Cohomology of Heisenberg-Virasoro conformal algebra

0.11 MB·

by  Lamei Yuan

Cohomology of Heisenberg-Virasoro conformal algebra 1
Lamei Yuan‡, Henan Wu†
‡ Academy of Fundamental and Interdisciplinary Sciences,
Harbin Institute of Technology, Harbin 150080, China
†School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
6 E-mail: [email protected], [email protected]
1
0 Abstract: Inthispaper,wecomputethecohomologyoftheHeisenberg-Virasoroconformalalgebra
2 with coefficients in its modules, and in particular with trivial coefficients both for the basic and

n reduced complexes.
a
J Keywords: Heisenberg-Virasoro conformal algebra, conformal module, cohomology

6
MR(2000) Subject Classification: 17B65, 17B68
2

]
A 1. Introduction
R
Thenotion of Lieconformalalgebra, introducedby Kacin[5], encodes anaxiomatic description
.
h of the operator product expansion of chiral fields in conformal field theory. In a more general
t context, a Lie conformal algebra is justan algebra in the pseudotensor category [1]. Closely related
a
m to vertex algebras, Lie conformal algebras have many applications in other areas of algebras and
[ integrable systems. In particular, they give us powerful tools for the study of infinite-dimensional

Lie (super)algebras and associative algebras (and their representations), satisfying the sole locality
1
v property [7]. The main examples of such Lie algebras are those based on the punctured complex
7 plane, such as the Virasoro algebra and the loop Lie algebras [4]. In addition, Lie conformal
1
algebras resemble Lie algebras in many ways [6, 9, 10, 12, 13]. A general cohomology theory of
9
6 conformal algebras with coefficients in an arbitrary conformal module was developed in [2], where
0
explicit computations of cohomologies for the Virasoro conformal algebra and current conformal
.
1 algebra were given. The low-dimensional cohomologies of the general Lie conformal algebras gc
N
0
werestudiedin[8]. Thecohomologies oftheW(2,2)-type conformalalgebra withtrivial coefficients
6
1 were completely determined in [11].
: In this paper, we study the cohomology of the Heisenberg-Virasoro conformal algebra, which
v
i was introduced in [10] as a Lie conformal algebra associated with the twisted Heisenberg-Virasoro
X
Lie algebra. By definition, the Heisenberg-Virasoro conformal algebra is free Lie conformal algebra
r
a of rank 2 with a C[∂]-basis {L,M} and satisfying
[L L] = (∂ +2λ)L, [L M]= (∂ +λ)M, [M L] = λM, [M M]= 0. (1.1)
λ λ λ λ
We denote by HV the Heisenberg-Virasoro conformal algebra. It is easily to see that HV contains
the Virasoro conformal algebra Vir as a subalgebra, which is a free C[∂]-module generated by L
such that
Vir =C[∂]L, [L L]= (∂ +2λ)L. (1.2)
λ
Moreover, HV hasanontrivialabelian conformalideal withonefreegenerator M as aC[∂]-module.
Thus it is neither simple nor semi-simple.
The paper is organized as follows. In Section 2, we recall the notions of Lie conformal algebra,
conformal module and cohomology of Lie conformal algebras, and present the main results of this
paper (see Theorem 2.6). Section 3 is devoted to the proof of the main theorem.
1
Corresponding author: Henan Wu ([email protected]).
1
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.
+
2. Preliminaries and Main results
In this section, we recall the definition of a Lie conformal algebra and a (conformal) module
over it and cohomology with coefficients in an arbitrary module. Then we list our main results of
this paper.
Definition 2.1 ([5]) A Lie conformal algebra R is a C[∂]-module endowed with a C-bilinear map
