k + 2α + 1)(2m − k − 2α − 1))^{p_{3}(s−mk)}.

As an immediate corollary, we get

Proposition A.1. The L-module M_{L}(α, γ) has a non-trivial submodule trivially intersecting the top T (α, γ) if and only if α ∈ \frac{1}{2}ℤ.

Remark. The above argument can be applied in the general case considered in [KM]. For the generalized Verma modules M_{L}(α, γ, c) for affine Kac-Moody algebra L̂ = ŝl_{2} at level c, we get

det G^{s,n} = ∏_{m=1}^{s} \begin{matrix}[ & \frac{s}{m} & ] \\ ∏ \\ k=1 \end{matrix} ((c + 2)(m(c + 2) − k + 2α + 1)(m(c + 2) − k − 2α − 1))^{p_{3}(s−mk)}.

As a consequence, we get that M_{L}(α, γ, c) has a non-trivial submodule trivially intersecting the top T (α, γ) if and only if 2α + 1 = ±(m(c + 2) − k) for some integer m, k ≥ 1 or c = −2 (cf., [F], Theorem 3.11).

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