Let $\mathcal{C}$ denote the ring ℂ[∂] of polynomials in the indeterminate ∂.

For the Lie conformal algebra $\mathcal{W}$, we have the following result.

An extension of a Lie conformal algebra $\mathcal{A}$ by an abelian Lie conformal algebra a is a short exact sequence of Lie conformal algebras $$0\to a\to \hat{\mathcal{A}}\to \mathcal{A}\to 0.$$ In this case $\hat{\mathcal{A}}$ is also called an extension of $\mathcal{A}$ by $\mathsf{a}$. The extension is said to be central if $$a\subseteq Z(\hat{\mathcal{A}})=\{x\in \hat{\mathcal{A}}|[x_{\lambda }y{]}_{\hat{\mathcal{A}}}=0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}\in \hat{\mathcal{A}}\},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\partial }a=0\text{.}$$ Consider the central extension $\hat{\mathcal{A}}$ of $\mathcal{A}$ by the trivial module ℂ. This means $\hat{\mathcal{A}}$ ≅ $\mathcal{A}$ ⊕ ℂ_$\mathsf{c}$, and $$[a_{\lambda }b{]}_{\hat{\mathcal{A}}}=[a_{\lambda }b{]}_{\mathcal{A}}+f_{\lambda }(a,b)c,\mathrm{f}\mathrm{o}\mathrm{r}a,b\in \mathcal{A},$$ where f_λ : $\mathcal{A}$ × $\mathcal{A}$ → ℂ[λ] is a bilinear map. The axioms (1)–(3) imply the following properties of the 2-cocycle f_λ(a, b): $$f_{\text{λ}}(a,b)=-f_{-\text{λ}-\partial }(b,a),$$(10) $$f_{\text{λ}}(\partial a,b)=-\text{λ}{f}_{\text{λ}}(b,a)=-f_{\text{λ}}(a,\partial b),$$(11) $$f_{\text{λ}+\mu }([a_{\text{λ}},b],c)=f_{\text{λ}}(a,\text{\hspace{0.17em}[}{b}_{\mu },c])=-f_{\mu }(b,\text{\hspace{0.17em}[}{a}_{\text{λ}}c]),$$(12) for all a, b, c ∈ $\mathcal{A}$. For any linear function f : $\mathcal{A}$ → ℂ, the map $${\psi }_f(a,b)=f([a_{\lambda }b]),\mathrm{f}\mathrm{o}\mathrm{r}a,b\in \mathcal{A}\text{,}$$(13) defines a trivial 2-cocycle. Let ${a^{\prime }}_{\text{λ}}(a,b)=a_{\text{λ}}(a,b)+{\psi }_f(a,b)$. The equivalent 2-cocycles a^'_λ(a, b) and a_λ(a, b) define isomorphic extensions.

In the following we determine the central extension $\hat{\mathcal{W}}$ of $\mathcal{W}$ by ℂ_c, i.e., $\hat{\mathcal{W}}$ = $\mathcal{W}$ © ℂ_$\mathsf{c}$, and the relations in (4) are replaced by $$[L_{\text{λ}}L]=(\partial +2\text{λ})L+a_{\text{λ}}(L,L)c,\text{\hspace{0.17em}[}{L}_{\text{λ}}M]=(\partial +2\text{λ})M+a_{\text{λ}}(L,M)c,\text{\hspace{0.17em}[}{M}_{\text{λ}}M]=a_{\text{λ}}(M,M)c,$$(14) and the others can be obtained by skew-symmetry. Applying the Jacobi identity for (L, L, L), we have $$(\text{λ}+2\mu )a_{\text{λ}}(L,L)-(\mu +2\text{λ})a_{\mu }(L,L)=(\text{λ}-\mu )a_{\text{λ}+\mu }(L,L).$$ Write $a_{\text{λ}}(L,L)={\displaystyle {\sum }_{i-0}^{i=n}{a}_i{\text{λ}}^i}\in ℂ[\text{λ}]$ 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 $$a_{\text{λ}}(L,L)=a_0+a_1\text{λ}+a_2{\text{λ}}^2+a_3{\text{λ}}^3.$$(15) Plugging this in (15) and comparing the similar terms, we obtain a₀ = a₂ = 0. Thus $$a_{\text{λ}}(L,L)=a_1\text{λ}+a_3{\text{λ}}^3.$$(16) To compute a_λ(L, M), we apply the Jacobi identity for (L, L, M) and obtain $$(\text{λ}+2\mu )a_{\text{λ}}(L,M)-(\mu +2\text{λ})a_{\mu }(L,M)=(\text{λ}-\mu )a_{\text{λ}+\mu }(L,M).$$ By doing similar discussions as those in the process of computing a_λ (L, M), we have $$a_{\text{λ}}(L,M)=b_1\text{λ}+b_3{\text{λ}}^3,\text{\hspace{0.17em}for\hspace{0.17em}some\hspace{0.17em}}{b}_1,b_3\in ℂ.$$(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 $$a_{\text{λ}}(M,M)=0.$$(18) From the discussions above, we obtain the following results.

