Jet Modules.

Yuly Billig

Abstract: In this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of such modules is given by spaces of jets of tensor fields.

0. Introduction.

In recent years there was a substantial progress in representation theory of infinitedimensional Lie algebras of rank n > 1, toroidal Lie algebras in particular. In this paper we turn our attention to another Lie algebra of rank n, the Lie algebra W_{n} of vector fields on an n-dimensional torus 𝕋^{n}.

An important class of irreducible representations for W_{n} has its origin in differential geometry – these are the modules of tensor fields on a torus. In addition to being modules for the Lie algebra of vector fields, tensor fields also admit multiplication by functions. For the torus, which is a flat manifold, the spaces of tensor fields are free modules of a finite rank over the commutative algebra of functions F(𝕋^{n}). We formalize this property in the definition of a category J of W_{n}-modules (cf., [R2]).

We also discuss another class of W_{n}-modules of a geometric nature – the modules of jets of tensor fields [S]. Jets of functions are used as a tool for the symmetry analysis for partial differential equations [O]. The action on a space of jets of the Lie algebra of vector fields, known under the term “prolongation of vector fields”, plays a key role in that theory.

From the algebraic point of view, jet modules are typically not irreducible, but are often indecomposable. The goal of the present paper is to classify indecomposable modules in category J.

Let us state our result in case n = 1, for the sake of simplicity of notations.

Theorem. There is a 1-1 correspondence between indecomposable W_{1}-modules J in category J and pairs (λ, U), where λ ∈ ℂ/ℤ and U is a finite-dimensional indecomposable module for a Lie algebra

W_{1}^{+} = Span ⟨z^{m} \frac{d}{dz} | m ≥ 1⟩ .

Such a correspondence is given by the tensor product decomposition

J = F(𝕋^{1}) ⊗ U,

where W_{1} acts according to the formula

(\frac{1}{2πi}e^{2πisx} \frac{d}{dx}) (e^{2πimx} ⊗ u) = (m + λ)e^{2πi(s+m)x} ⊗ u + ∑_{b≥1} \frac{s^{b}}{b!} e^{2πi(s+m)x} ⊗ ρ (z^{b} \frac{d}{dz} ) u.

AMS subject classification: 17B66, 58A20.

The sum in the right hand side is finite since for every finite-dimensional representation of W_{n}^{+}, ρ (z^{b} \frac{d}{dz} ) = 0 for b ≫ 1.

This result reduces the classification of modules in J to a problem in a completely finite-dimensional set-up – describing finite-dimensional representations of certain finitedimensional Lie algebras.

The case of rank one is of interest by itself since in this case we deal with the representations of the Virasoro algebra with a trivial action of the center, and the Virasoro algebra plays a prominent role in applications to physics. Only in the case n = 1 we have a complete classification of irreducible modules with finite-dimensional weight spaces [M]. Very little has been known about indecomposable W_{n}-modules even in rank 1 case.

The concept of a polynomial module, introduced in [BB] (see also [BZ]), has turned out to be extremely useful for the present paper. We prove that all modules in category J are polynomial modules, and this property allows us to establish the classification result. It is interesting that unlike all previously known examples, the degrees of the structure polynomials for the modules in J can be arbitrarily high.

The technique developed in this paper also allows us to recover Eswara Rao’s classification [R2] of irreducible modules in category J, significantly simplifying his proof.

The structure of the paper is the following. In Section 1 we discuss the module structure on the space of jets of tensor fields. Motivated by the construction of jet modules, we introduce a category J in Section 2, and state our main theorems at the end of the section. In Sections 3 and 4 we prove the classification results for the modules in category J. In the final Section 5, we give the analogous result for the semidirect product of W_{n} with a multi-loop algebra.

1. Jets of tensor fields.

The group of diffeomorphisms of a manifold and its Lie algebra of vector fields have several natural constructions of modules coming from differential geometry. Examples of such modules include the space of functions on a manifold and more generally, the space of tensor fields. In addition to these, one can also consider the spaces of jets of tensor fields which also admit a natural action of the group of diffeomorphisms and the Lie algebra of vector fields [S].

In this section we will review the construction of the bundle of jets of tensor fields and the module structure on the vector space of its sections.

First, let us discuss the notations that will be used in the paper. We denote the set of non-negative integers by ℤ_{+}, and consider a partial order on ℤ_{+}^{n} where α ≥ β whenever α − β ∈ ℤ_{+}^{n}. The standard basis of ℤ^{n} will be denoted by {ϵ_{1}, …, ϵ_{n}}. For a coordinate system {x^{i}}_{i=1,...,n} and α = (α_{1}, …, α_{n}) ∈ ℤ_{+}^{n}, we denote by f^{(α)}(x) the partial derivative (\frac{∂}{∂x^{1}} )^{α_{1}} … (\frac{∂}{∂x^{n}} )^{α_{n}} f(x). We will use notations |α| = α_{1} + … + α_{n}, α! = α_{1}! … α_{n}!,

( \begin{matrix}α \\ β \end{matrix} ) = \frac{α!}{β!(α−β)!}, (x−p)^{α} = (x^{1} −p^{1})^{α_{1}} … (x^{n} −p^{n})^{α_{n}}, etc. We will also use a convention

of dropping the summation symbol when we take a sum over matching upper and lower

indices: u^{i} \frac{∂}{∂x^{i}} = ∑_{i=1}^{n} u^{i} \frac{∂}{∂x^{i}} .

Let M be an n-dimensional C^{∞} manifold, which is allowed to be either real or complex. However the spaces of functions on M, vector fields, etc., will be always taken to be complex spaces.

We begin by recalling the definition of an N-jet of a function. Let f_{1}, f_{2} be two C^{∞} functions defined in a neighbourhood of a point p on manifold M. We say that f_{1} is equivalent to f_{2} at p, f_{1} ∼ f_{2}, if all partial derivatives of f_{1} and f_{2} at p of orders up to N are equal:

f_{1}^{(α)}(p) = f_{2}^{(α)}(p), for all 0 ≤ |α| ≤ N.

An equivalence class for this relation is called an N-jet of a function at p. Any N-jet at p has a unique Taylor polynomial representative:

∑_{0≤|α|≤N} f_{[α]}(p)(x − p)^{α}.

