A Lie conformal algebra of Block type

0.18 MB·

by  Lamei Yuan

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A Lie conformal algebra of Block type
Lamei Yuan
Academy of Fundamental and Interdisciplinary Science,
6 Harbin Institute of Technology, Harbin 150080, China
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[email protected]
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Abstract: The aim of this paper is to study a Lie conformal algebra of Block type. In this

n
paper, conformal derivation, conformal module of rank 1 and low-dimensional comohology
a
J of the Lie conformal algebra of Block type are studied. Also, the vertex Poisson algebra

6 structure associated with the Lie conformal algebra of Block type is constructed.
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Keywords: Lie conformal algebra, vertex Lie algebra, cohomology, vertex Poisson algebra

]
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MR(2000) Subject Classification: 17B65, 17B69
R
.
h
t 1 Introduction
a
m
[ The notion of Lie conformal algebra, introduced by Kac [9], encode an axiomatic description

1 of the operator product expansions of chiral fields in conformal field theory. It is a pow-
v erful tool for the study of infinite-dimensional Lie (super)algebras, associative algebras and
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their representations. Lie conformal algebras have been extensively studied, including the
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3 classification problem [5, 6], cohomology theory [2, 12] and representation theory [3].
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The Lie conformal algebras are closely related to vertex algebras. Primc [11] introduced
0
. and studied a notion of vertex Lie algebra, which is a special case of a more general notion of
1
0 local vertex Lie algebra [4]. As it was explained in [10], the notion of Lie conformal algebra
6
and the notion of vertex Lie algebra are equivalent. In this paper, we shall use Lie conformal
1
: algebra and vertex Lie algebra synonymously.
v
i With the notion of vertex Lie algebra, one arrives at the notion of vertex Poisson algebra,
X
which is a combination of a differential algebra structure and a vertex Lie algebra structure,
r
a satisfying a natural compatibility condition. The symmetric algebra of a vertex Lie algebra
is naturally a vertex Poisson algebra [7]. A general construction theorem of vertex Poisson
algebras was given in [10]. Applications of vertex Poisson algebras to the theory of integrable
systems were studied in [1].
In the present paper, we study a nonsimple Lie conformal algebra of infinite rank, which
is endowed with a C[∂]-basis {J |i ∈ Z+}, such that
i
[J J ] = ((i+1)∂ +(i+j +2)λ)J , for i,j ∈ Z+. (1.1)
iλ j i+j
The corresponding formal distribution Lie algebra is a Block type Lie algebra, which is the
associated graded Lie algebra of the filtered Lie algebra W [13, 14, 15, 16, 18]. Thus we
1+∞
call this Lie conformal algebra a Lie conformal algebra of Block type and denote it by B in
this paper. It is a conformal subalgebra of grgc studied in [17]. In addition, it contains the
1
Virasoro conformal algebra Vir = C[∂]J with [J J ] = (∂ +2λ)J as a subalgebra.
0 0λ 0 0
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The paper is organized as follows. In Section 2, we recall the notions of Lie conformal
algebra and vertex Lie algebra. In Section 3, we study conformal derivations of the Lie
conformal algebra of Block type B. In Section 4, we recall the notions of conformal module
and comohology of Lie conformal algebras. Then we study conformal module of rank 1 and
low-dimensional comohology of B with coefficients in B-modules. In Section 5, we equip a
vertex Lie algebra structure (Y ,∂) with B and establish an association of a vertex Poisson
−
algebra structure to the vertex Lie algebra (B,Y ,∂).
−
2 Preliminaries
Throughout this paper, all vector spaces 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.
Definition 2.1. A Lie conformal algebra R is a C[∂]-module with a C-bilinear map,
R ⊗R → C[λ]⊗R, a⊗b 7→ [a b],
λ
called the λ-bracket, and satisfying the following axioms (a,b,c ∈ R),
(conformal sesquilinearity) [∂a b] = −λ[a b], [a ∂b] = (∂ +λ)[a b], (2.1)
λ λ λ λ
(skew-symmetry) [a b] = −[b a], (2.2)
λ −λ−∂
(Jacobi identity) [a [b c]] = [[a b] c]+[b [a c]]. (2.3)
λ µ λ λ+µ µ λ
If we consider the expansion
λj
[a b] = (a b), (2.4)
λ (j)
j!
