Let $\mathcal{C}$ 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 → $\mathcal{C}$[λ] ⊗_{$\mathcal{C}$} W, denoted by ϕ_λ V → W, is called a conformal linear map, if $$\varphi (\partial \upsilon )=(\partial +\text{λ})(\varphi \upsilon ),for\upsilon \in V.$$ Denote by Chom(V, W) the space of conformal linear maps from $\mathcal{C}$-modules V to W. It can be made into an $\mathcal{C}$-module via $$(\partial \varphi {)}_{\text{λ}}\upsilon =-\text{λ}{\varphi }_{\text{λ}}\upsilon ,\text{\hspace{0.17em}for\hspace{0.17em}}\upsilon \in V\text{.}$$

Definition 2.2: Let $\mathcal{A}$ be a Lie conformal algebra. A conformal linear map d_λ : $\mathcal{A}$ → $\mathcal{A}$ is called a conformal derivation if $$d_{\lambda }([a_{\mu }b])=[(d_{\lambda }a{)}_{\lambda +\mu }b]+[a_{\mu }(d_{\lambda }b)],foralla,b\in \mathcal{A}\text{.}$$ Denote by CDer($\mathcal{A}$) the space of all conformal derivations of $\mathcal{A}$. For any a ∈ $\mathcal{A}$, one can define a linear map (ad a)_λ : $\mathcal{A}$ → $\mathcal{A}$ by (ada)_λ b = [a_λ] for all b ∈ $\mathcal{A}$. It is easy to check that (ada)_λ is a conformal derivation of $\mathcal{A}$. Any conformal derivation of this kind is called an inner derivation. The space of all inner derivations is denoted by CInn($\mathcal{A}$).

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

Proposition 2.3: Every conformal derivation of $\mathcal{W}$ is inner, namely, CDer($\mathcal{W}$) = CInn($\mathcal{W}$).

Remark 2.4: Proposition 2.3 is equivalent to H¹ ($\mathcal{W}$, $\mathcal{W}$) = 0.

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.

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 algebra $\mathcal{W}$ by ℂ_$\mathsf{c}$, such that $$[L_{\text{λ}}L]=(\partial +2\text{λ})L+a{\text{λ}}^{3}c,[L_{\text{λ}}M]=(\partial +2\text{λ})M+b{\text{λ}}^{3}c,[M_{\text{λ}}M]=0$$(19)(ii) There exists a unique nontrivial universal central extension of $\mathcal{W}$ by ℂ$\mathsf{c}$ ⊕ ℂ$\mathsf{c}$′, satisfying $$[L_{\text{λ}}L]=(\partial +2\text{λ})L+a{\text{λ}}^{3}c,[L_{\text{λ}}M]=(\partial +2\text{λ})M+{\text{λ}}^3{c}^{\prime },[M_{\text{λ}}M]=0\text{.}$$(20)

Remark 3.2: Theorem 3.1 (ii) implies that dim H²($\mathcal{W}$, ℂ) = 2.

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 $\mathcal{A}$ is a ℂ[∂]-module endowed with a bilinear map $\mathcal{A}$ ⊗ V → V[∂], a ⊗ v ↦ a_λ v satisfying the following axioms for a, b ∈ $\mathcal{A}$, v ∈ V: $$\begin{array}{l}{a}_{\text{λ}}(b_{\mu }\upsilon )-b_{\mu }(a_{\text{λ}}\upsilon )=[a_{\text{λ}}b{]}_{\text{λ}+\mu }\upsilon ,\\ (\partial a{)}_{\text{λ}}\upsilon =-\text{λ}{a}_{\text{λ}}\upsilon ,a_{\text{λ}}(\partial )=(\partial +\text{λ})a_{\text{λ}}\upsilon .\end{array}$$ An $\mathcal{A}$-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 $\mathcal{A}$ 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.

