Structures of W(2.2) Lie conformal algebra

Lamei Yuan; Henan Wu

doi:10.1515/math-2016-0054

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Structures of W(2.2) Lie conformal algebra

Lamei Yuan and Henan Wu

Published/Copyright: September 15, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

The purpose of this paper is to study W(2, 2) Lie conformal algebra, which has a free ℂ[∂]-basis {L, M} such that [LλL]=(∂+2λ)L,[LλM]=(∂+2λ)M,[MλM]=0 . In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.

Keywords: Conformal derivation; Central extension; Conformal module; Cohomology

MSC 2010: 05C50; 05C76; 05C30; 05C05

1 Introduction

A Lie conformai algebra is a ℂ[∂]-module R equipped with a λ-bracket [·λ·] which is a ℂ-bilinear map from R ⊗ R to ℂ[λ] ⊗ R, such that the following axioms hold for all a, b, c ∈ R:

[∂aλb]=−λ[aλb], [aλb], [aλ∂b]=(∂+λ)[aλb] (conformal sesquilinearity),(1)

[aλb]=−[b−λ−∂a](skew−symmetry),(2)

[aλ[bμc]]=[[aλb]λ+μc]+[bμ[aλc]] (Jacobi identity).(3)

In practice, the λ-brackets arise as generating functions for the singular part of the operator product expansion in two-dimensional conformal field theory [1]. Lie conformal algebras are closely related to vertex algebras and infinite-dimensional Lie (super)algebras satisfying the locality property [2]. In the past few years, semisimple Lie conformal algebras have been intensively studied. In particular, the classification of all finite semisimple Lie conformal (super)algebras were given in [3, 4]. Finite irreducible conformal modules over the Virasoro, the current and the Neveu-Schwarz conformal algebras were classified in [5]. The cohomology theory was developed by Bakalov, Kac, and Voronov in [6], where explicit computations of cohomologies of the Virasoro and the current conformal algebras were given. The aim of this paper is to study structures including derivations, central extensions, conformal modules and cohomologies of a non-semisimple Lie conformal algebra associated to the W-algebra W(2, 2). The method here is based on the theory of the Virasoro conformal modules [5] and the techniques developed in [3, 6]. Our discussions on the cohomology may be useful to do the same thing for the other non-semisimple Lie conformal algebras, such as the Schrodinger-Virasoro, the loop Virasoro and the loop Heisenberg-Virasoro conformal algebras studied in [7-9].

The Lie conformal algebra which we consider in this paper, denoted by W, is a free ℂ[3]-module of rank 2 generated by L, M, satisfying

[LλL]=(∂+2λ)L, LλM]|=(∂+2λ)M, [MλM]=0.(4)

The formal distribution Lie algebra corresponding to W is the centerless W-algebra W(2, 2) introduced in [10]. Besides, this Lie conformal algebra can be seen as a special case of a more general W(a, b) Lie conformal algebra studied in [11]. Obviously, the Lie conformal algebra W contains the Virasoro conformal algebra Vir as a subalgebra, namely,

Vir=C[∂]L,[LλL]=(∂+2λ)L.(5)

And it has a nontrivial abelian conformal ideal generated by M. Thus it is not semisimple.

The rest of the paper is organized as follows. In Sect. 2, we study conformal derivations of the Lie conformal algebra W. It turns out that all the conformal derivations of W are inner. In Sect. 3, we discuss central extensions of W and show that W has a unique nontrivial universal central extension. In Sect. 4, we determine all free nontrivial conformal W-modules of rank one. In Sect. 5, we compute cohomologies of W with coefficients in W-modules ℂ, ℂ_a and M_Δ,λ, respectively. Consequently, we have the basic and reduced cohomologies for all q ≥ 0 determined.

Throughout this paper, all vector spaces, linear maps and tensor products are over the complex field ℂ. We use notations ℤ for the set of integers and ℤ₊ for the set of nonnegative integers.

2 Conformai derivation

Let C denote the ring ℂ[∂] of polynomials in the indeterminate ∂.

Definition 2.1

Let V and W be two C-modules. A ℂ-linear map ϕ : V → C[λ] ⊗_C W, denoted by ϕ_λV → W, is called a conformal linear map, if

ϕ(∂υ)=(∂+λ)(ϕυ), for υ∈V.

Denote by Chom(V, W) the space of conformal linear maps from C-modules V to W. It can be made into an C-module via

(∂ϕ)λυ=−λϕλυ, for υ ∈ V.

Definition 2.2

Let A be a Lie conformal algebra. A conformal linear map d_λ : A → A is called a conformal derivation if

dλ([aμb])=[(dλa)λ+μb]+[aμ(dλb)],foralla,b∈A.

Denote by CDer(A) the space of all conformal derivations of A. For any a ∈ A, one can define a linear map (ad a)_λ : A → A by (ada)_λb = [a_λ] for all b ∈ A. It is easy to check that (ada)_λ is a conformal derivation of A. Any conformal derivation of this kind is called an inner derivation. The space of all inner derivations is denoted by CInn(A).

For the Lie conformal algebra W, we have the following result.

Proposition 2.3

Every conformal derivation of W is inner, namely, CDer(W) = CInn(W).

