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
1 Introduction
A Lie conformai algebra is a ℂ[∂]-module
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
The formal distribution Lie algebra corresponding to
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
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
Let V and W be two C-modules. A ℂ-linear map ϕ : V →
Denote by Chom(V, W) the space of conformal linear maps from
Let
Denote by CDer(
For the Lie conformal algebra
Every conformal derivation of
Let dλ be a conformal derivation of
where fi (λ, ∂) and hi (λ, ∂) for i = 1, 2 are polynomials in ℂ[λ, ∂]. Applying dλ to [Lμ L] = (∂ + 2μ)L, we have
Comparing the coefficients of the similar terms gives
Write
By replacing dλ by dλ — ad(a1, 1 (—∂)L)λ — ad(a2, 1 (—∂)M)λ, we can suppose a1, 1(λ) = a2, 1(λ) = 0. Then plugging fi (λ, ∂) = ai, 0(λ) into (7) gives ai, 0(λ) = 0 for i = 1, 2. Thus dλ (L) = 0 by (6). Fouthermore, applying dλ to [LμM] = (∂ + 2μ)M, we have
Comparing the coefficients of highest degree of λ in (9) gives hi (λ, ∂) = 0 for i = 1, 2. Hence dλ (M) = 0 by (6). This concludes the proof.
Proposition 2.3 is equivalent to H1 (
3 Central extension
An extension of a Lie conformal algebra
In this case
Consider the central extension
where fλ :
for all a, b, c ∈
defines a trivial 2-cocycle. Let
In the following we determine the central extension
and the others can be obtained by skew-symmetry. Applying the Jacobi identity for (L, L, L), we have
Write
Plugging this in (15) and comparing the similar terms, we obtain a0 = a2 = 0. Thus
To compute aλ(L, M), we apply the Jacobi identity for (L, L, M) and obtain
By doing similar discussions as those in the process of computing aλ (L, M), we have
Finally, applying the Jacobi identity for (L, M, M) yields (λ — μ)aλ+μ(M, M) = —(2λ + μ)aμ(M, M), which implies
From the discussions above, we obtain the following results.
(i) For any a, b ∈ ℂ with (a, b) ≠ (0, 0), there exists a unique nontrivial universal central extension of the Lie conformal algebra
(ii) There exists a unique nontrivial universal central extension of
Theorem 3.1 (ii) implies that dim H2(
4 Conformai module
Let us first recall the notion of a conformal module given in [5].
A (conformal) module V over a Lie conformal algebra
The vector space ℂ can be seen as a module (called the trivial module) over any conformal algebra
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):
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
All free nontrivial
Suppose that Lλv = f(∂, λ)v, Mλv = g(∂, λ)v, where f(∂, λ), g(∂, λ) ∈ ℂ [λ, ∂]. By the result of Virmodules, we have
On the other hand, it follows from Mλ(Mμu) = Mμ(Mλ v) that
This implies degλg(∂, λ) + deg∂g(∂, λ) = degλg(∂, λ), where the notation degλg(∂, 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, [LλM]λ + μv = (λ — μ)Mλ + μv 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]:
An n-cochain (n ∈ ℤ+) of a Lie conformal algebra
(i)
(ii) γ is skew-symmetric with respect to simultaneous permutations of ai's and λi's (skew-symmetry).
As usual, let
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
and called the basic complex. Moreover, define a (left) ℂ[∂]-module structure on
where ∂V denotes the action of ∂ on V. Then d∂ = ∂d and thus
is called the reduced complex.
The basic cohomology
For a q-cochain
The main results of this section are the following.
For the Lie conformal algebra
(i) For the trivial module ℂ,
(ii) If a ≠ 0, then dim Hq (
(iii) If a ≠ 0, then dim Hq (
(i) For any
Let
for X, Y ∈ {L, M}. By (26) and (4),
Letting λ = λ1 + λ2 in (27) gives
which implies that γλ(X) is divisible by λ. Define
Clearly,
Let ψ be a 2-cocycle. For X ∈
Letting λ3 = 0 and λ1 + λ1 = λ gives (λ — 2λ2)ψλ1, 0(X, L) = λ ψλ1, λ2(X, L). Hence, ψλ1, 0 is divisible by λ. Define a 1-cochain f by
Set γ = ψ + df, which is also a 2-cocycle. By (29),
By (30), we have
Thus γλ1,λ2(L, L) = 0. Similarly, by (31),
which gives γλ1,λ2(L, M)=0 and so γλ1,λ2(M, L)=0 . Finally,
Setting λ1 = 0 in (32) gives
γλ2, 0(M, M) = 0
and thus
γλ1λ2(M, M) = 0
. This shows γ = 0. Hence
To determine higher dimensional cohomologies (for q ≥ 3), we define an operator
for X1, ⋯, Xq–1 ∈ {L, M}. By (22), (33) and skew-symmetry of γ,
By the fact that [Xi λi, L] = (∂ + 2λi)Xi and conformal antilinearity of γ, [Xi λiL] can be replaced by (λi, — λ)Xi, in (34). Thus, equality (34) can be rewritten as
where deg γ is the total degree of γ in λ1, ⋯, λq. 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
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
Thus
For q = 3, we need to consider four cases for k, i.e., k = 0,1,2,3. Let
This gives γλ1λ2λ3 (M, M, M) = 0. In the case of k = 1, we have
Note that γλ1λ2λ3(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
where a2, a2, a3, a4 ∈ ℂ. Plugging (38) into (37) gives a4 = 0, a3 = 2a2, a1 = —a2. Therefore,
Note that ϕ1 is a coboundary of
Similarly, suppose that
where b1, b2, b3, b4 ϕ ℂ. Substituting (40) into the following equality
gives b4 = b1 + b2. Hence,
On the other hand, there is a 2-cochain
Thus γλ1,λ2,λ3 (L, L, M) in (41) is equivalent to a constant factor of
For q = 4, three cases (i.e., k = 1,2,3) should be taken into account. Let
where c, c1, c2, c3, e1, e2 ∈ ℂ. And there exist three 3-cochains of degree 3
such that
where
Moreover,
For q = 5, we need to consider k = 2, 3. Let
where
such that
where
Furthermore, there exists another one 4-cochains of degree 4
such that
For q = 6, one only needs to consider the case when k = 3. One can check that
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 γλ1, λ2, λ3, λ4, λ5 (L, L, M, M, M) in (54), whose coboundary is zero. Therefore, dim
According to [6, Proposition 2.1], the map
It remains to compute H•(
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]):
where ıq, πq are induced bı, π respectively and wq is the q—th connecting homomorphism. Given
Therefore,
Then (25) follows from (65). Moreover, we can give a basis for
Hq (
Thus the pre-image of ∂φ under the connecting homomorphism ωp is ωq—1 (∂φ) = r(∂φ).
Finally, we finish our proof by giving a basis of
Hq (
This gives
Hence,
Therefore,
(ii) Define an operator
for
Let
(iii) In this case,
for
If γ is a reduced q-cocycle, it follows from (69) that
This completes the proof of Theorem 5.3.
Denote by Lie(
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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