j∈PZ+
the coefficients of λj are called the j-product satisfying a b = 0 for n sufficiently large,
j! (n)
and the axioms (2.1)–(2.3) can be written in terms of them as follows:
∂a b = −na b, a ∂b = ∂(a b)+na b, (2.5)
(n) (n−1) (n) (n) (n−1)
1
a b = − (−1)n+i ∂ib a, (2.6)
(n) (n+i)
i!
i∈PZ+
m
a b c = b a c+ m (a b) c. (2.7)
(m) (n) (n) (m) i (i) (m+n−i)
iP=0(cid:0) (cid:1)
In terms of generating functions, Mirko Primc in [11] presented an equivalent definition
of a Lie conformal algebra under the name of vertex Lie algebra (see also [10]). Let V be
any vector space. Following [11], for a formal series
f(x ,··· ,x ) = u(m ,··· ,m )x−m1−1···x−mn−1 ∈ V[[x±1,··· ,x±1]],
1 n 1 n 1 n 1 n
m1,··P·,mn∈Z
we set
Singf(x ,··· ,x ) = u(m ,··· ,m )x−m1−1···x−mn−1. (2.8)
1 n 1 n 1 n
m1,···P,mn∈Z+
Clearly, for 1 ≤ i ≤ n,
∂ ∂
Singf(x ,··· ,x ) = Sing f(x ,··· ,x ). (2.9)
1 n 1 n
∂x ∂x
i i
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Definition 2.2. A vertex Lie algebra is a vector space A equipped with a linear operator ∂
called the derivation and a linear map
Y (·,z) : A → z−1(EndA)[[z−1]], a 7→ Y (a,z) = a z−n−1,
− − n≥0 (n)
P
satisfying the following conditions for a,b ∈ A, n ∈ Z+:
a b = 0 for n sufficiently large, (2.10)
(n)
d
[∂,Y (a,z)] = Y (∂a,z) = Y (a,z), (2.11)
− − −
dz
Y (a,z)b = Sing ez∂Y (b,−z)a , (2.12)
− −
(cid:0) (cid:1)
and the half Jacobi identity holds:
z −z z −z
Sing z−1δ( 1 2)Y (a,z )Y (b,z )−z−1δ( 2 1)Y (b,z )Y (a,z )
0 z − 1 − 2 0 −z − 1 − 2
(cid:16) 0 0 (cid:17)
z −z
= Sing z−1δ( 1 0)Y (Y (a,z )b,z ) . (2.13)
2 z − − 0 2
(cid:16) 2 (cid:17)
Relation (2.12) is called the half skew-symmetry. It was shown in [11] that the half Jacobi
identity (2.13) amounts to the following half commutator formula:
Y (a,z )Y (b,z )−Y (b,z )Y (a,z ) = Sing (z −z )−i−1Y (a b,z ) . (2.14)
− 1 − 2 − 1 − 2 i∈Z+ 1 2 − (i) 2
(cid:16) (cid:17)
P
As it was explained in [10, Remark 2.6], the notion of Lie conformal algebra is equivalent
to the notion of vertex Lie algebra. We often denote a vertex Lie algebra by (A,Y ,∂) and
−
refer to (Y ,∂) as the vertex Lie algebra structure. A vertex Lie algebra (A,Y ,∂) is said to
− −
be free, if A is a free C[∂]-module over a vector space V, namely, A = C[∂]V ∼= C[∂]⊗C V.
3 Conformal derivation
Let C denote the ring C[∂] of polynomials in the indeterminate ∂.
Definition 3.1. Let V and W be two C-modules. A linear map φ : V → C[λ] ⊗C W,
denoted by φ : V → W, is called a conformal linear map, if
λ
φ (∂v) = (∂ +λ)(φ v), for v ∈ V. (3.1)
λ λ
ThespaceofconformallinearmapsbetweenC-modulesV andW isdenotedbyChom(V,W)
and it can be made into an C-module via
(∂φ) v = −λφ v, for v ∈ V.
λ λ
Definition 3.2. Let R be a Lie conformal algebra. A conformal linear map d : R → R is
λ
called a conformal derivation of R if
d [a b] = [(d a) b]+[a (d b)], for a,b ∈ R. (3.2)
λ µ λ λ+µ µ λ
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The space of all conformal derivations of R is denoted by CDer(R). For any a ∈ R,
one can define a conformal derivation (ada) : R → R by (ada) b = [a b] for b ∈ R. Such
λ λ λ
conformal derivation is called inner. Denote by CInn(R) the space of all conformal inner
derivations of R.
