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# Structures of W(2.2) Lie conformal algebra

Lamei Yuan
• Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin 150080, China
• Email:
/ Henan Wu
• Corresponding author
• School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China
• Email:
Published Online: 2016-09-15 | DOI: https://doi.org/10.1515/math-2016-0054

## 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 $\left[{L}_{\text{λ}}L\right]=\left(\partial +2\text{λ}\right)L,\left[{L}_{\text{λ}}M\right]=\left(\partial +2\text{λ}\right)M,\left[{M}_{\text{λ}}M\right]=0$ . 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.

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

## 1 Introduction

A Lie conformai algebra is a ℂ[∂]-module $\mathcal{R}$ equipped with a λ-bracket [·λ·] which is a ℂ-bilinear map from $\mathcal{R}$$\mathcal{R}$ to ℂ[λ] ⊗ $\mathcal{R}$, such that the following axioms hold for all a, b, c$\mathcal{R}$: $[∂aλb]=−λ[aλb], [aλb], [aλ∂b]=(∂+λ)[aλb] (conformal sesquilinearity),$(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 $\mathcal{W}$ 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 $\mathcal{W}$ 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 $\mathcal{W}$. It turns out that all the conformal derivations of $\mathcal{W}$ are inner. In Sect. 3, we discuss central extensions of $\mathcal{W}$ and show that $\mathcal{W}$ has a unique nontrivial universal central extension. In Sect. 4, we determine all free nontrivial conformal $\mathcal{W}$-modules of rank one. In Sect. 5, we compute cohomologies of $\mathcal{W}$ with coefficients in $\mathcal{W}$-modules ℂ, ℂa and MΔ,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 $\mathcal{C}$ denote the ring ℂ[∂] of polynomials in the indeterminate ∂.

Definition 2.1: Let V and W be two C-modules. A-linear map ϕ : V$\mathcal{C}$[λ] ⊗$\mathcal{C}$ W, denoted by ϕλ V → 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 $\mathcal{C}$-modules V to W. It can be made into an $\mathcal{C}$-module via $(∂ϕ)λυ=−λϕλυ, for υ ∈ V.$

Definition 2.2: Let $\mathcal{A}$ be a Lie conformal algebra. A conformal linear map dλ : $\mathcal{A}$$\mathcal{A}$ is called a conformal derivation if $dλ([aμb])=[(dλa)λ+μb]+[aμ(dλb)],foralla,b∈A.$ Denote by CDer($\mathcal{A}$) the space of all conformal derivations of $\mathcal{A}$. For any a$\mathcal{A}$, one can define a linear map (ad a)λ : $\mathcal{A}$$\mathcal{A}$ by (ada)λ b = [aλ] 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.

## 3 Central extension

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 $0→a→A^→A→0.$ In this case $\stackrel{\mathcal{^}}{\mathcal{A}}$ is also called an extension of $\mathcal{A}$ by $\mathsf{a}$. The extension is said to be central if $a⊆Z(A^)={x∈A^|[xλy]A^=0forally∈A^},and∂a=0.$ Consider the central extension $\stackrel{\mathcal{^}}{\mathcal{A}}$ of $\mathcal{A}$ by the trivial module ℂ. This means $\stackrel{\mathcal{^}}{\mathcal{A}}$$\mathcal{A}$ ⊕ ℂ$\mathsf{c}$, and $[aλb]A^=[aλb]A+fλ(a,b)c,fora,b∈A,$ where fλ : $\mathcal{A}$ × $\mathcal{A}$ → ℂ[λ] is a bilinear map. The axioms (1)(3) imply the following properties of the 2-cocycle fλ(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$\mathcal{A}$. For any linear function f : $\mathcal{A}$ → ℂ, the map $ψf(a,b)=f([aλb]),fora,b∈A,$(13) defines a trivial 2-cocycle. Let ${{a}^{\prime }}_{\text{λ}}\left(a,b\right)={a}_{\text{λ}}\left(a,b\right)+{\psi }_{f}\left(a,b\right)$. The equivalent 2-cocycles a'λ(a, b) and aλ(a, b) define isomorphic extensions.

In the following we determine the central extension $\stackrel{\mathcal{^}}{\mathcal{W}}$ of $\mathcal{W}$ by ℂc, i.e., $\stackrel{\mathcal{^}}{\mathcal{W}}$ = $\mathcal{W}$ © ℂ$\mathsf{c}$, 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 ${a}_{\text{λ}}\left(L,L\right)={\sum }_{i-0}^{i=n}{a}_{i}{\text{λ}}^{i}\in ℂ\left[\text{λ}\right]$ with an ≠ 0. Then, assuming n > 1 and equating the coefficients of λn in (15), we get 2μan = (n — 1)μan 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 aλ(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 aλ (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)aλ+μ(M, M) = —(2λ + μ)aμ(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 algebra $\mathcal{W}$ by$\mathsf{c}$, such that $[LλL]=(∂+2λ)L+aλ3c,[LλM]=(∂+2λ)M+bλ3c,[MλM]=0$(19)(ii) There exists a unique nontrivial universal central extension of $\mathcal{W}$ by ℂ$\mathsf{c}$ ⊕ ℂ$\mathsf{c}$′, satisfying $[LλL]=(∂+2λ)L+aλ3c,[LλM]=(∂+2λ)M+λ3c′,[MλM]=0.$(20)