Proposition 4.2: All free nontrivial $\mathcal{W}$-modules of rank one over ℂ[∂] are the following ones: $$M_{\Delta ,\alpha }=ℂ[\partial ]\upsilon ,L_{\text{λ}}\upsilon =(\partial +\alpha +\Delta \text{λ})\upsilon ,M_{\text{λ}}\upsilon =0forsom\text{e\hspace{0.17em}}\Delta \text{,\hspace{0.17em}}\alpha \in ℂ\text{.}$$

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

Definition 5.1: An n-cochain (n ∈ ℤ₊) of a Lie conformal algebra $\mathcal{A}$ with coefficients in an $\mathcal{A}$-module V is a ℂ-linear map $$\gamma :{\mathcal{A}}^{\otimes n}\to v[{\lambda }_1,\cdots ,{\lambda }_n],a_1\otimes \cdots \otimes a_n\mapsto {\gamma }_{{\lambda }_1,\cdots ,{\lambda }_n}(a_1,\cdots ,a_n)$$ satisfying the following conditions:(i) ${\gamma }_{{\text{λ}}_1,\cdots ,{\text{λ}}_n}(a_1,\cdots ,\partial a_i\cdots ,a_n)=-{\text{λ}}_i{\gamma }_{{\text{λ}}_1},\cdots ,{\text{λ}}_n(a_1,\cdots ,a_n)$ (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 $\mathcal{A}$^⊗0 = ℂ, so that a 0-cochain is an element of V. Denote by ${\tilde{C}}^n(\mathcal{A},V)$ the set of all n-cochains. The differential d of an n-cochain γ is defined as follows: $$\begin{array}{l}{(}_d(a_1,\cdots ,a_{n+1}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=}{\displaystyle \sum _{i=1}^{n+1}{\left(-1\right)}^{i+1}{a}_{i_{{\text{λ}}_i}}{\gamma }_{{\text{λ}}_1,\cdots ,{\widehat{\text{λ}}}_i,\cdots ,{\text{λ}}_{n+1}}(a_{1,}\cdots ,{\widehat{a}}_i,\cdots ,a_{n+1})}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+}{\displaystyle \sum _{\underset{i<j}{i,j=1}}^{N+1}{(-1)}^{i+j}}{\gamma }_{{\text{λ}}_i+{\text{λ}}_j,{\text{λ}}_{1,}\cdots {\widehat{\text{λ}}}_j,\cdots ,{\text{λ}}_{n+1}}([a_{i_{{\text{λ}}_i}}{a}_j],a_1,\cdots ,{\widehat{a}}_i,\cdots ,{\widehat{a}}_j,\cdots ,a_{n+1}),\end{array}$$(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 $\mathcal{A}$ 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.

Definition 5.2: The basic cohomology ${\tilde{\text{H}}}^{•}(\mathcal{A},V)$ of a Lie conformal algebra $\mathcal{A}$ with coefficients in an $\mathcal{A}$-module V is the cohomology of the basic complex ${\tilde{\text{C}}}^{•}(\mathcal{A},V)$ and the (reduced) cohomology ${\text{H}}^{•}(\mathcal{A},V)$ is the cohomology of the reduced complex ${\text{C}}^{•}(\mathcal{A},V)$ .

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.

Theorem 5.3: For the Lie conformal algebra $\mathcal{W}$, the following statements hold.(i) For the trivial module ℂ, $$\dim {\tilde{\text{H}}}^q(\mathcal{W},ℂ)=\{\begin{array}{l}1ifq=0,4,5,6\\ 2ifq=3\\ 0\text{\hspace{0.17em}otherwise}\end{array}$$(24) and $$\dim {\text{H}}^q(\mathcal{W},ℂ)=\{\begin{array}{l}1ifq=0,6\\ \begin{array}{l}2ifq=2,4,5\\ \text{3\hspace{0.17em}}ifq=3\end{array}\\ 0\text{\hspace{0.17em}otherwise}\text{.}\end{array}$$(25)(ii) If a ≠ 0, then dim H^q ($\mathcal{W}$, ℂ_a) = 0, for q ≥ 0.(iii) If a ≠ 0, then dim H^q ($\mathcal{W}$, M_Δ,α) = 0, for q ≥ 0.

Remark 5.4: Denote by Lie($\mathcal{W}$)— the annihilation Lie algebra of $\mathcal{W}$. Note that Lie($\mathcal{W}$)— 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], ${\tilde{\text{H}}}^q(\mathcal{W},ℂ)\cong {\text{H}}^q(\text{Lie(}\mathcal{W}{\text{)}}_{-,}ℂ)$ . Thus we have determined the cohomology of Lie($\mathcal{W}$)_— with trivial coefficients.

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).