Proposition 3.3. Every conformal derivation of the Lie conformal algebra B is inner.
Proof. Let d be any conformal derivation of B. Denote L = J . Assume that there exists
0
a finite subset I = {i ,··· ,i } ⊆ Z+ such that d L = n f (∂,λ)J , where f (∂,λ) ∈
1 n λ j=1 ij ij ij
C[∂,λ]. Condition (3.2) requires dλ([LµL]) = [Lµ(dλL)]+P[(dλL)λ+µL]. This is equivalent to
n n
(∂ +λ+2µ) f (∂,λ)J − (∂ +(i +2)µ)f (∂ +µ,λ)J
ij ij j ij ij
j=1 j=1
P P
n
= ((i +1)∂ +(i +2)(λ+µ))f (−λ−µ,λ)J . (3.3)
j j ij ij
j=1
P
For each j,
(∂ +λ+2µ)f (∂,λ)−(∂ +(i+2)µ)f (∂ +µ,λ)
ij ij
= ((i +1)∂ +(i +2)(λ+µ))f (−λ−µ,λ). (3.4)
j j ij
Write f (λ,∂) = m a (λ)∂k with a (λ) 6= 0. Then, assuming m > 1, if we equate
ij k=0 ij,k ij,m
terms of degree mPin ∂, we have (λ−i−mµ)aij,m(λ) = 0 and thus aij,m(λ) = 0. This contra-
dicts a (λ) 6= 0. Thus deg f (λ,∂) ≤ 1, and f (λ,∂) = a (λ)+a (λ)∂. Substituting
ij,m ∂ ij ij ij,0 ij,1
it into (3.4) gives a (λ) = ij+2λa (λ). Therefore,
ij,0 ij+1 ij,1
n a (λ)
d L = ij,1 ((i +1)∂ +(i +2)λ)J .
λ i +1 j j ij
jP=1 j
Replacing d by d −(adh) with h = n aij,1(−∂)J , we get d (L) = 0.
λ λ λ j=1 ij+1 ij λ
l P
Fork > 0, assume thatd J = f (∂,λ)J .Applying d to[L J ] = (∂+(k+2)µ)J
λ k i=1 ki ki λ µ k k
and using dλ(L) = 0, we obtain P
l l
(∂ +λ+(k +2)µ) f (∂,λ)J = (∂ +(k +2)µ)f (∂ +µ,λ)J , (3.5)
ki ki i ki ki
i=1 i=1
P P
and thus
(∂ +λ+(k +2)µ)f (∂,λ) = (∂ +(k +2)µ)f (∂ +µ,λ), for 1 ≤ i ≤ l. (3.6)
ki i ki
Comparing the highest degree of λ gives f (∂,λ) = 0 for 1 ≤ i ≤ l. Hence d (J ) = 0 for
ki λ k
k > 0. This concludes the proof. (cid:3)
4 Cohomology
Definition 4.1. A module M over a Lie conformal algebra R is a C[∂]-module endowed
with a bilinear map
R ⊗M → M[[λ]], a⊗v 7→ a v
λ
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such that (a,b ∈ R, v ∈ M)
a (b v)−b (a v) = [a b] v, (4.1)
λ µ µ λ λ λ+µ
(∂a) v = −λa v, a (∂v) = (∂ +λ)a v, (4.2)
λ λ λ λ
If a v ∈ M[λ] for all a ∈ R, v ∈ M, then the R-module M is said to be conformal. If M is
λ
finitely generated as C[∂]-module, then M is simply called finite.
Since we only consider conformal modules, we will simply shorten the term “conformal
module” to “module”. The one-dimensional vector space C can be viewed as a module
(called the trivial module) over any conformal algebra R with both the action of ∂ and the
action ofR being zero. In addition, fora fixed nonzero complex constant a, thereis a natural
C[∂]-module C , which is the one-dimensional vector space C such that ∂v = av for v ∈ C .
a a
Then C becomes an R-module on which all elements of R act by zero.
a
For the Virasoro conformal algebra Vir, it was proved in [3] that all free nontrivial Vir-
modules of rank 1 are the following ones (∆,α ∈ C):
M = C[∂]v, L v = (∂ +α+∆λ)v. (4.3)
∆,α λ
The module M is irreducible if and only if ∆ 6= 0, the module M contains a unique
∆,α 0,α
nontrivial submodule (∂ + α)M isomorphic to M , and the modules M with ∆ 6= 0
0,α 1,α ∆,α
exhaust all finite irreducible nontrivial conformal Vir-modules.