The set of N-jets at p forms a finite-dimensional vector space with coordinates {f_{[α]}(p) | 0 ≤ |α| ≤ N} and a basis {(x − p)^{α} | 0 ≤ |α| ≤ N}. If we now let p vary over M, we get the vector bundle of N-jets of functions. Let us denote by J_{N}(M) the space of sections of this bundle.

Now we are going to describe the action of the group of diffeomorphisms Diff(M) on J_{N}(M). Suppose for a diffeomorphism φ ∈ Diff(M) we have φ(p) = q. Set ψ = φ^{−1}. Let {x^{i}} be a coordinate system near p, and {y^{i}} be a coordinate system near q.

Let F be a section of an N-jet bundle, F ∈ J_{N}(M), with its value at p given by the equivalence class

F (p) ∼ ∑_{0≤|α|≤N} f_{[α]}(p)(x − p)^{α}.

Then the value of the section φF at point q is defined by the jet

φF (q) ∼ ∑_{0≤|α|≤N} f_{[α]}(p)(ψ(y) − ψ(q))^{α}

∼ ∑_{0≤|α|≤N} f_{[α]}(p) (∑_{β>0} \frac{1}{β!}ψ^{(β)}(q)(y − q)^{β})^{α} .  (1.1)

The coordinates of φF at q are then computed by expanding the last expression in powers of (y − q), and dropping terms of degrees greater than N.

If we pass to the infinitesimal action, we will obtain the action of the Lie algebra of vector fields Vect(M) on J_{N}(M).

For a vector field \overline{u} = u^{i}(x)\frac{∂}{∂x^{i}} we consider the corresponding flow φ_{ϵ} ∈ Diff(M),

φ_{ϵ}^{i} = x^{i} − ϵu^{i}(x) + o(ϵ).

We now assume that the point p and its image q under φ_{ϵ} are in the same chart with coordinates {x^{i}}. Then for the inverse, ψ_{ϵ} = φ_{ϵ}^{−1} ∈ Diff(M), we have

ψ_{ϵ}^{i} = x^{i} + ϵu^{i}(x) + o(ϵ),

and (1.1) becomes

φ_{ϵ}F (q) ∼ ∑_{0≤|α|≤N} f_{[α]}(ψ_{ϵ}(q)) (∑_{β>0} \frac{1}{β!}ψ_{ϵ}^{(β)}(q)(x − q)^{β})^{α}

∼ F (q) + ϵ ∑_{0≤|α|≤N} u^{i}(q)\frac{∂f_{[α]}}{∂x^{i}} (q)(x − q)^{α}

+ϵ ∑_{0≤|α|≤N} f_{[α]}(q) ∑_{j=1}^{n} α_{j} ∑_{β>0} \frac{1}{β!}(u^{j})^{(β)}(q)(x − q)^{α−ϵ_{j}+β} + o(ϵ).

Thus the action of Vect(M) on J_{N}(M) is given by the formula

\overline{u}F (q) ∼ ∑_{0≤|α|≤N} u^{i}(q)\frac{∂f_{[α]}}{∂x^{i}} (q)(x − q)^{α}

+ ∑_{0≤|α|≤N} f_{[α]}(q) ∑_{j=1}^{n} α_{j} ∑_{β>0} \frac{1}{β!}(u^{j})^{(β)}(q)(x − q)^{α−ϵ_{j}+β}.  (1.2)

We can see from these formulas that the subspace of sections with derivatives up to order ℓ vanishing everywhere

{F ∈ J_{N}(M) | f_{[α]} ≡ 0 for all 0 ≤ |α| ≤ ℓ}

is a submodule for the actions of both Diff(M) and Vect(M). The factor-module in this case is isomorphic to the space J_{ℓ}(M) of sections of the bundle of ℓ-jets.

Now let us look at a more general case of tensor fields. A tensor field of type (s, k) is a section of the corresponding tensor bundle. Suppose that a tensor field F is given in a chart with local coordinates {x^{i}} by expression

F (x) = f_{(j_{1}...j_{k})}^{(i_{1}...i_{s})}(x)dx^{j_{1}} … dx^{j_{k}} \frac{∂}{∂x^{i_{1}}} … \frac{∂}{∂x^{i_{s}}} .

Then the diffeomorphism φ ∈ Diff(M) with φ^{−1} = ψ acts on F according to the formula

φF (y) =

f_{(j_{1}...j_{k})}^{(i_{1}...i_{s})}(ψ(y))\frac{∂ψ^{j_{1}}}{∂y^{j′_{1}}} … \frac{∂ψ^{j_{k}}}{∂y^{j′_{k}}} \frac{∂φ^{i′_{1}}}{∂x^{i_{1}}} (ψ(y)) … \frac{∂φ^{i′_{s}}}{∂x^{i_{s}}} (ψ(y)) dy^{j′_{1}} … dy^{j′_{k}} \frac{∂}{∂y^{i′_{1}}} … \frac{∂}{∂y^{i′_{s}}} . (1.3)

The corresponding action of the Lie algebra Vect(M) can be conveniently encoded using representations of the Lie algebra gl_{n}. Let us explain this construction. The tensor bundle in question is a tensor product of s copies of the tangent bundle and k copies of the cotangent bundle. For a given coordinate system {x^{i}} there is an action of the Lie algebra gl_{n} on the cotangent space, where we set {dx^{i}} as the standard basis of the natural gl_{n}- module. The tangent space becomes the conatural module for gl_{n}. The action of the elementary matrices E_{q}^{p} (a matrix that has entry 1 in position (p, q) and zeros elsewhere) on the tangent and cotangent spaces is given by the formulas:

E_{q}^{p}dx^{i} = δ_{q}^{i}dx^{p},

E_{q}^{p} \frac{∂}{∂x^{i}} = −δ_{i}^{p} \frac{∂}{∂x^{q}} .

The fiber V = Span ⟨dx^{j_{1}} … dx^{j_{k}} \frac{∂}{∂x^{i_{1}}} … \frac{∂}{∂x^{is}} |1 ≤ j_{1}, …, j_{k}, i_{1}, … i_{s} ≤ n⟩ of the (s, k) tensor bundle gets the structure of a gl_{n}-module as a tensor product.