R⊗R → C[λ]⊗R, a⊗b 7→ [a b], and satisfying the following axioms (a,b,c ∈ R),
λ
[∂a b] = −λ[a b], [a ∂b]= (∂ +λ)[a b] (conformal sesquilinearity), (2.1)
λ λ λ λ
[a b] = −[b a] (skew-symmetry), (2.2)
λ −λ−∂
[a [b c]] = [[a b] c]+[b [a c]] (Jacobi identity). (2.3)
λ µ λ λ+µ µ λ
Definition 2.2 ([3]) A moduleV over a Lie conformal algebra A is a C[∂]-module endowed with a
C-bilinear map A⊗V → V[[λ]], a⊗v 7→ a v, satisfying the following relations for a,b ∈ A, v ∈ V,
λ
a (b v)−b (a v) = [a b] v,
λ µ µ λ λ λ+µ
(∂a) v = −λa v, a (∂v) = (∂ +λ)a v.
λ λ λ λ
If a v ∈ V[λ] for all a ∈ A, v ∈ V, then V is called conformal. If V is finitely generated over C[∂],
λ
then V is simply called finite.
Since we only consider conformal modules, we will simply shorten the term “conformal module”
to “module”. The vector space C is viewed as a trivial module with trivial actions of ∂ and A.
For a fixed nonzero complex constant a, there is a natural C[∂]-module C , such that C = C and
a a
∂v = av for v ∈ C . Then C becomes an A-module with A acting by zero.
a a
For theVirasoro conformalalgebra Vir(cf. (1.2)), itwas shownin[3]thatall thefreenontrivial
Vir-modules of rank 1 over C[∂] are the following ones (∆,α ∈ C):
M = C[∂]v, L v = (∂ +α+∆λ)v. (2.4)
∆,α λ
ThemoduleM is irreducibleifandonly if ∆ 6= 0. ThemoduleM contains auniquenontrivial
∆,α 0,α
submodule (∂ +α)M isomorphic to M . Moreover, the modules M with ∆ 6= 0 exhaust all
0,α 1,α ∆,α
finite irreducible nontrivial Vir-modules.
From the proof of [10, Theorem 4.5 (1)], we have
Proposition 2.3 All free nontrivial HV-modules of rank 1 over C[∂] are the following ones:
M = C[∂]v, L v = (∂+α+∆λ)v, M v = βv, for some ∆,α,β ∈ C.
∆,α,β λ λ
Definition 2.4 ([2]) An n-cochain (n ∈ Z ) of a Lie conformal algebra A with coefficients in an
+
A-module V is a C-linear map
γ : A⊗n → V[λ ,··· ,λ ], a ⊗···⊗a 7→ γ (a ,··· ,a )
1 n 1 n λ1,···,λn 1 n
satisfying the following conditions:
(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 (skew-
i i
symmetry).
2
As usual, let A⊗0 = C, so that a 0-cochain is an element of V. Denote by C˜n(A,V) the set of
all n-cochains. The differential d of an n-cochain γ is defined as follows:
(dγ) (a ,··· ,a )
λ1,···,λn+1 1 n+1
n+1
= (−1)i+1a γ (a ,··· ,aˆ,··· ,a )
iP=1 iλi λ1,···,λˆi,···,λn+1 1 i n+1
n+1
+ (−1)i+jγ ([a a ],a ,··· ,aˆ,··· ,aˆ,··· ,a ),(2.5)
i,j=P1;i<j λi+λj,λ1,···,λˆi,···,λˆj,···,λn+1 iλi j 1 i j n+1
where γ is linearly extended over the polynomials in λ . In particular, if γ ∈V is a 0-cochain, then
i
(dγ) (a) = a γ.
λ λ
It was shown in [2] that the operator d preserves the space of cochains and d2 = 0. Thus the
cochains of a Lie conformal algebra A with coefficients in an A-module V form a complex, called
the basic complex and will be denoted by
C˜•(A,V)= C˜n(A,V). (2.6)
nL∈Z+
Moreover, define a (left) C[∂]-module structure on C˜•(A,V) by
n
(∂γ) (a ,··· ,a ) = (∂ + λ )γ (a ,··· ,a ),
λ1,···,λn 1 n V iP=1 i λ1,···,λn 1 n
where ∂ denotes the action of ∂ on V. Then d∂ = ∂d and thus ∂C˜•(A,V) ⊂ C˜•(A,V) forms a
V
subcomplex. The quotient complex
C•(A,V) = C˜•(A,V)/∂C˜•(A,V) = Cn(A,V)
nL∈Z+
is called the reduced complex.
Definition 2.5 The basic cohomology H˜•(A,V) of a Lie conformal algebra A with coefficients in
an A-module V is the cohomology of the basic complex C˜•(A,V) and the (reduced) cohomology
H•(A,V) corresponds to the reduced complex C•(A,V).
The following theorem is our main results of this paper.
Theorem 2.6 For the Heisenberg-Virasoro conformal algebra HV, the following statements hold.
(1) For the trivial module C,
1 if q = 0,