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

The vector space ℂ can be seen as a module (called the trivial module) over any conformal algebra $\mathcal{A}$ with both the action of ∂ and the action of $\mathcal{A}$ 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 $\mathcal{A}$-module, where $\mathcal{A}$ 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): $$M_{\Delta ,\alpha }=ℂ[\partial ]\upsilon ,L_{\text{λ}}\upsilon =(\partial +\alpha +\Delta \text{λ})\upsilon .$$(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 $\mathcal{W}$-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.

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

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 $\mathcal{A}$ with coefficients in its module V form a complex, which is denoted by $${\tilde{C}}^{\bullet }(\mathcal{A},V)=\underset{n\in {\mathbb{Z}}_{+}}{\oplus }{\tilde{C}}^n(\mathcal{A},V)\text{,}$$(23) and called the basic complex. Moreover, define a (left) ℂ[∂]-module structure on ${\tilde{C}}^{\bullet }(\mathcal{A},V)$ by $${(\partial \gamma )}_{{\text{λ}}_1}{}_{,\cdots ,{\text{λ}}_n}(a_1,\cdots ,a_n)=({\partial }_v+{\displaystyle \sum _{i=1}^n{\text{λ}}_i}){\gamma }_{{\text{λ}}_1,\cdots ,{\text{λ}}_n}(a_1,\cdots ,a_n),$$ where ∂_V denotes the action of ∂ on V. Then d∂ = ∂d and thus $\mathrm{\partial }{\tilde{C}}^{\bullet }(\mathcal{A},V)\subset {\tilde{C}}^{\bullet }(\mathcal{A},V)$ forms a subcomplex. The quotient complex $$C^{\bullet }(\mathcal{A},V)={\tilde{C}}^{\bullet }(\mathcal{A},V)/\mathrm{\partial }{\tilde{C}}^{\bullet }(\mathcal{A},V)=\underset{n\in {\mathbb{Z}}_{+}}{\oplus }{c}^n(\mathcal{A},V)$$ is called the reduced complex.

For a q-cochain $\gamma \in {\tilde{C}}^q(\mathcal{A},V)$ we call γ a q-cocycle if d(γ) = 0; a q-coboundary if there exists a (q — 1)-cochain $\varphi \in {\tilde{C}}^{q-1}(\mathcal{A},V)$ such that γ = d(ϕ). Two cochains γ₁ and γ₂ are called equivalent if γ₁ — γ₂ is a coboundary. Denote by ${\tilde{D}}^q(\mathcal{A},V)$ and ${\tilde{B}}^q(\mathcal{A},V)$ the spaces of q-cocycles and q-boundaries, respectively. By Definition 5.2, $${\tilde{\text{H}}}^q(\mathcal{A},V)={\tilde{D}}^q(\mathcal{A},V)/{\tilde{B}}^q(\mathcal{A},V)=(\text{equivalent\hspace{0.17em}classes\hspace{0.17em}of\hspace{0.17em}}q-\text{cocycles})\text{.}$$ The main results of this section are the following.

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).