In local coordinates {x^{i}}, any tensor field of type (s, k) is a linear combination of f(x)v, where v ∈ V. The action of a vector field \overline{u} = u^{i}(x)\frac{∂}{∂x^{i}} is then given by the (Lie derivative) formula:

\overline{u} (f(x)v) = (u^{i}(x)\frac{∂f}{∂x^{i}} ) v + (f(x)\frac{∂u^{i}}{∂x^{j}} ) E_{i}^{j}v.  (1.4)

Just as in the case of functions, we can introduce the bundle of N-jets of tensor fields. The fiber at a point p is spanned by the jets

v^{(α)} = (x − p)^{α}v, v ∈ V, 0 ≤ |α| ≤ N.

Again, let us define the action of Diff(M) on the space J_{N}T^{(s,k)}(M) of sections of this bundle. If we have a section F with value at a point p given by an equivalence class of a local tensor field

F (p) ∼ ∑_{0≤|α|≤N} f_{(j_{1}...j_{k})[α]}^{(i_{1}...i_{s})}(p)(x − p)^{α}dx^{j_{1}} … dx^{j_{k}} \frac{∂}{∂x^{i_{1}}} … \frac{∂}{∂x^{i_{s}}}

then φF is a section with value at q = φ(p) given by the equivalence class of the image of this local tensor field under the action of φ according to (1.3):

φF (q) ∼ ∑_{0≤|α|≤N} f_{(j_{1}...j_{k})[α]}^{(i_{1}...i_{s})}(ψ(q))(ψ(y) − ψ(q))^{α}×

×\frac{∂ψ^{j_{1}}}{∂y^{j′_{1}}} … \frac{∂ψ^{j_{k}}}{∂y^{j′_{k}}} \frac{∂φ^{i′_{1}}}{∂x^{i_{1}}} (ψ(y)) … \frac{∂φ^{i′_{s}}}{∂x^{i_{s}}} (ψ(y)) dy^{j′_{1}} … dy^{j′_{k}} \frac{∂}{∂y^{i′_{1}}} … \frac{∂}{∂y^{i′_{s}}} .

Infinitesimal variant of the above formula yields the action of the Lie algebra Vect(M) on J_{N}T^{(s,k)}(M):

\overline{u} (f(x)v^{(α)}) = (u^{i}(x)\frac{∂f(x)}{∂x_{i}}) v^{(α)} + ∑_{j=1}^{n} α_{j} ∑_{β>0} \frac{1}{β!}f(x)(u^{j})^{(β)}(x)v^{(α−ϵ_{j}+β)}

+ ∑_{j=1}^{n} ∑_{β≥0} \frac{1}{β!}f(x)(u^{i})^{(β+ϵ_{j})}(x) (E_{i}^{j}v)^{(α+β)} .  (1.5)

We would like to point out some properties of the module J_{N}T^{(s,k)}(M). In addition to being a module for the Lie algebra Vect(M), it is also a module over a commutative algebra F(M) of functions on M. Moreover (1.5) shows that the two structures are compatible in the following way:

\overline{u} (f(x)F) = (\overline{u}f(x)) F + f(x) (\overline{u}F) , \overline{u} ∈ Vect(M), f ∈ F(M), F ∈ J_{N}T^{(s,k)}(M).

We see that vector fields act as derivations of the multiplication of the jets of tensor fields by functions. This motivates the definition of a category of modules that will be introduced in the next section.

2. Category J.

For the rest of the paper the manifold will be an n-dimensional torus, M = 𝕋^{n} = ℝ^{n}/ℤ^{n}. The algebra of functions on a torus has the Fourier basis {e^{2πimx}|m ∈ ℤ^{n}}. The Lie algebra Vect(𝕋^{n}), also denoted as W_{n}, is a free module over F(𝕋^{n}) of rank n with a basis (as an F(𝕋^{n})-module) \{d_{j} = \frac{1}{2πi} \frac{∂}{∂x^{j}} |j = 1, …, n\}. A basis of W_{n} over ℂ is given by

\{d_{j}(s) = \frac{1}{2πi}e^{2πisx} \frac{∂}{∂x^{j}} |s ∈ ℤ^{n}, j = 1, …, n\} .

The subspace spanned by {d_{j}}_{j=1,...,n} is a Cartan subalgebra in W_{n}.

Note that because the torus is a flat manifold, all bundles considered in Section 1 are trivial. Equivalently, the spaces of sections of these bundles are free modules over F(𝕋^{n}).

Let us define the following category J of W_{n}-modules:

Definition. A W_{n}-module J belongs to category J if the following properties hold: (J1) The action of d_{j}, j = 1, …, n, on J is diagonalizable. (J2) J is a free F(𝕋^{n})-module of a finite rank. (J3) For any \overline{u} ∈ W_{n}, f ∈ F(𝕋^{n}), w ∈ J,

\overline{u} (fw) = (\overline{u}f) w + f (\overline{u}w) .

Strictly speaking, we should denote this category J_{n}, but we will omit the subscript most of the time. All the Vect(𝕋^{n})-modules discussed in Section 1 belong to category J.

The goal of this paper is to classify indecomposable and irreducible modules in this category. Irreducible modules in J have been already classified by Eswara Rao [R2], but here we simplify the proof and make it more lucid. A class of indecomposable W_{n}-modules has been constructed in [R1].

Remark. When we talk about a submodule for a module in category J we mean a subspace which is invariant under the action of both W_{n} and F(𝕋^{n}). Also, in this paper an indecomposable module will be understood to be non-zero.

The concept of a polynomial module will play a central role in our proof. The definition of a polynomial module was given in [BB] in a general setup. Here we adapt that definition for the particular case of W_{n}. We will show below that one can choose a basis v_{1}, … v_{k} of J over F(𝕋^{n}) in such a way that the action of W_{n} is as follows:

d_{j}(s) (e^{2πimx}v_{r}) = ∑_{ℓ=1}^{k} f_{jrℓ}(s, m)e^{2πi(m+s)x}v_{ℓ}, s, m ∈ ℤ^{n}.  (2.1)

We say that module J is a polynomial module if the structure constants f_{jrℓ}(s, m) are polynomials in s, m ∈ ℤ^{n}.

The desired classification of modules in J will be obtained in three steps by proving the following theorems (see the next section for more detailed statements).