dimH˜q(HV,C)=  3 if q = 3, (2.7)
 2 if q = 4,
0 otherwise,



1 if q = 0,

3 if q = 2,

dimHq(HV,C)=  5 if q = 3, (2.8)

2 if q = 4,

 0 otherwise.



(2) H•(HV,C )= 0 if a 6= 0.
a
3
(3) H•(HV,M )= 0 if α 6= 0.
∆,α
Remark 2.7 Theorem2.6(1) inparticular showsthatthereisauniquenontrivial universalcentral
extension of the Heisenberg-Virasoro conformal algebra HV by a three-dimensional center CC ⊕
1
CC ⊕ CC , which agrees with that of the twisted Heisenberg-Virasoro Lie algebra. The three
2 3
independent reduced 2-cocycle φ¯ , φ¯ and φ¯ are determined by (3.19)–(3.21) respectively, and the
1 2 3
corresponding universal central extension HV of HV is given by
g
λ3
[L L]= (∂ +2λ)L+ C ,
λ 1
12
[L M] = (∂ +λ)M +λ2C ,
λ 2
[M L] = λM −λ2C ,
λ 2
[M M] = λC ,
λ 3
where C ,C ,C are nonzero central elements of HV with ∂C = 0, i = 1,2,3.
1 2 3 i
g
Remark 2.8 Denote by Lie(HV) the annihilation Lie algebra of HV. It can be easily checked
−
that Lie(HV) is isomorphic to the subalgebra spanned by {L ,M −1≤ n ∈Z} of the centerless
− n n
(cid:12)
Heisenberg-Virasoro algebra. Since H˜q(HV,C) ∼= Hq(Lie(HV) ,C)(cid:12), we have actually determined
−
the cohomology group of Lie(HV) with trivial coefficients (cf. [2]).
−
3. Proof of Theorem 2.6
In this section, we prove Theorem 2.6, which will be done by several lemmas.
Keep notations in the previous section. For γ ∈ C˜q(A,V), we call γ a q-cocycle if d(γ) = 0; a
q-coboundary if there exists a (q −1)-cochain φ ∈ C˜q−1(A,V) such that γ = d(φ). Two cochains
γ and γ are called equivalent if γ −γ is a coboundary. Denote by D˜q(A,V) and B˜q(A,V) the
1 2 1 2
spaces of q-cocycles and q-boundaries, respectively. By Definition 2.5,
H˜q(A,V) = D˜q(A,V)/B˜q(A,V) = {equivalent classes of q-cocycles}.
Lemma 3.1 H˜0(HV,C) = H0(HV,C)= C.
Proof. For any γ ∈ C˜0(HV,C) = C, (dγ) (X) = X γ = 0 for X ∈ HV. This means D˜0(HV,C) =
λ λ
C and B˜0(HV,C) = 0. Thus H˜0(HV,C) = C and H0(HV,C) = C since ∂C = 0. (cid:3)
Let γ ∈ C˜q(HV,C) with q > 0. By Definition 2.4, γ is determined by its value on X ⊗···⊗X
1 q
with X ∈ {L,M}. Without loss of generality, we always assume that the first k variables are L
i
and the last q−k variables are M in γ (X ,··· ,X ), since γ is skew-symmetric. Thus we can
λ1,···,λq 1 q
regard γ (X ,··· ,X ) as a polynomial in λ ,··· ,λ , which is skew-symmetric in λ ,··· ,λ
λ1,···,λq 1 q 1 q 1 k
and also skew-symmetric in λ ,··· ,λ . Therefore, γ (X ,··· ,X ) is divisible by
k+1 q λ1,···,λq 1 q
(λ −λ )× (λ −λ ),
i j i j
1≤iQ<j≤k k+1≤Qi<j≤q
whose polynomial degree is k(k−1)/2+(q−k)(q−k−1)/2.
Following [2], we define an operator τ :C˜q(HV,C) → C˜q−1(HV,C) by
∂
(τγ) (X ,··· ,X ) = (−1)q−1 γ (X ,··· ,X ,L)| , (3.1)
λ1,···,λq−1 1 q−1 ∂λ λ1,···,λq−1,λ 1 q−1 λ=0
4
where X = ··· = X = L, X = ··· = X = M. By (2.5), (3.1) and skew-symmetry of γ,
1 k k+1 q−1
((dτ +τd)γ) (X ,··· ,X )
λ1,···,λq 1 q
∂ q
= (−1)q (−1)i+q+1γ ([X L],X ,··· ,Xˆ ,··· ,X )|
∂λiP=1 λi+λ,λ1,···,λˆi,···,λq iλi 1 i q λ=0
∂ q
= γ (X ,··· ,X ,[X L],X ,··· ,X )| . (3.2)
∂λiP=1 λ1,···,λi−1,λi+λ,λi+1,···,λq 1 i−1 iλi i+1 q λ=0
By (1.1) andconformalantilinearity of γ,[X L]can bereplaced byeither (λ −λ)X whenX = L
iλi i i i
or by λ X when X = M in (3.2). Thus, equality (3.2) becomes
i i i
((dτ +τd)γ) (X ,··· ,X )
λ1,···,λq 1 q
∂ k
= (λ −λ)γ (X ,··· ,X ,X ,X ,··· ,X )|
∂λiP=1 i λ1,···,λi−1,λi+λ,λi+1,···,λq 1 i−1 i i+1 q λ=0
∂ q
+ λ γ (X ,··· ,X ,X ,X ,··· ,X )|
∂λ P i λ1,···,λi−1,λi+λ,λi+1,···,λq 1 i−1 i i+1 q λ=0
i=k+1
= (degγ−k)γ (X ,··· ,X ), (3.3)
λ1,···,λq 1 q
where degγ is the total degree of γ in λ ,··· ,λ . Therefore, only those homogeneous cochains
1 q
whose degree as a polynomial is equal to k contribute to the cohomology of C˜•(HV,C). Consider
the quadratic inequality
k(k−1) (q−k)(q−k−1)
+ ≤ k,
2 2
whose discriminant is ∆ = −4k2+12k+1. Since ∆ ≥ 0 has k = 0,1,2 and 3 as the only integral
k k
solutions, we have
0,1 if q = 1,