Proof

Let d_λ be a conformal derivation of W. Then

dλL=f1(λ,∂)L+f2(λ,∂)M, dλM=h1(λ,∂)L+h2(λ,∂)M,(6)

where f_i (λ, ∂) and h_i (λ, ∂) for i = 1, 2 are polynomials in ℂ[λ, ∂]. Applying d_λ to [L_{μ L]} = (∂ + 2μ)L, we have

dλ[LμL]=[(dλL)λ+μL]+[Lμ(dλL)] = (∂+2λ+2μ)f1(λ,−λ−μ)L+(∂+2λ+2μ)f2(λ,−λ−μ)M +(∂+2μ)f1(λ,∂+μ)L+(∂+2μ)f2(λ,∂+μ)M = dλ((∂+2μ)L) =(∂+λ+2μ)(f1(λ, ∂)L+f2(λ, ∂)M).

Comparing the coefficients of the similar terms gives

(∂+λ+2μ)fi(λ,∂)−(∂+2μ)fi(λ,∂+μ)=(∂+2λ+2μ)fi(λ,−λ−μ), for i=1,2.(7)

Write fi(λ,∂)∑j=0nai,j(λ)∂j with a_{i, n}(λ) ≠ 0. Assume n > 1. Equating the coefficients of ∂ⁿ in (7) gives (λ – nμ)a_{i, n}(λ) = 0. Thus a_{i, n}(λ) = 0, a contradiction. Therefore,

fi(λ,∂)=ai,0(λ)+ai,1(λ)∂, for i=1,2.(8)

By replacing d_λ by d_λ — ad(a_{1, 1} (—∂)L)_λ — ad(a_{2, 1} (—∂)M)_λ, we can suppose a_{1, 1}(λ) = a_{2, 1}(λ) = 0. Then plugging f_i (λ, ∂) = a_{i, 0}(λ) into (7) gives a_{i, 0}(λ) = 0 for i = 1, 2. Thus d_λ (L) = 0 by (6). Fouthermore, applying d_λ to [L_μM] = (∂ + 2μ)M, we have

(∂+λ+2μ)hi(λ,∂)=(∂+2μ)hi(λ,∂+μ) for i=1,2.(9)

Comparing the coefficients of highest degree of λ in (9) gives h_i (λ, ∂) = 0 for i = 1, 2. Hence d_λ (M) = 0 by (6). This concludes the proof.

Remark 2.4

Proposition 2.3 is equivalent to H¹ (W, W) = 0.

3 Central extension

An extension of a Lie conformal algebra A by an abelian Lie conformal algebra a is a short exact sequence of Lie conformal algebras

0→a→A^→A→0.

In this case A^ is also called an extension of A by a. The extension is said to be central if

a⊆Z(A^)={x∈A^|[xλy]A^=0forally∈A^},and∂a=0.

Consider the central extension A^ of A by the trivial module ℂ. This means A^ ≅ A ⊕ ℂ_c, and

[aλb]A^=[aλb]A+fλ(a,b)c,fora,b∈A,

where f_λ : A × A → ℂ[λ] is a bilinear map. The axioms (1)–(3) imply the following properties of the 2-cocycle f_λ(a, b):

fλ(a,b)=−f−λ−∂(b,a),(10)

fλ(∂a,b)=−λfλ(b,a)=−fλ(a,∂b),(11)

fλ+μ([aλ,b],c)=fλ(a, [bμ,c])=−fμ(b, [aλc]),(12)

for all a, b, c ∈ A. For any linear function f : A → ℂ, the map

ψf(a,b)=f([aλb]),fora,b∈A,(13)

defines a trivial 2-cocycle. Let a′λ(a,b)=aλ(a,b)+ψf(a,b). The equivalent 2-cocycles a^'_λ(a, b) and a_λ(a, b) define isomorphic extensions.

In the following we determine the central extension W^ of W by ℂ_c, i.e., W^ = W © ℂ_c, and the relations in (4) are replaced by

[LλL]=(∂+2λ)L+aλ(L,L)c, [LλM]=(∂+2λ)M+aλ(L,M)c, [MλM]=aλ(M,M)c,(14)

and the others can be obtained by skew-symmetry. Applying the Jacobi identity for (L, L, L), we have

(λ+2μ)aλ(L,L)−(μ+2λ)aμ(L,L)=(λ−μ)aλ+μ(L,L).

Write aλ(L,L)=∑i−0i=naiλi∈ℂ[λ] with a_n ≠ 0. Then, assuming n > 1 and equating the coefficients of λⁿ in (15), we get 2μa_n = (n — 1)μa_n and thus n = 3. Then

aλ(L,L)=a0+a1λ+a2λ2+a3λ3.(15)

Plugging this in (15) and comparing the similar terms, we obtain a₀ = a₂ = 0. Thus

aλ(L,L)=a1λ+a3λ3.(16)

To compute a_λ(L, M), we apply the Jacobi identity for (L, L, M) and obtain

(λ+2μ)aλ(L,M)−(μ+2λ)aμ(L,M)=(λ−μ)aλ+μ(L,M).

By doing similar discussions as those in the process of computing a_λ (L, M), we have

aλ(L,M)=b1λ+b3λ3, for some b1,b3∈ℂ.(17)

Finally, applying the Jacobi identity for (L, M, M) yields (λ — μ)a_λ+μ(M, M) = —(2λ + μ)a_μ(M, M), which implies

aλ(M,M)=0.(18)

From the discussions above, we obtain the following results.

Theorem 3.1

(i) For any a, b ∈ ℂ with (a, b) ≠ (0, 0), there exists a unique nontrivial universal central extension of the Lie conformal algebraWby ℂ_c, such that

[LλL]=(∂+2λ)L+aλ3c,[LλM]=(∂+2λ)M+bλ3c,[MλM]=0(19)

(ii) There exists a unique nontrivial universal central extension of W by ℂc ⊕ ℂc′, satisfying

[LλL]=(∂+2λ)L+aλ3c,[LλM]=(∂+2λ)M+λ3c′,[MλM]=0.(20)

Proof

(i) By (16)—(18), replacing L, M respectively by L — ½a₁c, M — ½b₁c and noticing that ∂c = 0, we can suppose a₁ = b₁ = 0. This shows (19). The universality of the extension follows from [12] and the fact that W is perfect, namely, [W_λW] = W[λ].