Theorem 1. Let n = 1. Every W_{1}-module in category J_{1} is a polynomial module. We will see that with very little effort one can derive from Theorem 1 its generalization

to an arbitrary rank:

Theorem 2. Every W_{n}-module in category J is a polynomial module.

From this Theorem we almost immediately deduce the classification of the indecomposable modules in category J. In order to state this result, we would need to introduce the Lie algebra W_{n}^{+}.

Consider the Lie algebra of derivations of the algebra of polynomials in n variables:

Der (ℂ[z_{1}, …, z_{n}]) = Span ⟨z^{α} \frac{∂}{∂z_{j}} | α ∈ ℤ_{+}^{n}, j = 1, …, n⟩ .

The Lie algebra W_{n}^{+} is defined as a subalgebra in Der (ℂ[z_{1}, …, z_{n}]):

W_{n}^{+} = Span ⟨z^{α} \frac{∂}{∂z_{j}} | α ∈ ℤ_{+}^{n}\\{0\}, j = 1, …, n⟩ .

Theorem 3. There is a 1-1 correspondence between indecomposable modules in category J and pairs (λ, U), where λ ∈ ℂ^{n}/ℤ^{n} and U is an indecomposable finite-dimensional module for W_{n}^{+}.

3. Structure of indecomposable modules.

Finiteness condition (J2) implies that every module in category J is a finite direct sum of indecomposable submodules. This allows us to restrict our attention to indecomposable modules.

Let us write the weight decomposition of an indecomposable module J with respect to the Cartan subalgebra of W_{n}:

J = ⊕_{μ∈ℂ^{n}} J_{μ}, where J_{μ} = {w ∈ J|d_{j}w = μ_{j}w} .

It is easy to see that d_{j}(s)J_{μ} ⊆ J_{μ+s} and e^{2πisx}J_{μ} ⊆ J_{μ+s}, thus the weights of J are split into ℤ^{n}-cosets in ℂ^{n}, and the submodules corresponding to distinct cosets form a direct sum. Since we assumed J to be indecomposable, its weight lattice is a single coset λ + ℤ^{n}, λ ∈ ℂ^{n}. Let us denote the weight space J_{λ} by U. It is easy to see that the basis of U is also a basis of J as an F(𝕋^{n})-module. Thus, by (J2), the space U is finite-dimensional and J = F(𝕋^{n}) ⊗ U.

Since e^{2πisx}U = J_{λ+s}, we may identify each weight space of J with the same finitedimensional space U. The operator

d_{j}(s) : U → e^{2πisx}U

induces an endomorphism D_{j}(s) : U → U, such that

d_{j}(s)|_{U} = e^{2πisx}D_{j}(s).

In particular we have D_{j}(0) = λ_{j}Id .

The finite-dimensional operator D_{j}(s) ∈ End U completely determines the action of d_{j}(s) on J since by (J3),

d_{j}(s) (e^{2πimx}v) = (d_{j}(s)e^{2πimx}) v + e^{2πimx}d_{j}(s)v = e^{2πi(m+s)x} (m_{j}Id + D_{j}(s)) v. (3.1)

Now we see that the action of d_{j}(s) is written in the form (2.1) The structure constants f_{jrℓ} are encoded in the operators m_{j}Id + D_{j}(s). The dependence on m here is clearly polynomial, so the statement that J is a polynomial W_{n}-module is equivalent to the claim that the dependence of the family of operators {D_{j}(s)} on s is polynomial.

Theorem 2. An indecomposable W_{n}-module J = F(𝕋^{n}) ⊗ U in category J is a polynomial module. The action of W_{n} can be written as follows:

d_{j}(s) (e^{2πimx}v) = e^{2πi(m+s)x} (m_{j}Id + D_{j}(s)) v,

where operators D_{j}(s) ∈ End U have a polynomial dependence on s ∈ ℤ^{n} and D_{j}(0) = λ_{j}Id .

Proving Theorem 2 directly would be rather technical. Instead, we will derive it from its special case of n = 1, which is Theorem 1. The proof of Theorem 1 will be deferred to the next section.

First of all, let us write down the commutator relations between operators D_{j}(s): Lemma 1.

[D_{j}(s), D_{k}(m)] = m_{j} (D_{k}(s + m) − D_{k}(m)) − s_{k} (D_{j}(s + m) − D_{j}(s)) .  (3.2)

This Lemma can be derived in a straightforward way from the commutator relations in W_{n}

[d_{j}(s), d_{k}(m)] = m_{j}d_{k}(s + m) − s_{k}d_{j}(s + m)  (3.3)

and (3.1). The details of this computation are left as an exercise.

Proof of Theorem 2. We assume that the claim of the theorem holds in rank one case, n = 1.

Consider the following subalgebras in W_{n}, each isomorphic to W_{1}:

W_{1}^{(j)} = Span ⟨\frac{1}{2πi}e^{2πisx^{j}} \frac{∂}{∂x^{j}} | s ∈ ℤ⟩ .

The subspace ⊕_{m∈ℤ} e^{2πimx^{j}}U ⊂ J is a W_{1}^{(j)}-module which belongs to category J_{1}. Applying

Theorem 1, we get that the family of operators {D_{j}(sϵ_{j})} has a polynomial dependence on s ∈ ℤ.

Without the loss of generality we may restrict ourselves to proving that D_{1}(s_{1}, …, s_{n}) is a polynomial in s_{1}, …, s_{n}. This is of course equivalent to showing that D_{1}(s_{1}, 1 + s_{2}, …, 1 + s_{n}) is a polynomial in s_{1}, s_{2}, …, s_{n}. We will prove by induction the claim that D_{1}(s_{1}, 1 + s_{2}, …, 1 + s_{j}, 1, …, 1) is a polynomial in s_{1}, s_{2}, …, s_{j}.

Let us establish the basis of induction. From (3.2) we get that

[D_{1}(s_{1}ϵ_{1}), D_{1}(0, 1, … 1)] = −s_{1}D_{1}(s_{1}, 1, …, 1) + s_{1}D_{1}(s_{1}ϵ_{1}),

and so

s_{1}D_{1}(s_{1}, 1, …, 1) = s_{1}D_{1}(s_{1}ϵ_{1}) − [D_{1}(s_{1}ϵ_{1}), D_{1}(0, 1, … 1)] .