Proposition 4.2. All free nontrivial B-modules of rank 1 are as follows (∆,α ∈ C):
M = C[∂]v, J v = (∂ +α+∆λ)v, J v = 0, for i > 0.
∆,α 0λ iλ
Proof. By (4.3), J v = (∂ + α + ∆λ)v for some ∆,α ∈ C. By [17, Lemma 5.1], we can
0λ
suppose that k is the smallest nonnegative integer such that J v 6= 0, J v = 0. Assume
kλ k+1λ
k > 0 and write J v = g(λ,∂)v, for some g(λ,∂) ∈ C[λ,∂]. Since [J J ] v = 0,
kλ kλ k λ+µ
g(λ,∂)g(µ,λ+∂) = g(µ,∂)g(λ,µ+∂). (4.4)
This implies deg g(λ,∂)+deg g(λ,∂) = deg g(λ,∂). Thus deg g(λ,∂) = 0. Then we have
λ ∂ λ ∂
g(λ,∂) = g(λ) for some g(λ) ∈ C[λ]. The fact that [J J ] v = ((k+1)λ−µ)J v yields
0λ k λ+µ kλ+µ
((k +1)λ−µ)g(λ+µ)v = (∂ +α+∆λ)g(µ)v−(∂ +µ+α+∆λ)g(µ)v = −µg(µ)v,
which gives g(µ) = 0. Hence, J v = 0, a contradiction. Thus k = 0 and J v = 0. It
kλ 1λ
follows immediately that J v = 0 for all i ≥ 1. (cid:3)
iλ
In the following we study cohomology of the Lie conformal algebra B with coefficients in
B-modules C, C and M , respectively. For completeness, we shall present the definition
a ∆,α
of cohomology of Lie conformal algebras given in [2].
Definition 4.3. An n-cochain (n ∈ Z+) of a Lie conformal algebra R with coefficients in
an R-module M is a C-linear map
γ : R⊗n → M[λ ,··· ,λ ], a ⊗···⊗a 7→ γ (a ,··· ,a )
1 n 1 n λ1,···,λn 1 n
satisfying
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(1) γ (a ,··· ,∂a ,··· ,a ) = −λ γ (a ,··· ,a ) (conformal antilinearity),
λ1,···,λn 1 i n i λ1,···,λn 1 n
(2) γ is skew-symmetric with respect to simultaneous permutations of a ’s and λ ’s,
i i
namely,
γ (a ,··· ,a ,a ,a ,α ,··· ,a )
λ1,···,λi−1,λi+1,λi,λi+2,···,λn 1 i−1 i+1 i i+2 n
= −γ (a ,··· ,a ,a ,a ,a ,··· ,a ). (4.5)
λ1,···,λi−1,λi,λi+1,λi+2,···,λn 1 i−1 i i+1 i+2 n
As usual, let R⊗0 = C, so that a 0-cochain is an element of M. Denote by C˜n(R,M)
the set of all n-cochains. The differential d of an n-cochain γ is defined by
(dγ) (a ,··· ,a )
λ1,···,λn+1 1 n+1
n+1
= (−1)i+1a γ (a ,··· ,aˆ,··· ,a )
iλi λ1,···,λˆi,···,λn+1 1 i n+1
i=1
P
n+1
+ (−1)i+jγ [a a ],a ,··· ,aˆ,··· ,aˆ,··· ,a ,(4.6)
λi+λj,λ1,···,λˆi,···,λˆj,···,λn+1 iλi j 1 i j n+1
i,j=P1;i<j (cid:0) (cid:1)
where γ isextended linearlyover thepolynomialsinλ . Inparticular, ifγ ∈ M isa0-cochain,
i
then (dγ) (a) = a γ.