Remark 3.2

Theorem 3.1 (ii) implies that dim H²(W, ℂ) = 2.

4 Conformai module

Let us first recall the notion of a conformal module given in [5].

Definition 4.1

A (conformal) module V over a Lie conformal algebra A is a ℂ[∂]-module endowed with a bilinear mapA ⊗ V → V[∂], a ⊗ v ↦ a_λv satisfying the following axioms for a, b ∈ A, v ∈ V:

aλ(bμυ)−bμ(aλυ)=[aλb]λ+μυ,(∂a)λυ=−λaλυ, aλ(∂)=(∂+λ)aλυ.

An A-module V is called finite if it is finitely generated over ℂ[∂].

The vector space ℂ can be seen as a module (called the trivial module) over any conformal algebra A with both the action of ∂ and the action of A being zero. For a fixed nonzero complex constant a, there is a natural ℂ[∂]-module ℂ_a, which is the one-dimensional vector space ℂ such that ∂v = av for v ∈ ℂ_a. Then ℂ_a becomes an A-module, where A acts as zero.

For the Virasoro conformal algebra Vir, it is known from [3] that all the free nontrivial Vir-modules of rank one over ℂ[∂] are the following ones (Δ, α ∈ ℂ):

MΔ,α=ℂ[∂]υ, Lλυ=(∂+α+Δλ)υ.(21)

The module M_{Δ, α} is irreducible if and only if Δ ≠ 0. The module M_{0, α} contains a unique nontrivial submodule (∂ + α)M_{0, α} isomorphic to M_{1, α}. Moreover, the modules M_{Δ, α} with Δ ≠ 0 exhaust all finite irreducible nontrivial Vir-modules.

The following result describes the free nontrivial W-modules of rank one. Similar result for the more general W(a, b) Lie conformal algebra was given in [11]. We aim to consider it in details in the W(2, 2) case.

Proposition 4.2

All free nontrivial W-modules of rank one over ℂ[∂] are the following ones:

MΔ,α=ℂ[∂]υ, Lλυ=(∂+α+Δλ)υ, Mλυ=0 for some Δ, α ∈ℂ.

Proof

Suppose that L_λv = f(∂, λ)v, M_λv = g(∂, λ)v, where f(∂, λ), g(∂, λ) ∈ ℂ [λ, ∂]. By the result of Virmodules, we have

f(∂, λ)=∂+α+Δλ, for some α, Δ ∈ℂ.

On the other hand, it follows from M_λ(M_μu) = M_μ(M_λ v) that

g(∂, λ)g(∂+λ,μ)=g(∂, λ)g(∂+μ,λ).

This implies deg_λg(∂, λ) + deg_∂g(∂, λ) = deg_λg(∂, λ), where the notation deg_λg(∂, λ) stands for the highest degree of λ in g(∂, λ). Thus deg_∂g(∂, λ) = 0 and so g(∂, λ) = g(λ) for some g(λ) ∈ ℂ[λ]. Finally, [L_λM]_{λ + μ}v = (λ — μ)M_{λ + μ}v gives (λ — μ)g(λ + μ) = —μg(μ) which yields g(λ) = 0. This proves the result.

5 Cohomology

For completeness, we recall the following definition from [6]:

Definition 5.1

An n-cochain (n ∈ ℤ₊) of a Lie conformal algebra A with coefficients in an A-module V is a ℂ-linear map

γ:A⊗n→v[λ1,⋯,λn],a1⊗⋯⊗an↦γλ1,⋯,λn(a1,⋯,an)

satisfying the following conditions:

(i) γλ1,⋯,λn(a1,⋯,∂ai⋯,an)=−λiγλ1,⋯,λn(a1,⋯,an)(conformal antilinearity),

(ii) γ is skew-symmetric with respect to simultaneous permutations of a_i's and λ_i's (skew-symmetry).

As usual, let A^⊗0 = ℂ, so that a 0-cochain is an element of V. Denote by C~n(A,V) the set of all n-cochains. The differential d of an n-cochain γ is defined as follows:

(dγ)λ1,⋯,λn+1(a1,⋯,an+1 =∑i=1n+1(−1)i+1aiλiγλ1,⋯,λ^i,⋯,λn+1(a1,⋯,a^i,⋯,an+1) +∑i,j=1i<jN+1(−1)i+jγλi+λj,λ1,⋯λ^j,⋯,λn+1([aiλiaj],a1,⋯,a^i,⋯,a^j,⋯,an+1),(22)

where γ is linearly extended over the polynomials in λ_i. In particular, if γ ∈ V is a 0-cochain, then (dγ)_λ(a) = a_λγ.

It is known from [6] that the operator d preserves the space of cochains and d² = 0. Thus the cochains of a Lie conformal algebra A with coefficients in its module V form a complex, which is denoted by

C~∙(A,V)=⊕n∈Z+C~n(A,V),(23)

and called the basic complex. Moreover, define a (left) ℂ[∂]-module structure on C~∙(A,V) by

(∂γ)λ1,⋯,λn(a1,⋯,an)=(∂v+∑i=1nλi)γλ1,⋯,λn(a1,⋯,an),

where ∂_V denotes the action of ∂ on V. Then d∂ = ∂d and thus ∂C~∙(A,V)⊂C~∙(A,V) forms a subcomplex. The quotient complex

C∙(A,V)=C~∙(A,V)/∂C~∙(A,V)=⊕n∈Z+cn(A,V)

is called the reduced complex.