The right hand side is manifestly a polynomial in s_{1}, so is the left hand side. Note also that the right hand side vanishes at s_{1} = 0 because D_{1}(0) = λ_{1}Id . Hence this polynomial has a factor of s_{1}, and thus D_{1}(s_{1}, 1, …, 1) is a polynomial in s_{1}, which proves the basis of induction.

Let us now prove the inductive step. Again from (3.2) we get

D_{1}(s_{1}, 1 + s_{2}, … 1 + s_{j−1}, 1 + s_{j}, 1, …, 1) =

[D_{j}(s_{j}ϵ_{j}), D_{1}(s_{1}, 1 + s_{2}, … 1 + s_{j−1}, 1, …, 1)] + D_{1}(s_{1}, 1 + s_{2}, … 1 + s_{j−1}, 1, …, 1).

By induction assumption, the right hand side is a polynomial in s_{1}, …, s_{j−1}, s_{j}, and so is the left hand side. This completes the induction and Theorem 2 is now proved (under assumption of validity of Theorem 1).

Now given that D_{j}(s) is a polynomial function with a constant term D_{j}(0) = λ_{j}Id , we can expand it in a finite sum:

D_{j}(s) = λ_{j}Id + ∑_{α∈ℤ_{+}^{n}\\{0\}}^{finite} \frac{s^{α}}{α!} D_{j}^{(α)},  (3.4)

where the operators D_{j}^{(α)} ∈ End U are independent of s and D_{j}^{(α)} = 0 for all but finitely many α ∈ ℤ_{+}^{n}\{0}.

Theorem 3. (a) The operators D_{j}^{(α)} ∈ End U yield a finite-dimensional representation on space U of a Lie algebra

W_{n}^{+} = Span ⟨z^{α} \frac{∂}{∂z_{j}} | α ∈ ℤ_{+}^{n}\\{0\}, j = 1, …, n⟩ ,

given by ρ (z^{α} \frac{∂}{∂z_{j}} ) = D_{j}^{(α)}.

(b) There is a 1-1 correspondence between indecomposable modules in category J and pairs (λ, U), where λ ∈ ℂ^{n}/ℤ^{n} and U is an indecomposable finite-dimensional module for W_{n}^{+}. This correspondence is given by the tensor product decomposition J = F(𝕋^{n}) ⊗ U, and the action of W_{n} is

\frac{1}{2πi}e^{2πisx} \frac{∂}{∂x^{j}} (e^{2πimx}v) = (m_{j} + λ_{j})e^{2πi(m+s)x}v + ∑_{β>0} \frac{s^{β}}{β!} e^{2πi(m+s)x}ρ (z^{β} \frac{∂}{∂z_{j}} ) v. (3.5)

(c) There is a 1-1 correspondence between irreducible modules in category J and pairs (λ, V), where λ ∈ ℂ^{n}/ℤ^{n} and V is an irreducible gl_{n}(ℂ)-module. The action of W_{n} on J = F(𝕋^{n}) ⊗ V is as follows:

\frac{1}{2πi}e^{2πisx} \frac{∂}{∂x^{j}} (e^{2πimx}v) = (m_{j} + λ_{j})e^{2πi(m+s)x}v + ∑_{p=1}^{n} s_{p}e^{2πi(m+s)x}ρ (E_{j}^{p}) v. (3.6)

Proof. Let us determine the commutator relations between operators D_{j}^{(α)}. We have

[D_{j}(s), D_{k}(m)] = ∑_{α,β∈ℤ_{+}^{n}\\{0\}} \frac{s^{α}m^{β}}{α!β!} [D_{j}^{(α)}, D_{k}^{(β)}] .

On the other hand, by Lemma 1,

[D_{j}(s), D_{k}(m)] = ∑_{γ∈ℤ_{+}^{n}\\{0\}} m_{j} \frac{(s + m)^{γ} − m^{γ}}{γ!} D_{k}^{(γ)} − ∑_{γ∈ℤ_{+}^{n}\\{0\}} s_{k} \frac{(s + m)^{γ} − s^{γ}}{γ!} D_{j}^{(γ)}.

Two polynomials have equal values whenever their coefficients coincide. If we equate the coefficients at \frac{s^{α}m^{β}}{α!β!} , α, β ∈ ℤ_{+}^{n}\\{0\}, we get

[D_{j}^{(α)}, D_{k}^{(β)}] = β_{j}D_{k}^{(α+β−ϵ_{j})} − α_{k}D_{j}^{(α+β−ϵ_{k})}.  (3.7)

We see that these are precisely the commutator relations in the algebra W_{n}^{+}:

[z^{α} \frac{∂}{∂z_{j}} , z^{β} \frac{∂}{∂z_{k}} ] = β_{j}z^{α+β−ϵ_{j}} \frac{∂}{∂z_{k}} − α_{k}z^{α+β−ϵ_{k}} \frac{∂}{∂z_{j}} ,

so the map ρ : W_{n}^{+} → End U, given by ρ (z^{α} \frac{∂}{∂z_{j}} ) = D_{j}^{(α)}, is a representation of W_{n}^{+}. This completes the proof of part (a).

Let us prove (b). We have already seen that an indecomposable module J in category J yields a coset of weights λ+ℤ^{n} ⊂ ℂ^{n}, and a finite-dimensional representation of W_{n}^{+}. It is easy to check that the W_{n}^{+}-module U is independent of the choice of the weight λ in the coset. Conversely, the commutator relations (3.7) imply (3.2) and together with (3.1) give (3.3), provided of course that the right hand side in (3.4) is finite. Thus we need to show that for a finite-dimensional representation (U, ρ) of the Lie algebra W_{n}^{+}, we will have that D_{j}^{(α)} = ρ (z^{α} \frac{∂}{∂z_{j}} ) = 0 for all but finitely many α ∈ ℤ_{+}^{n}\\{0\}. This will follow from the following simple

Lemma 2. Let L be a Lie algebra, and (U, ρ) its finite-dimensional representation. Suppose that for x, y_{1}, y_{2}, … ∈ L we have

[x, y_{k}] = ν_{k}y_{k}, ν_{k} ∈ ℂ k = 1, 2, ….

Then there are at most (dim U)^{2} − dim U + 1 distinct eigenvalues for which ρ(y_{k}) ≠ 0. Proof of the Lemma. In representation ρ we have

[ρ(x), ρ(y_{k})] = ν_{k}ρ(y_{k}).