λ λ
It is proved in [2] that the operator d preserves the space of cochains and d2 = 0. Thus
the cochains of a Lie conformal algebra R with coefficients in R-module M form a complex,
which is called the basic complex and will be denoted by
C˜•(R,M) = C˜n(R,M).
nL∈Z+
Moreover, define a C[∂]-module structure on C˜•(R,M) by
n
(∂γ) (a ,··· ,a ) = (∂ + λ )γ (a ,··· ,a ), (4.7)
λ1,···,λn 1 n M i λ1,···,λn 1 n
i=1
P
where ∂ denotes the action of ∂ on M. Then d∂ = ∂d and thus ∂C˜•(R,M) ⊂ C˜•(R,M)
M
forms a subcomplex. The quotient complex
C•(R,M) = C˜•(R,M)/∂C˜•(R,M) = Cn(R,M)
nL∈Z+
is called the reduced complex.
Definition 4.4. The basis cohomology H˜•(R,M) of a Lie conformal algebra R with coeffi-
cients in R-module M is the cohomology of the basis complex C˜•(R,M) and the (reduced)
cohomology H•(R,M) is the cohomology of the reduced complex C•(R,M).
Remark 4.5. The basic cohomology H˜•(R,M) is naturally a C[∂]-module, whereas the
reduced cohomology H•(R,M) is a complex vector space.
For a q-cochain γ ∈ C˜q(R,M), we call γ a q-cocycle if d(γ) = 0; a q-coboundary or a
trivial q-cocycle if there is a (q − 1)-cochain φ ∈ C˜q−1(R,M) such that γ = d(φ). Two
cochains γ and γ are called equivalent if γ −γ is a coboundary. Denote by D˜q(R,M) and
1 2 1 2
B˜q(R,M) the spaces of q-cocycles and q-boundaries, respectively. By Definition 4.4,
H˜q(R,M) = D˜q(R,M)/B˜q(R,M) = {equivalent classes of q-cocycles}.
The main results of this section are the following theorem.
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Theorem 4.6. For the Lie conformal algebra B, the following statements hold.
(1) For the trivial module C, we have
dimH˜q(B,C) = 1 if q = 0,
(cid:26) 0 if q = 1,or 2,
and
1 if q = 0,or 2,
dimHq(B,C) =
(cid:26) 0 if q = 1.
(2) If a 6= 0, then dimH•(B,C ) = 0.
a
(3) If α 6= 0, then dimH•(B,M ) = 0.
∆,α
Proof. (1) Since a 0-cochainγ is anelement of C and (dγ) (a) = a γ = 0 for a ∈ B, we have
λ λ
D˜0(B,C) = C˜0(B,C) = C and B˜0(B,C) = 0. Thus H˜0(B,C) = D˜0(B,C)/B˜0(B,C) = C,
and H0(B,C) = C because ∂C˜0(B,C) = ∂C = 0.
Let γ ∈ C˜1(B,C) and dγ ∈ ∂C˜2(B,C), namely, there is φ ∈ C˜2(B,C) such that
d(γ) = ∂φ. By (4.6), (4.7) and ∂C = 0,
γ ([a b]) = −(dγ) (a,b) = −(∂φ) (a,b) = −(λ +λ )φ (a,b), a,b ∈ B,(4.8)
λ1+λ2 λ1 λ1,λ2 λ1,λ2 1 2 λ1,λ2
By (1.1), (4.8) and Definition 4.3 (1),
((i+1)λ −λ )γ (J ) = −(λ +λ )φ (J ,J ), i ≥ 0. (4.9)
1 2 λ1+λ2 i 1 2 λ1,λ2 0 i
Setting λ = λ +λ in (4.9) gives
1 2
((i+1)λ−(i+2)λ )γ (J ) = −λφ (L,J ), i ≥ 0,
2 λ i λ1,λ2 i
which implies that γ (J ) is divisible by λ. We can define a 1-cochain γ′ ∈ C˜1(B,C) by
λ i
γ′(J ) = λ−1γ (J ), for i ≥ 0. (4.10)
λ i λ i
Since ∂C = 0, γ = ∂γ′ ∈ ∂C˜1(B,C). Hence H1(B,C) = 0. If γ is a 1-cocycle, namely, φ = 0
in (4.8), then (4.9) gives γ = 0. Thus H˜1(B,C) = 0.