Remark 3.2: Theorem 3.1 (ii) implies that dim H2($\mathcal{W}$, 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 $\mathcal{A}$ is a ℂ[∂]-module endowed with a bilinear map $\mathcal{A}$VV[∂], avaλ v satisfying the following axioms for a, b ∈ $\mathcal{A}$, v ∈ V: $aλ(bμυ)−bμ(aλυ)=[aλb]λ+μυ,(∂a)λυ=−λaλυ, aλ(∂)=(∂+λ)aλυ.$ 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 ℂa, which is the one-dimensional vector space ℂ such that ∂v = av for v ∈ ℂa. Then ℂa 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): $MΔ,α=ℂ[∂]υ, Lλυ=(∂+α+Δλ)υ.$(21)

The module MΔ, α is irreducible if and only if Δ ≠ 0. The module M0, α contains a unique nontrivial submodule (∂ + α)M0, α isomorphic to M1, α. Moreover, the modules MΔ, 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: $MΔ,α=ℂ[∂]υ, Lλυ=(∂+α+Δλ)υ, Mλυ=0 for some Δ, α ∈ℂ.$

## 5 Cohomology

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 $γ:A⊗n→v[λ1,⋯,λn],a1⊗⋯⊗an↦γλ1,⋯,λn(a1,⋯,an)$ satisfying the following conditions:(i) ${\gamma }_{{\text{λ}}_{1},\cdots ,{\text{λ}}_{n}}\left({a}_{1},\cdots ,\partial {a}_{i}\cdots ,{a}_{n}\right)=-{\text{λ}}_{i}{\gamma }_{{\text{λ}}_{1}},\cdots ,{\text{λ}}_{n}\left({a}_{1},\cdots ,{a}_{n}\right)$ (conformal antilinearity),(ii) γ is skew-symmetric with respect to simultaneous permutations of ai's and λi's (skew-symmetry).As usual, let $\mathcal{A}$⊗0 = ℂ, so that a 0-cochain is an element of V. Denote by ${\stackrel{~}{C}}^{n}\left(\mathcal{A},V\right)$ 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(22) where γ is linearly extended over the polynomials in λi. In particular, if γ ∈ V is a 0-cochain, then (dγ)λ(a) = aλγ.

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 $C~∙(A,V)=⊕n∈Z+C~n(A,V),$(23) and called the basic complex. Moreover, define a (left) ℂ[∂]-module structure on ${\stackrel{~}{C}}^{\bullet }\left(\mathcal{A},V\right)$ 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 $\mathrm{\partial }{\stackrel{~}{C}}^{\bullet }\left(\mathcal{A},V\right)\subset {\stackrel{~}{C}}^{\bullet }\left(\mathcal{A},V\right)$ 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 cohomology ${\stackrel{˜}{\text{H}}}^{•}\left(\mathcal{A},V\right)$ of a Lie conformal algebra $\mathcal{A}$ with coefficients in an $\mathcal{A}$-module V is the cohomology of the basic complex ${\stackrel{˜}{\text{C}}}^{•}\left(\mathcal{A},V\right)$ and the (reduced) cohomology ${\text{H}}^{•}\left(\mathcal{A},V\right)$ is the cohomology of the reduced complex ${\text{C}}^{•}\left(\mathcal{A},V\right)$ .

For a q-cochain $\gamma \text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{\stackrel{˜}{C}}^{q}\left(\mathcal{A},V\right)$ we call γ a q-cocycle if d(γ) = 0; a q-coboundary if there exists a (q — 1)-cochain $\varphi \in {\stackrel{˜}{C}}^{q-1}\left(\mathcal{A},V\right)$ such that γ = d(ϕ). Two cochains γ1 and γ2 are called equivalent if γ1 — γ2 is a coboundary. Denote by ${\stackrel{˜}{D}}^{q}\left(\mathcal{A},V\right)$ and ${\stackrel{˜}{B}}^{q}\left(\mathcal{A},V\right)$ 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 $\mathcal{W}$, 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 Hq ($\mathcal{W}$, ℂa) = 0, for q ≥ 0.(iii) If a ≠ 0, then dim Hq ($\mathcal{W}$, MΔ,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 {Ln, Mn| — 1 ≤ n ∈ ℤ of the centerless W-algebra W (2, 2). By [6, Corollary 6.1], ${\stackrel{˜}{\text{H}}}^{q}\left(\mathcal{W},ℂ\right)\cong {\text{H}}^{q}\left(\text{Lie(}\mathcal{W}{\text{)}}_{-,}ℂ\right)$ . Thus we have determined the cohomology of Lie($\mathcal{W}$) 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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Accepted: 2016-08-01

Published Online: 2016-09-15

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, Export Citation