However an element ρ(x) in the Lie algebra gl(U) may have at most (dim U)^{2} − dim U + 1 distinct eigenvalues in the adjoint representation, which implies the claim of the Lemma.

To apply this lemma, we consider the element E = z_{1} \frac{∂}{∂z_{1}} + … z_{n} \frac{∂}{∂z_{n}} ∈ W_{n}^{+} . The following relations hold:

[E, z^{α} \frac{∂}{∂z_{j}} ] = (|α| − 1) z^{α} \frac{∂}{∂z_{j}} .

Then by Lemma 2, ρ (z^{α} \frac{∂}{∂z_{j}} ) = 0 for all but finitely many α.

To complete the proof of part (b) of the theorem, we note that an indecomposable W_{n}-module in category J yields an indecomposable W_{n}^{+}-module and vice versa.

Now let us prove part (c) and assume that J is an irreducible module in category J. From part (b) we know that such a module is determined by a pair (λ, V), where λ ∈ ℂ^{n}/ℤ^{n} and V is a finite-dimensional W_{n}^{+}-module. The module V has to be irreducible, because we have a correspondence between W_{n}^{+}-submodules S ⊂ V and W_{n}-submodules F(𝕋^{n}) ⊗ S ⊂ J.

Let us show that irreducible finite-dimensional W_{n}^{+}-modules V are just irreducible gl_{n}(ℂ)-modules. Let V_{ν} be an eigenspace of the operator ρ(E) in V corresponding to eigenvalue ν. It is easy to see that

ρ (z^{α} \frac{∂}{∂z_{j}} ) V_{ν} ⊆ V_{ν+|α|−1}.

This implies that ⊕_{k=1}^{∞} V_{ν+k} is a W_{n}^{+}-submodule in V. However V is irreducible, which

implies that V_{ν+k} = (0) for all k = 1, 2, …, and hence

ρ (z^{α} \frac{∂}{∂z_{j}} ) = 0 for all |α| > 1.

Thus the ideal

W_{n}^{++} = Span ⟨z^{α} \frac{∂}{∂z_{j}} | α ∈ ℤ_{+}^{n}, |α| > 1, j = 1, …, n⟩

vanishes in every finite-dimensional irreducible W_{n}^{+}-module. But W_{n}^{+}/W_{n}^{++} ≅ gl_{n}(ℂ), and the claim of part (c) of the theorem follows.

Next, let us give an example of a family of indecomposable finite-dimensional W_{n}^{+}- modules.

Example. Let V be an irreducible finite-dimensional gl_{n}(ℂ)-module and let

Ṽ = ℂ[z_{1}, …, z_{n}] ⊗ V.

We can define on Ṽ the structure of a tensor module for the Lie algebra Der ℂ[z_{1}, …, z_{n}] and its subalgebra W_{n}^{+} (cf., (1.4), see also [Ru]):

z^{β} \frac{∂}{∂z_{j}} (z^{α}v) = α_{j}z^{α+β−ϵ_{j}}v + ∑_{k=1}^{n} β_{k}z^{α+β−ϵ_{k}}(E_{j}^{k}v).

As a W_{n}^{+}-module, this tensor module has finite-dimensional factors

V^{(N)} = Ṽ/ ⟨z^{α} ⊗ V | |α| > N⟩ .

Using the correspondence of Theorem 3(b), we construct a representation of the Lie algebra Vect(𝕋^{n}) on space F(𝕋^{n}) ⊗ V^{(N)} (setting λ = 0):

\frac{1}{2πi}e^{2πisx} \frac{∂}{∂x^{j}} (e^{2πimx}z^{α}v) = m_{j}e^{2πi(m+s)x}z^{α}v

+α_{j} ∑_{β>0} \frac{s^{β}}{β!} e^{2πi(m+s)x}z^{α+β−ϵ_{j}}v + ∑_{β>0} \frac{s^{β}}{β!} ∑_{k=1}^{n} β_{k}e^{2πi(m+s)x}z^{α+β−ϵ_{k}}E_{j}^{k}v.  (3.8)

Comparing (3.8) with (1.5) we see that the module F(𝕋^{n})⊗V^{(N)} is in fact isomorphic to a module of sections of N-jets of tensor fields corresponding to the gl_{n}-module V. The isomorphism is given by z^{α}v = (2πi)^{|α|}v^{(α)}.

Remark. It is curious to point out that for this family of polynomial modules, the degree of the structure polynomials is equal to N + 1, and thus could be arbitrarily high. In the previously known examples (see [BB] and [BZ]) the degree was at most 3.

4. Rank one case.

In this section we will give the proof of Theorem 1. Since we will deal exclusively with the case n = 1, we may simplify notations denoting x^{1} as x, d_{1}(s) as d(s), etc. Let J be an indecomposable module in category J_{1}. As in Section 3 we see that J = F(𝕋^{1}) ⊗ U. Then (3.1) becomes

d(s) (e^{2πimx}v) = e^{2πi(m+s)x} (mId + D(s)) v, s, m ∈ ℤ.  (4.1)

The operators D(s) ∈ End U satisfy the commutator relations (cf. (3.2)):

[D(s), D(m)] = (m − s)D(s + m) − mD(m) + sD(s).  (4.2)

Theorem 1. Every W_{1}-module J in category J_{1} is a polynomial module.

Proof. Just as in Section 3, we will assume J to be indecomposable, and so (4.1) holds. We will prove this theorem by showing that a family of operators {D(s)} on a finite-dimensional space U satisfying (4.2) must have a polynomial dependence on s ∈ ℤ.

The relations (4.2) define an infinite-dimensional Lie algebra L with basis {D(s)|s ∈ ℤ}, and we are studying a finite-dimensional representation ρ of L on space U.

Our strategy will be the following. First we will find eigenvectors of D(−1) in the adjoint representation of L. We shall see that these eigenvectors are difference derivatives of D(s) with respect to s. By applying Lemma 2, we will conclude that higher order derivatives of D(s) vanish, which means that D(s) is a polynomial in s.