Let ψ ∈ D˜2(B,C) be a 2-cocycle. We have
0 = (dψ) (J ,J ,J )|
λ1,λ2,λ3 i 0 0 λ3=0
= −(λ −(i+1)λ )ψ (J ,J )| +(λ −(i+1)λ )ψ (J ,J )|
1 2 λ1+λ2,λ3 i 0 λ3=0 1 3 λ1+λ3,λ2 i 0 λ3=0
−(λ −λ )ψ (J ,J )|
2 3 λ2+λ3,λ1 0 i λ3=0
= −((λ −(i+1)λ )ψ (J ,J )+(λ +λ )ψ (J ,J ). (4.11)
1 2 λ1+λ2,0 i 0 1 2 λ1,λ2 i 0
Setting λ = λ + λ in (4.11) gives ((λ − (i + 2)λ )ψ (J ,J ) = λψ (J ,J ). Thus
1 2 2 λ,0 i 0 λ1,λ2 i 0
ψ (J ,J ) is divisible by λ. Define a 1-cochain f by
λ,0 i 0
f (J ) = λ−1ψ (J ,J )| , for i ≥ 0.
λ1 i 1 λ1,λ i 0 λ=0
Set γ = ψ +df, which is also a 2-cocycle. For all i ≥ 0,
γ (J ,J )| = ψ (J ,J )| −λ f (J ) = 0. (4.12)
λ1,λ i 0 λ=0 λ1,λ i 0 λ=0 1 λ1 i
8
By (4.12) and (4.11) with γ in place of ψ, we have (λ + λ )γ (J ,J ) = 0. Therefore
1 2 λ1,λ2 i 0
γ (J ,J ) = 0 = γ (J ,J ). With this,
λ1,λ2 i 0 λ1,λ2 i 0
0 = (dγ) (J ,J ,J )|
λ1,λ2,λ 0 i k λ=0
= −γ ([J J ],J )| +γ ([J J ],J )| −γ ([J J ],J )|
λ1+λ2,λ 0λ1 i k λ=0 λ1+λ,λ2 0λ1 k i λ=0 λ2+λ,λ1 iλ2 k 0 λ=0
= −((i+1)λ −λ )γ (J ,J )−(k +1)λ γ (J ,J ). (4.13)
1 2 λ1+λ2,0 i k 1 λ2,λ1 i k
Setting λ = 0 in (4.13) gives γ (J ,J ) = 0 and thus γ (J ,J ) = 0. This proves γ = 0.
1 λ2,0 i k λ2,λ1 i k
Hence H˜2(B,C) = 0.
It remains to compute H2(B,C). Following [12], we define a linear map σγ : B⊗q →
C[λ ,··· ,λ ] for q ≥ 2 by
1 q−1
(σγ)( a ⊗···⊗a ) = γ (a ,··· ,a )| , a ,··· ,a ∈ B. (4.14)
1 n λ1,···,λq 1 q λq=−λ1−···−λq−1 1 q
We define σγ = γ if q = 0 and σγ(a ) = γ (a )| if q = 1. Set C′q(B,C) = {σγ|γ ∈
1 λ1 1 λ1=0
C˜q(B,C)}. Obviously, σ : C˜q(B,C) → C′q(B,C) is a surjective map. If γ ∈ ∂C˜q(B,C) =
( q λ )C˜q(B,C), then σγ = 0. That is, σ factors to a map σ : Cq(B,C) → C′q(B,C).
i=1 i
PWe claim that σ : Cq(B,C) → C′q(B,C) is an isomorphism as vector spaces. In fact, if
σγ = 0 for a q-cochain γ, then, by (4.14), γ (a ,··· ,a ) as a polynomial in λ has a
λ1,···,λq 1 q q
root λ = − q−1λ , namely, it is divided by q λ . Thus we get a q-cochain
q i=1 i i=1 i
P P
γ′ (a ,··· ,a ) = ( q λ )−1γ (a ,··· ,a ), (4.15)
λ1,···,λq 1 q i=1 i λ1,···,λq 1 q
P
and γ = ( q λ )γ′ ∈ ∂C˜q(B,C), which proves that σ is injective. Hence the claim is true.
i=1 i
In thePfollowing we can identify Cq(B,C) with C′q(B,C). We still call an element in
C′q(B,C) a reduced q-cochain. By defining the operator d′ : C′q(B,C) → C′q+1(B,C)
by d′(σγ) = σdγ, we have similar notions of reduced q-cocycle and q-coboundary. For
convenience, we will abbreviate γ (a ,··· ,a )| to γ (a ,··· ,a ).