Definition 5.2

The basic cohomologyH˜•(A,V)of a Lie conformal algebra A with coefficients in an A-module V is the cohomology of the basic complexC˜•(A,V)and the (reduced) cohomologyH•(A,V)is the cohomology of the reduced complexC•(A,V) .

For a q-cochain γ ∈ C˜q(A,V) we call γ a q-cocycle if d(γ) = 0; a q-coboundary if there exists a (q — 1)-cochain ϕ∈C˜q−1(A,V) such that γ = d(ϕ). Two cochains γ₁ and γ₂ are called equivalent if γ₁ — γ₂ is a coboundary. Denote by D˜q(A,V) and B˜q(A,V) the spaces of q-cocycles and q-boundaries, respectively. By Definition 5.2,

H˜q(A,V)=D˜q(A,V)/B˜q(A,V)=(equivalent classes of q−cocycles).

The main results of this section are the following.

Theorem 5.3

For the Lie conformal algebra W, the following statements hold.

(i) For the trivial module ℂ,

dimH˜q(W,ℂ)={1 if q=0,4,5,62 if q=3 0 otherwise(24)

and

dimHq(W,ℂ)={1 if q=0,62 if q=2,4,5 3 if q=30 otherwise.(25)

(ii) If a ≠ 0, then dim H^q (W, ℂ_a) = 0, for q ≥ 0.

(iii) If a ≠ 0, then dim H^q (W, M_Δ,α) = 0, for q ≥ 0.

Proof

(i) For any γ∈C˜0(W,ℂ)=ℂ , we have (dγ)_λ(X) = X_λγ = 0 for X ∈ W. This means D˜0(W,ℂ)=ℂ and B˜0(W,ℂ)=0.. Thus H˜0(W,ℂ)=ℂ and H0(W,ℂ)=ℂ since ∂ℂ = 0.

Let γ∈C˜1(W,ℂ) be such that dγ∈∂C˜2(W,ℂ) , namely, there is ϕ∈C˜2(W,ℂ) such that

γλ1+λ2([Xλ1Y])=−(dγ)λ1,λ2(X,Y)=−(∂ϕ)λ1,λ2(X,Y)=−(λ1+λ2)ϕλ1,λ2(X,Y),(26)

for X, Y ∈ {L, M}. By (26) and (4),

(λ1−λ2)γλ1+λ2(X)=−(λ1+λ2)ϕλ1,λ2(L,X), X∈{L,M}.(27)

Letting λ = λ₁ + λ₂ in (27) gives

(λ−2λ2)γλ(X)=−λϕλ1,λ2(L,X), X∈{L,M},(28)

which implies that γ_λ(X) is divisible by λ. Define

γ′λ(X)=λ−1γλ(X), X∈{L,M}.

Clearly, γ′∈C˜1(W,ℂ) and γ= ∂γ′∈∂C˜1(W,ℂ). Thus Hⁱ (W, ℂ) = 0. If γ is a 1-cocycle (this means ϕ = 0), then (28) gives γ = 0. Hence, H˜1(W,ℂ)=0 .

Let ψ be a 2-cocycle. For X ∈ W, we have

0=(dψ)λ1,λ2,λ3(X,L,L)=−(λ1−λ2)ψλ1+λ1,λ3(X,L)+(λ1−λ3)ψλ1+λ1,λ3(X,L)−(λ2−λ3)ψλ1+λ1,λ3(L,X).

Letting λ₃ = 0 and λ₁ + λ₁ = λ gives (λ — 2λ₂)ψ_{λ_{1, 0}}(X, L) = λ ψ_{λ₁, λ₂}(X, L). Hence, ψ_{λ_{1, 0}} is divisible by λ. Define a 1-cochain f by

fλ1(L)=λ1−1ψλ1,λ2(L,L)|λ=0 fλ1(M)=λ1−1ψλ1,λ(M,L)|λ=0.(29)

Set γ = ψ + df, which is also a 2-cocycle. By (29),

γλ1,λ(L,L)|λ=0=ψλ1,λ(L,L)|λ=0−λ1fλ1(L)=0,(30)

γλ1,λ(M,L)|λ=0=ψλ1,λ(M,L)|λ=0−λ1fλ1(M)=0.(31)

By (30), we have

0=(dγ)λ1,λ2,λ(L,L,L)|λ=0=−γλ1+λ2,λ([Lλ1L],L)|λ=0+γλ1+λ2,λ([Lλ1L],L)|λ=0−γλ2+λ,λ1([Lλ2L],L)|λ=0=λ1γλ1,λ2(L,L)−λ2γλ2,λ1(L,L)=(λ1+λ2)γλ1,λ2(L,L).

Thus γ_λ₁,λ₂(L, L) = 0. Similarly, by (31),

0=(dγ)λ1,λ2,λ(L,M,L)|λ=0=(λ1+λ2)γλ1,λ2(L,M),

which gives γ_λ₁,λ₂(L, M)=0 and so γ_λ₁,λ₂(M, L)=0 . Finally,

0=(dγ)λ1,λ2,λ(L,M,M)|λ=0=−(λ1−λ2)γλ1,λ2,0(M,M)+λ1γλ1,λ2(M,M).(32)

Setting λ₁ = 0 in (32) gives γλ_{2, 0}(M, M) = 0 and thus γλ₁λ₂(M, M) = 0 . This shows γ = 0. Hence H˜2(W,ℂ)=0 . According to Theorem 3.1 (ii), dim H²(W, ℂ) = 2.