Let us define the operator of a difference derivative. Let

f : ℤ → A

be a function of an integer variable with values in an abelian group A (in our case A is the vector space L). The difference derivative ∂f is a function ∂f : ℤ → A, defined by ∂f(s) = f(s+1)−f(s). By iteration, we can also define higher order difference derivatives of f. It is easy to see that

∂^{m}f(s) = ∑_{k=0}^{m}(−1)^{m−k} ( \begin{matrix}m \\ k \end{matrix} ) f(s + k).  (4.3)

Let us now additionally assume that A is a module over ℚ, so that we can define interpolation polynomials. For any N + 1 distinct points r_{1}, …, r_{N+1} ∈ ℤ and any set of values a_{1}, …, a_{N+1} ∈ A, there exists a unique polynomial f(t) ∈ A[t] of degree at most N such that f(r_{j}) = a_{j} for j = 1, …, N + 1 ([vdW], section 22).

Lemma 3. Fix s ∈ ℤ.

(a) If ∂^{m}f(s) = 0 for all m ≥ 0 then f(r) = 0 for all r ≥ s.

(b) Suppose ∂^{m}f(s) = 0 for all m > N. Let g(t) be the interpolation polynomial of degree at most N defined by g(j) = f(j) for j = s, s + 1, …, s + N. Then f(r) = g(r) for all r ≥ s.

Proof of the Lemma. Part (a) can be easily proved by induction using (4.3). To prove part (b), we consider the function h(r) = f(r) − g(r). Since h(s) = … = h(s + N) = 0, we get from (4.3) that ∂^{m}h(s) = 0 for m = 0, …, N. For m > N we have ∂^{m}f(s) = 0 by assumption and ∂^{m}g(s) = 0 since it is a polynomial of degree at most N. Thus ∂^{m}h(s) = 0 for all m ≥ 0, and according to part (a) of the Lemma, f(r) = g(r) for all r ≥ s. This completes the proof of the Lemma.

Now consider the following elements in the Lie algebra L:

y_{m} = ∂^{m+1}D(−1) = ∑_{k=0}^{m+1}(−1)^{m+1−k} ( \begin{matrix}m + 1 \\ k \end{matrix} ) D(−1 + k).

Lemma 4. (a) For m ≥ 0, y_{m} is an eigenvector for adD(−1) with eigenvalue ν_{m} = −m.

(b) For m, k ≥ 0, [y_{k}, y_{m}] = (m − k)y_{m+k}.

Proof. In the calculation below we will use two elementary properties of binomial coefficients:

∑_{k=0}^{m}(−1)^{m−k} ( \begin{matrix}m \\ k \end{matrix} ) = 0 for m ≥ 1, and (r + 1) ( \begin{matrix}m \\ r + 1 \end{matrix} ) = (m − r) ( \begin{matrix}m \\ r \end{matrix} ) .

Now,

[D(−1), y_{m−1}] = ∑_{k=0}^{m}(−1)^{m−k} ( \begin{matrix}m \\ k \end{matrix} ) [D(−1), D(−1 + k)]

= ∑_{k=0}^{m}(−1)^{m−k} ( \begin{matrix}m \\ k \end{matrix} ) (kD(−2 + k) − (k − 1)D(−1 + k) − D(−1))

= − ∑_{{r=0 \atop r=k−1}}^{m−1}(−1)^{m−r}(r + 1) ( \begin{matrix}m \\ r + 1 \end{matrix} ) D(−1 + r) − ∑_{k=0}^{m}(−1)^{m−k}(k − 1) ( \begin{matrix}m \\ k \end{matrix} ) D(−1 + k)

− ∑_{k=0}^{m}(−1)^{m−k} ( \begin{matrix}m \\ k \end{matrix} ) D(−1)

= − ∑_{r=0}^{m}(−1)^{m−r}(m − r) ( \begin{matrix}m \\ r \end{matrix} ) D(−1 + r) − ∑_{r=0}^{m}(−1)^{m−r}(r − 1) ( \begin{matrix}m \\ r \end{matrix} ) D(−1 + r)

= (−m + 1) ∑_{r=0}^{m}(−1)^{m−r} ( \begin{matrix}m \\ r \end{matrix} ) D(−1 + r) = (−m + 1)y_{m−1}.

Part (b) of the lemma will not be used in this paper and its proof is left as an exercise.

Now we are ready to complete the proof of Theorem 1. Combining Lemma 4(a) with Lemma 2, we conclude that there exists N such that ρ(y_{m}) = ρ(∂^{m+1}D(−1)) = 0 for all m ≥ N. Then by Lemma 3(b), there exists an End U-valued polynomial g(s) such that ρ(D(r)) = g(r) for all r ≥ −1. It only remains to prove that ρ(D(r)) = g(r) for all r ∈ ℤ.

To achieve this, we take p = 2, 3, 4, … and consider a subalgebra L_{p} ⊂ L:

L_{p} = Span ⟨D(pk)|k ∈ ℤ⟩ .

It is easy to see that the map θ_{p} : L_{p} → L defined by θ_{p}(D(pk)) = pD(k), is an isomorphism. Thus everything we proved for L is also valid for L_{p}. This means that there exists a polynomial g_{p}(s) such that ρ(D(pr)) = g_{p}(pr) for all r ≥ −1. Since the values of the polynomials g(s) and g_{p}(s) coincide at infinitely many points, we conclude that g_{p}(s) = g(s). Taking now r = −1 and letting p = 2, 3, …, we get that ρ(D(−p)) = g(−p). Thus ρ(D(r)) = g(r) for all r ∈ ℤ. Theorem 1 is now proved.

5. Modules for the semidirect product of Vect(𝕋^{n}) with a multi-loop algebra.

Let 𝔤̇ be a finite-dimensional Lie algebra over ℂ. Consider a multi-loop Lie algebra

𝔤̃ = F(𝕋^{n}) ⊗ 𝔤̇.

The Lie algebra W_{n} = Vect(𝕋^{n}) acts in a natural way on the multi-loop algebra, so we can form the semidirect product

𝔤 = W_{n} ⊕ 𝔤̃.

We define the category J consisting of 𝔤-modules J satisfying (J1)-(J3) and also

(J4)  g̃(fv) = f(g̃v), for g̃ ∈ 𝔤̃, f ∈ F(𝕋^{n}), v ∈ J.

Theorem 4. (a) Every 𝔤-module in category J is a polynomial module.