λ1,···,λq 1 q λq=−λ1−···−λq−1 λ1,···,λq−1 1 q
Let ψ′ = σψ ∈ C′2(B,C) be a reduced 2-cochain. By (4.6) and (4.14),
(dψ′) (a ,a ,a )
λ1,λ2 1 2 3
= −ψ′ ([a a ],a )+ψ′ ([a a ],a )−ψ′ ([a a ],a ), (4.16)
λ1+λ2 1λ1 2 3 −λ2 1λ1 3 2 −λ1 2λ2 3 1
for a ,a ,a ∈ B. Define a reduce 1-cochain f′ = σf ∈ C′1(B,C) by
1 2 3
d
f′(J ) = (i+2)−1 ψ′(J ,J )| , for i ≥ 0. (4.17)
i dλ λ i 0 λ=0
Note that f′(a) = f (a)| . Thus f′ is simply a linear function from B to C, satisfying
λ λ=0
f′(∂a) = f (∂a)| = −λf (a)| = 0, and
λ λ=0 λ λ=0
(df′) (a ,a ) = −f′([a a ]), for a , a ∈ B. (4.18)
λ 1 2 1λ 2 1 2
If ψ′ is a reduced 2-cocycle, then γ′ = ψ′ +df′ is a reduced 2-cocycle, equivalent to ψ′. By
(4.17) and (4.18),
d
γ′(J ,J )| = 0, for i ≥ 0. (4.19)
dλ λ i 0 λ=0
9
This, along with (4.16) and (4.5), gives
∂
0 = (dγ′) (J ,J ,J )|
∂λ λ1,λ i k 0 λ=−λ1
∂
= −γ′ ([J J ],J )+γ′ ([J J ],J )−γ′ ([J J ],J ) |
∂λ λ1+λ iλ1 k 0 −λ iλ1 0 k −λ1 kλ 0 i λ=−λ1
(cid:0) (cid:1)
∂
= ((i+1)(λ+λ )+λ )γ′ (J ,J )−((k +1)(λ+λ )+λ)γ′ (J ,J ) | .
∂λ 1 1 −λ i k 1 −λ1 k i λ=−λ1
(cid:0) (cid:1)
∂
= (i+k +3)γ′ (J ,J )−λ γ′ (J ,J ). (4.20)
λ1 i k 1∂λ λ1 i k
1
Thus,
γ′(J ,J ) = c λi+k+3, for some c ∈ C. (4.21)
λ i k i,k i,k
By (4.16) with γ′ in place of ψ′ and (4.21),
0 = −γ′ ([J J ],J )+γ′ ([J J ],J )−γ′ ([J J ],J )
λ1+λ2 iλ1 j k −λ2 iλ1 k j −λ1 jλ2 k i
= −((j +1)λ −(i+1)λ )c (λ +λ )i+j+k+3
1 2 i+j,k 1 2
+((i+1)λ +(i+k +2)λ )c (−λ )i+j+k+3
2 1 i+k,j 2
−((j +1)λ +(j +k +2)λ )c (−λ )i+j+k+3. (4.22)
1 2 j+k,i 1
Taking i = j = 0 in (4.22), we get
c (λ −λ )(λ +λ )k+3 = c (λ +(k +2)λ )(−λ )k+3 −(λ +(k +2)λ )(−λ )k+3(4..23)
0,k 1 2 1 2 k,0 2 1 2 1 2 1
(cid:0) (cid:1)
Setting λ = 0 in (4.23) gives c = (−1)kc . Comparing coefficients of λ2λk+2 in (4.23)
2 0,k k,0 1 2
gives c = c = 0 for k ≥ 1. Setting i = 0 in (4.22) and comparing coefficients of λj+k+4,
0,k k,0 1
we obtain c = 0 for j,k ≥ 1. Thus, by (4.21), there exists a nonzero complex number c,
j,k
such that
γ′(J ,J ) = cλ3, γ′(J ,J ) = 0, for i, j ≥ 1. (4.24)
λ 0 0 λ i j
Therefore, dimH2(B,C) = 1, which proves (1).