To determine higher dimensional cohomologies (for q ≥ 3), we define an operator τ:C˜q(W,ℂ)→C˜q−1(W,ℂ) by

(τγ)λ1,⋯,λq−1(X1,⋯,Xq−1)=(−1)q−1∂∂λγλ1,⋯,λq−1,λ(X1,⋯,Xq−1,L)|λ=0,(33)

for X₁, ⋯, X_q–1 ∈ {L, M}. By (22), (33) and skew-symmetry of γ,

((dτ+τd)γ)λ1,⋯,λq(X1,⋯Xq)=(−1)q∂∂λ∑i=1q(−1)i+q+1γλ1+λ,λ1,⋯,λ^i⋯,λq([XiλiL],X1,⋯X^i,⋯,Xq)|λ=0=∂∂λi∑i=1qγλ1,⋯,λi−1,λi+λ,λi+1,⋯λq(X1,⋯,Xi−1[XiλiL],X1+1,⋯Xq)|λ=0.(34)

By the fact that [X_{i λ_i}, L] = (∂ + 2λ_i)X_i and conformal antilinearity of γ, [X_{i λ_i}L] can be replaced by (λ_i, — λ)X_i, in (34). Thus, equality (34) can be rewritten as

((dτ+τd)γ)λ1,⋯,λq(X1,⋯Xq)=∂∂λ∑i=1q(λi−λ)γλ1,⋯,λi−1⋯,λi+λ,λi+1⋯,λq(X1,⋯,Xi−1,Xi,Xi+1⋯,Xq)|λ=0=(degγ−q)γλ1,⋯,λq(X1,⋯,Xq),(35)

where deg γ is the total degree of γ in λ₁, ⋯, λ_q. As it was explained in [6], only those homogeneous cochains, whose degree as a polynomial is equal to their degree as a cochain, contribute to the cohomology of C˜•(W,ℂ). Without loss of generality, we always assume that the first k variables are L and the last q — k variables are M in γ_{λ₁, ⋯ λ_q} (X₁ ⋯ X_q), so that γ_{λ₁, ⋯ λ_q} (X₁ ⋯ X_q) as a polynomial in λ₁, ⋯ λ_q is skew-symmetric in λ₁, ⋯ λ_k and also skew-symmetric in λ_{k + 1}, ⋯ λ_q. Therefore, it is divisible by

Π1≤i<j≤k(λ1−λj)×Πk+1≤i<j≤q(λi−λj),

whose polynomial degree is k(k — 1)/2 + (q — k)(q — k — 1)/2. Consider the quadratic inequality k(k — 1)/2 + (q — k)(q — k — 1)/2 ≤ q, whose discriminant is —4k² + 12k + 9. Since —4k² + 12k + 9 ≥ 0 has k = 0,1,2 and 3 as the only integral solutions, we have

q={0,1,2,3, if k=0,1,2,3,4, if k=1,2,3,4,5, if k=2,3,4,5,6, if k=3.(36)

Thus H˜q(W,ℂ)=0 for q ≥ 7. It remains to compute H˜q(W,ℂ)=0 for q = 3,4,5,6.

For q = 3, we need to consider four cases for k, i.e., k = 0,1,2,3. Let γ∈D~3(W,C) be a 3-cocycle. A direct computation shows that

0=(dγ)λ1,λ2,λ3,λ(M,M,M,L)|λ=0=−(λ1+λ2+λ3)γλ1,λ2,λ3(M,M,M).

This gives γ_{λ₁λ₂λ₃} (M, M, M) = 0. In the case of k = 1, we have

0=(dγ)λ1,λ2,λ3,λ(L,M,M,L)|λ=0 = (λ1−λ2)γ0,λ3,λ1,+λ2(L,M,M)−(λ1−λ3)γ0,λ2,λ1,+λ3(L,M,M) −(λ1+λ2+λ3)γλ1,λ2,λ3(L,M,M).(37)

Note that γ_{λ₁λ₂λ₃}(L, M, M) is a homogeneous polynomial of degree 3 and skew-symmetric in λ₂ and λ₃. Thus it is divisible by λ₂ — λ₃. We can suppose that

γλ1,λ2,λ3(L,M,M)=(λ2−λ3)(a1λ12+a2(λ22+λ32)+a3λ2λ3+a4λ1(λ2+λ3)),(38)

where a₂, a₂, a₃, a₄ ∈ ℂ. Plugging (38) into (37) gives a₄ = 0, a₃ = 2a₂, a₁ = —a₂. Therefore,

γλ1,λ2,λ3(L,M,M)=a2ϕ1= where ϕ1=(λ2−λ3)(λ1+λ2+λ3)(−λ1+λ2+λ3).(39)

Note that ϕ₁ is a coboundary of γ¯λ1,λ2(M,M)=λ22−λ12 . In fact,

(dγ¯)λ1,λ2,λ3(L,M,M)=−(λ1−λ2)γλ1+λ2,λ3(M,M)+(λ1+λ3)γλ1+λ3,λ2(M,M)=−(λ1−λ2)(λ32−(λ1+λ2)2)+(λ1−λ3)(λ22−(λ1+λ3)2)=ϕ1.