(b) There exists a 1-1 correspondence between indecomposable 𝔤-modules in category J and pairs (λ, U), where λ ∈ ℂ^{n}/ℤ^{n} and U is a finite-dimensional indecomposable module for the semidirect product

𝔤^{+} = W_{n}^{+} ⊕ ℂ[z_{1}, …, z_{n}] ⊗ 𝔤̇.

This correspondence is given by the tensor product decomposition J = F(𝕋^{n}) ⊗ U, and 𝔤 acts according to (3.5) and

(e^{2πisx}g) (e^{2πimx}v) = ∑_{β≥0} \frac{s^{β}}{β!} e^{2πi(m+s)x}ρ (z^{β}g) v, g ∈ 𝔤̇, v ∈ U.  (5.1)

(c) Irreducible 𝔤-modules in category J are in a 1-1 correspondence with pairs (λ, V), where λ ∈ ℂ^{n}/ℤ^{n} and V is a finite-dimensional irreducible gl_{n}(ℂ) ⊕ 𝔤̇-module. The action of 𝔤 on J = F(𝕋^{n}) ⊗ V is given by (3.6) and

(e^{2πisx}g) (e^{2πimx}v) = e^{2πi(m+s)x}(gv), g ∈ 𝔤̇, v ∈ V.  (5.2)

Let us outline the proof of Theorem 4. In the same way as in the discussion at the beginning of Section 3, we can show that an indecomposable 𝔤-module in category J is a tensor product J = F(𝕋^{n}) ⊗ U with dim U < ∞, and the action of 𝔤 given by (3.1) and for g ∈ 𝔤̇:

(e^{2πisx}g) (e^{2πimx}v) = e^{2πi(m+s)x}g(s)v,

for some operators g(s) ∈ End U, s ∈ ℤ^{n}. In order to prove that J is a polynomial module, we have to show that the family of operators {g(s)} has a polynomial dependence on s.

From the commutator relation in 𝔤,

[\frac{1}{2πi}e^{2πisx} \frac{∂}{∂x^{j}} , e^{2πimx}g] = m_{j}e^{2πi(s+m)x}g,

we get that

[D_{j}(s), g(m)] = m_{j} (g(s + m) − g(m)) .  (5.3)

Applying the above equality to g(1, …, 1) we see that

g(1 + s_{1}, …, 1 + s_{n}) = [D_{j}(s), g(1, …, 1)] + g(1, …, 1).

By Theorem 2, the operators {D_{j}(s)} depend on s polynomially, hence the same is true for {g(s)}.

To prove part (b), we expand the polynomial g(s):

g(s) = ∑_{β≥0}^{finite} \frac{s^{β}}{β!} g^{(β)},

with g^{(β)} ∈ End U and g^{(β)} = 0 for β ≫ 0.

From the expansions of the relations

[g(s), h(m)] = [g, h](s + m), g, h ∈ 𝔤̇,

and (5.3), we see that

[g^{(α)}, h^{(β)}] = [g, h]^{(α+β)},  (5.4)
[D_{j}^{(α)}, g^{(β)}] = β_{j}g^{(α+β−ϵ_{j})}.  (5.5)

This shows that every indecomposable 𝔤-module J ∈ J yields a finite-dimensional 𝔤^{+}- module U.

Using the same technique as in the proof of Theorem 3 (b), we can show that for any finite-dimensional 𝔤^{+}-module U, ρ (z^{α}g) = 0 for α ≫ 0. This proves that the correspondence is bijective.

Finally, to prove part (c) of the theorem, we note that irreducible 𝔤-modules J correspond to irreducible 𝔤^{+}-modules V. Let us show that V is in fact a gl_{n}(ℂ) ⊕ 𝔤̇-module. Let V_{ν} be an eigenspace for ρ(E), E = z_{1} \frac{∂}{∂z_{1}} + … z_{n} \frac{∂}{∂z_{n}}. It is easy to see that

ρ (z^{α} \frac{∂}{∂z_{j}} ) V_{ν} ⊂ V_{ν+|α|−1}, and ρ (z^{β}g) V_{ν} ⊂ V_{ν+|β|}.

Thus ⊕_{k=1}^{∞} V_{ν+k} is a 𝔤^{+}-submodule in V. Irreducibility of V implies that V_{ν+k} = (0) for all

k ≥ 1, and hence the ideal

g^{++} = Span ⟨z^{α} \frac{∂}{∂z_{j}} , z^{β}g|α, β ∈ ℤ_{+}^{n}, |α| > 1, |β| ≥ 1, g ∈ 𝔤̇, j = 1, …, n⟩

vanishes in every finite-dimensional irreducible 𝔤^{+}-module V. The claim of part (c) follows from the fact that 𝔤^{+}/𝔤^{++} ≅ gl_{n}(ℂ) ⊕ 𝔤̇.

Acknowledgments. This research is supported by the Natural Sciences and Engineering Research Council of Canada.

I thank Mikhail Kochetov for helpful discussions. I also thank Eswara Rao for keeping me updated on his research. In particular, this work was strongly influenced by Eswara Rao’s papers [R1] and [R2].

References:

[BB] S. Berman, Y. Billig, Irreducible representations for toroidal Lie algebras. J.Algebra 221 (1999), 188-231.

[BZ] Y. Billig, K. Zhao, Weight modules over exp-polynomial Lie algebras. J. Pure Appl. Algebra 191 (2004), 23-42.

[R1] S. Eswara Rao, A new class of modules for derivations of Laurent polynomial ring in n variables. Preprint, 2003.

[R2] S. Eswara Rao, Partial classification of modules for Lie algebra of diffeomorphisms of d-dimensional torus. J. Math. Phys. 45 (2004), 3322-3333.

[M] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro algebra. Invent. Math. 107 (1992), 225-234.

[O] P.J. Olver, Applications of Lie groups to differential equations. Graduate Texts in Math. 107, Springer-Verlag, New York, 1986.

[Ru] A.N. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of Cartan type. Math. USSR Izv. 8 (1974), 836-866.

[S] D.J. Saunders, The geometry of jet bundles. London Math. Soc. Lecture Notes Ser.

142, Cambridge Univ. Press, Cambridge, 1989.

[vdW] B.L. van der Waerden, Modern Algebra. Frederick Ungar Publ., New York, 1949.

School of Mathematics and Statistics Carleton University Ottawa, Ontario, K1S 5B6, Canada.