(2) Define an operator τ : C˜q(B,C ) → C˜q−1(B,C ) by
a a
(τγ) (a ,··· ,a ) = (−1)q−1γ (a ,··· ,a ,J )| , (4.25)
λ1,···,λq−1 1 q−1 λ1,···,λq−1,λ 1 q−1 0 λ=0
for a ,··· ,a ∈ B. By the fact that ∂C˜q(B,C ) = (a+ q λ )C˜q(B,C ) and (4.25),
1 q−1 a i=1 i a
P
q
((dτ +τd)γ) (J ,··· ,J ) = λ γ (J ,··· ,J )
λ1,···,λq n1 nq i=1 i λ1,···,λq n1 nq
≡ (cid:0)−Paγ (cid:1) (J ,··· ,J ) (mod ∂C˜q(B,C ).(4.26)
λ1,···,λq n1 nq a
Suppose that γ is a q-cochain such that dγ ∈ ∂C˜q+1(B,C ), namely, there is a (q + 1)-
a
cochain φ such that dγ = (a+ q+1λ )φ. By (4.25), τdγ = (a+ q λ )τφ ∈ ∂C˜q(B,C ).
i=1 i i=1 i a
By(4.26), γ ≡ −d(a−1τγ) is a rPeduced coboundary because a 6= 0P. Thus Hq(B,Ca) = 0 for
q ≥ 0. This proves (2).
10
(3) Note that ∂C˜q(B,M ) = (∂ + q λ )C˜q(B,M ). Similarly to the proof of (2),
∆,α i=1 i ∆,α
we define an operator κ : Cq(B,M∆,α) →PCq−1(B,M∆,α) by
(κγ) (a ,··· ,a ) = (−1)q−1γ (a ,··· ,a ,J )| ,
λ1,···,λq−1 1 q−1 λ1,···,λq−1,λ 1 q−1 0 λ=0
for a ,··· ,a ∈ B. By Theorem 4.2,
1 q−1
((dκ+κd)γ) (J ,··· ,J )
λ1,···,λq n1 nq
q
= J γ (J ,··· ,J )| + λ γ (J ,··· ,J )
0λ λ1,···,λq n1 nq λ=0 i λ1,···,λq n1 nq
(cid:0)iP=1 (cid:1)
q
= ∂ +α+ λ γ (J ,··· ,J )
i λ1,···,λq n1 nq
(cid:0) iP=1 (cid:1)
≡ αγ (J ,··· ,J ) (mod ∂C˜q(B,M )). (4.27)
λ1,···,λq n1 nq ∆,α
If γ is a reduced q-cocycle, then there is a (q + 1)-cochain ϕ such that dγ = ∂ϕ = (∂ +
q+1λ )ϕ. In this case, κdγ = (∂+ q λ )κφ ∈ ∂C˜q(B,M ). It follows from (4.27) that
i=1 i i=1 i ∆,α
Pγ ≡ d(α−1κγ) is a reduced q-cobounPdary, since we assume α 6= 0. Hence Hq(B,M∆,α) = 0
for q ≥ 0.
(cid:3)
This completes the proof of Theorem 4.6.
Corollary 4.7. There is a unique nontrivial universal central extension B˜ = B⊕Cc of the
Lie conformal algebra B, satisfying
[J J ] = (∂ +2λ)J +λ3c,
0λ 0 0
[J J ] = ((i+1)∂ +(i+j +2)λ)J , for i,j > 0.
iλ j i+j
Remark 4.8. The formal distribution Lie algebra corresponding to B˜ is a well-known Lie
algebra of Block type studied in [16, 18].
B
5 Vertex Poisson algebra structure associated to
Denote V = CJ . Thus the Lie conformal algebra of Block type B is a free C[∂]-
i∈Z+ i
module over VL. By (2.4), the λ-bracket (1.1) is equivalent to the following j-products
J J = (i+1)∂J , J J = (i+k +2)J , J J = 0, (5.1)
i(0) k i+k i(1) k i+k i(n) k
for i,k ∈ Z+, n ≥ 2. Define a linear map Y (·,z) from V to z−1(EndV)[[z−1]] by
−
Y (J ,z) = J z−n−1, for i ≥ 0. (5.2)
− i i(n)
nP∈Z+
From (5.1) and (5.2), we have
Y (J ,z)J = (i+1)∂J z−1 +(i+k +2)J z−2, for i,k ∈ Z+. (5.3)
− i k i+k i+k
Extending the map Y−(·,z) to the whole B = C[∂]⊗C V by
m
Y (f(∂)J ,z) ∂mJ = f(d/dz) (−1)l∂m−l(d/dz)lY (J ,z)J , (5.4)
− i k − i k
(cid:0) (cid:1) lP=0