Similarly, suppose that

γλ1,λ2,λ3(L,M,M)=(λ1−λ2)(b1(λ12+λ22)+b2λ32+b3(λ1+λ2)λ3+b4λ1λ2),(40)

where b₁, b₂, b₃, b₄ ϕ ℂ. Substituting (40) into the following equality

0=(dγ)λ1,λ2,λ3,λ(L,L,M,L)|λ=0 =(λ1−λ2)γλ1+λ2,0,λ3(L,L,M)+(λ1−λ3)γλ2,0,λ1+λ3(L,L,M) −(λ2−λ3)γλ1,0,λ2+λ3(L,L,M)−(λ1+λ2+λ3)γλ1,λ2,λ3(L,L,M)

gives b₄ = b₁ + b₂. Hence,

γλ1,λ2,λ3(L,L,M)=(λ1−λ2)(b1(λ12+λ22)+b2λ32+(b1+b2)λ1λ2+b3(λ1+λ2)λ3).(41)

On the other hand, there is a 2-cochain γ¯λ1,λ2(L,M)=b1λ12+b2λ1λ2 such that

(dγ¯)λ1,λ2,λ3(L,L,M)+γλ1,λ2,λ3(L,L,M)=−(b1+b2−b3)(λ1−λ2)(λ1+λ2)λ3.(42)

Thus γ_{λ₁,λ₂,λ₃} (L, L, M) in (41) is equivalent to a constant factor of χ:=χλ1,λ2,λ3(L,L,M)=(λ1−λ2)(λ1+λ2)λ3, which is not a coboundary. By [6, Theorem 7.1], Λ3:=γλ1,λ2,λ3(L,L,L)=(λ1−λ2)(λ1−λ3)(λ2−λ3) (up to a constant factor) is a 3-cocycle, but not a coboundary. Therefore, dimH˜3(W,ℂ)=2. Specifically, H˜3(W,ℂ)=ℂχ⊕ℂΛ3

For q = 4, three cases (i.e., k = 1,2,3) should be taken into account. Let γ∈D˜2(W,ℂ) be a 4-cocycle. By using the method of undetermined coefficients and doing similar calculations to the case when q = 3, we obtain

γλ1,λ2,λ3λ4(L,M,M,M)=c(λ2−λ3)(λ3−λ4)(λ2−λ4)(λ2+λ3+λ4),(43)

γλ1,λ2,λ3λ4(L,M,M,M)=(λ1−λ2)(λ3−λ4)(c1(λ12−λ22) + c2(λ3+λ4)2 +(c1+c2)λ1λ2+c3(λ1+λ2)(λ3+λ4)),(44)

γλ1,λ2,λ3λ4(L,M,M,M)=(λ1−λ2)(λ2−λ3)(λ1−λ3)(e1(λ1+λ2+λ3)+e2λ4),(45)

where c, c₁, c₂, c₃, e₁, e₂ ∈ ℂ. And there exist three 3-cochains of degree 3

γ˜λ1,λ2,λ3(M,M,M)=(λ1−λ2)(λ1−λ3)(λ2−λ3),(46)

γ˜λ1,λ2,λ3(L,M,M)=(λ2−λ3)(c1λ12+c2λ1)(λ2+λ3)),(47)

γ˜λ1,λ2,λ3(L,M,M)=(λ1−λ2)(λ12+c2λ22),(48)

such that

γλ1,λ2,λ3,λ4(L,M,M,M)−c(dγ˜)λ1,λ2,λ3,λ4(L,M,M,M)=0,(49)

γλ1,λ2,λ3,λ4(L,L,M,M)+(dγ˜)λ1,λ2,λ3,λ4(L,L,M,M)=(c3−c1−c2)ψ1(50)

γλ1,λ2,λ3,λ4(L,L,L,M)+e2(dγ˜)λ1,λ2,λ3,λ4(L,L,L,M)=(e3−e1)ψ2,(51)

where

ψ1:=ψ1λ1,λ2,λ3,λ4(L,L,M,M)=(λ1−λ2)(λ1+λ2)(λ3−λ4)(λ3−λ4),(52)

ψ2:=ψ2λ1,λ2,λ3,λ4(L,L,L,M)=(λ1−λ2)(λ2−λ3)(λ1−λ3)(λ1+λ2+λ3).(53)

Moreover, 4ψ1=−(dψ¯)λ1,λ2,λ3,λ4(L,L,M,M) with ψ¯λ1,λ2,λ3,λ4(L,M,M)=(λ2−λ3)(3λ12−(λ22+λ32)) . This, together with (49)-(53), gives H˜4(W,ℂ)=ℂψ2 .

For q = 5, we need to consider k = 2, 3. Let γ∈D˜5(W,ℂ) be a 5-cocycle. We obtain

γλ1,λ2,λ3,λ4,λ5(L,L,M,M,M)=(λ1−λ2)(λ3−λ4)(λ3−λ5)(λ4−λ5)×(a¯(λ1+λ2)+a¯(λ3+λ4+λ5)),(54)

γλ1,λ2,λ3,λ4,λ5(L,L,L,M,M)=(λ1−λ2)(λ1−λ3)(λ2−λ3)(λ4−λ5)×(b¯1(λ1+λ2+λ3)+b¯2(λ4+λ5)),(55)

where a¯1,a¯2,b¯,b¯2∈ℂ . On the other hand, there exist two 4-cochains of degree 4

γ¯λ1,λ2,λ3,λ4(L,M,M,M)=λ1(λ2−λ3)(λ3−λ4)(λ2−λ4),

γ¯λ1,λ2,λ3,λ4(L,L,M,M)=(λ1−λ2)(λ3−λ4)(λ12+λ22),

such that

γλ1,λ2,λ3,λ4,λ5(L,L,M,M,M)+a1(dγ¯)λ1,λ2,λ3,λ4,λ5(L,L,M,M,M)=(a2−a1)φ1,(56)

γλ1,λ2,λ3,λ4,λ5(L,L,L,M,M)+b1(dγ¯)λ1,λ2,λ3,λ4,λ5(L,L,L,M,M)=(b2−b1)φ1,(57)

where

φ1:=φ1λ1,λ2,λ3,λ4,λ5(L,L,M,M,M)=(λ1−λ2)(λ3−λ4)(λ3−λ5)(λ4−λ5)(λ3+λ4+λ5),(58)

φ2:=φ2λ1,λ2,λ3,λ4,λ5(L,L,M,M,M)=(λ1−λ2)(λ1−λ3)(λ2−λ2)(λ4−λ5)(λ4+λ5).(59)

Furthermore, there exists another one 4-cochains of degree 4

φ¯λ1,λ2,λ3,λ4(L,L,M,M)=(λ1−λ2)(λ3−λ4)(λ1λ2−λ3λ4),

such that 2φ2=(dφ¯)λ1,λ2,λ3,λ4,λ5(L,L,L,M,M), namely, ‘₂ is a coboundary. By (56) and (57), dimH˜5(W,ℂ)=1 , and H˜5(W,ℂ)=ℂφ1 .

For q = 6, one only needs to consider the case when k = 3. One can check that

Λ:=γλ1,λ2,λ3,λ4,λ5,λ6(L,L,L,M,M)=(λ1−λ2)(λ2−λ3)(λ1−λ3)(λ4−λ5)(λ4−λ6)(λ5−λ6)(60)

is a 6-cocycle. It is not a coboundary. Because it can be the coboundary of a 5-cochain of degree 5, which must be a constant factor of γ_{λ₁, λ₂, λ₃, λ₄, λ₅} (L, L, M, M, M) in (54), whose coboundary is zero. Therefore, dimH˜6(W,ℂ)=1 and H˜6(W,ℂ)=ℂΛ . This proves (24).

According to [6, Proposition 2.1], the map γ↦∂γ gives an isomorphism H˜q(W,ℂ)≅Hq(∂C˜•) for q ≥ 1. Therefore

Hq(∂C˜•)={ℂ(∂χ)⊕ℂ(Λ3)if q=3,ℂ(∂ψ2)if q=4,ℂ(∂φ1)if q=5,ℂ(∂Λ)if q=6,0otherwise.(61)

It remains to compute H^•(W, ℂ) . This is based on the short exact sequence of complexes

0→∂C˜•→ιC˜•→πC•→0(62)

where ı and π are the embedding and the natural projection, respectively. The exact sequence (62) gives the following long exact sequence of cohomology groups (cf. [6]):

⋯→Hq(∂C˜•)→ιqH˜q(W, ℂ)→πqHq(W, ℂ)→ωq →Hq+1(∂C˜•)→ιq+1H˜q+1(W, ℂ)→πq+1Hq+1(W, ℂ)→⋯(63)

where ı_q, π_q are induced bı, π respectively and w_q is the q—th connecting homomorphism. Given ∂γ∈Hq(∂C˜•) with a nonzero element γ∈H˜q(W,ℂ) , then ιq(∂γ)=∂γ∈H˜q(W,ℂ) . Since deg (∂γ) = deg (γ) + 1 = q + 1, we have ∂γ=0∈H˜q(W,ℂ) . Then the image of ι₃ is zero for any q ∈ ℤ+ . Because ker(π_q) = im(ι_q) = {0} and im(ωq)=ker(ιq+1)=Hq+1(∂C˜•) , we obtain the following short exact sequence

0→H˜q(W, ℂ)→πqHq(W, ℂ)→ωqHq+1(∂C˜•)→0.(64)

Therefore,

dimHq(W,ℂ)=dimH˜q(W,ℂ)+dimHq+1(∂C˜•), for all q≥0.(65)

Then (25) follows from (65). Moreover, we can give a basis for H^q (W, ℂ). Indeed, any basis of H^q (W, ℂ) can be obtained by combining the images of a basis of H˜q(W,ℂ) with the pre-images of a basis of H˜q+1(W,ℂ) . For a nonzero ∂φ∈Hq+1(∂C˜•) with φ being a (q + 1)-cocycle, (35) gives

d(τ(∂φ))=(dτ+τd)(∂φ)=(deg(∂φ)−(q−1)(∂φ)=((q+2)−(q+1))(∂φ)=∂φ.(66)

Thus the pre-image of ∂φ under the connecting homomorphism ω_p is ω_q^—1 (∂φ) = r(∂φ).

Finally, we finish our proof by giving a basis of H^q (W, ℂ) for q = 2,..., 6. For q = 2, we have known that H˜2(W,ℂ)=0 and H3(∂C˜•)=ℂ(∂χ)⊕ℂ(∂Λ3). By (33) and (42),

χ¯:=(τ(∂χ))λ1,λ2(L,M)=(−1)2∂∂λ(∂χ)λ1,λ2λ(L,M,L)|λ=0=−∂∂λ(λ1+λ2+λ)(λ12−λ2)λ2|λ=0=−λ12λ2,Λ¯3:=(τ(∂Λ3))λ1,λ2(L,L)=(−1)2∂∂λ(∂Λ3)λ1,λ2,λ(L,L,L)|λ=0=∂∂λ(λ1+λ2+λ)(λ1−λ2)(λ2−λ)(λ1−λ)|λ=0=−λ13+λ23.

This gives H2(W,ℂ)=ℂχ¯⊕ℂΛ¯3 . For q = 3, by (33), (53) and (61),

ψ¯:=(τ(∂ψ2))λ1,λ2,λ3(L,L,M)=(−1)3∂∂λ(∂ψ2)λ1,λ2,λ3,λ(L,L,M,L)|λ=0=∂∂λ(λ1+λ2+λ3+λ)(λ1−λ2)(λ1−λ)(λ2−λ)(λ1+λ2+λ)|λ=0=−λ14−λ13λ3+λ23(λ2+λ3).

Hence, H3(W,ℂ)=H˜3(W,ℂ)⊕ℂψ¯=ℂχ⊕ℂΛ3⊕ℂψ¯ . By (33), (58), (60) and (61),

φ¯ : = (τ(∂φ1))λ1,λ2,λ3,λ4(L,M,M,M) = (−1)4∂∂λ(∂φ1)λ1,λ2,λ3,λ4,λ(L,M,M,M,L)|λ=0 = −∂∂λ(λ1+λ2+λ3+λ4+λ)(λ1−λ)(λ3−λ2)(λ3−λ4)(λ4−λ2)(λ2+λ3+λ4)|λ=0 =(λ2−λ3)(λ2−λ4)(λ3−λ4)(λ2+λ3+λ4)2,Λ¯:=(τ(∂Λ))λ1,λ2,λ3,λ4,λ5(L,L,M,M,M) = (−1)5∂∂λ(∂Λ)λ1,λ2,λ3,λ4,λ5,λ(L,L,M,M,M,L)|λ=0 =∂∂λ(∑i=05λi+λ)(λ1−λ2)(λ1−λ)(λ2−λ)(λ4−λ5)(λ4−λ3)(λ5−λ3)|λ=0 =−(λ1−λ2)(λ3−λ4)(λ3−λ5)(λ4−λ5)(λ12+λ1λ2+λ22+(λ1+λ2)(λ3+λ4+λ5)) ≡ λ1λ2(λ1−λ2)(λ3−λ4)(λ3−λ5)(λ4−λ5)(Mod ∂C~5(W,C)).

Therefore, H4(W,ℂ)=ℂψ2⊕ℂφ¯,H5(W,ℂ)=ℂφ1⊕ℂΛ¯ and H6(W,ℂ)=ℂΛ . Thus (i) is proved.

(ii) Define an operator τ2:C˜q(W,ℂa)→C˜q−1(W,ℂa) by

(τ2γ)λ1,⋯,λq−1(X1,⋯Xq−1)=(−1)q−1γλ1,⋯,λq−1,λ(X1,⋯,Xq−1,L)|λ=0,(67)

for X1,⋯,Xq−1∈{L,M} . By the fact that ∂C˜q(W,ℂa)=(a+∑i=1qλ1)C˜q(W,ℂa) , we have

((dτ2+τ2d)γ)λ1,⋯,λq(X1,⋯,Xq)=(∑i=1qλi)γλ1,⋯,λq(X1,⋯,Xq)≡−aγλ1,⋯λq(X1,⋯,Xq)(mod∂C~q(W,Ca)).(68)

Let γ∈C˜q(W,ℂa) be a q-cochain such that dγ∈∂C˜q+1(W,ℂa) namely, there is a (q + 1)-cochain ϕ such that dγ=(a+∑i=1q+1λi)ϕ . By (67), τ2dγ=(a+∑i=1qλi)τ2ϕ∈∂C˜q(W,ℂa) . It follows from (68) that γ≡−d(a−1τ2γ) is a reduced coboundary. This proves (ii).

(iii) In this case, ∂C˜q(W,MΔ,α)=(∂+∑i=1qλi)C˜q(W,MΔ,α) . As in the proof of (ii), we define an operator τ3:Cq(W,MΔ,α)→Cq−1(W,MΔ,α) by

(τ3γ)λ1,⋯,λq−1(X1,⋯Xq−1)=(−1)q−1γλ1,⋯,λq−1,λ(X1,⋯,Xq−1,L)|λ=0,

for X1,⋯Xq−1∈{L,M} . Then

((dτ3+τ3d)γ)λ1,⋯,λq(X1,⋯Xq)=Lλγλ1,⋯,λq(X1,⋯,Xq)|λ=0+(∑i=1qλi)γλ1,⋯,λq(X1,⋯,Xq)=(∂+α+∑i=1qλi)γλ1,⋯λq(X1,⋯,Xq)≡αγλ1,⋯λq(X1,⋯,Xq)(mod∂C~q(W,MΔ,α)).(69)

If γ is a reduced q-cocycle, it follows from (69) that γ≡d(α−1τ3γ) is a reduced coboundary, since α ≠ 0. Thus Hq(W,MΔ,α)=0 for all q ≥ 0.

This completes the proof of Theorem 5.3.

Remark 5.4

Denote by Lie(W)— the annihilation Lie algebra of W. Note that Lie(W)— is isomorphic to the subalgebra spanned by {L_n, M_n| — 1 ≤ n ∈ ℤ of the centerless W-algebra W (2, 2). By [6, Corollary 6.1], H˜q(W,ℂ)≅Hq(Lie(W)−,ℂ) . Thus we have determined the cohomology of Lie(W)_—with trivial coefficients.

Acknowledgement

The authors would like to thank the referees for helpful suggestions. This work was supported by National Natural Science Foundation grants of China (11301109, 11526125) and the Research Fund for the Doctoral Program of Higher Education (20132302120042).

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Received: 2016-5-5

Accepted: 2016-8-1

Published Online: 2016-9-15

Published in Print: 2016-1-1

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