1,2 if q = 2,
k =  (3.4)
 1,2,3 if q = 3,
3,4 if q = 4.



In particular, H˜q(W,C) = 0 for q ≥5.
Lemma 3.2 Theorem 2.6 (1) holds.
Proof. It needs to compute H˜q(HV,C) for q = 1,2,3,4.
For q = 1, we need to consider k = 0,1 by (3.4). Let γ be a 1-cocycle. From the discussions
below (3.3), we know γ (M) should be a constant, whereas γ (L) should be a constant factor of λ.
λ λ
By dγ = 0, it is easy to check that both γ (M) and γ (L) are zero. Hence, H˜1(HV,C) =0.
λ λ
For q = 2, we need to consider k = 1,2 by (3.4). If γ ∈ H˜2(HV,C), then deg(γ (L,L)) = 2
λ1,λ2
and deg(γ (L,M)) = 1 as polynomials in λ ,λ . By skew-symmetry of γ, γ (L,L) should
λ1,λ2 1 2 λ1,λ2
be a constant factor of λ2 −λ2, which is a coboundary of a 1-cochain of the form ϕ (L) = λ .
1 2 λ1 1
Assume that γ (L,M) = aλ +bλ for some a,b ∈ C. A straightforward computation shows
λ1,λ2 1 2
that
(dγ) (L,L,M) = −aλ (λ +λ )(λ −λ )+aλ λ −aλ λ . (3.5)
λ1,λ2,λ3 1 1 2 1 2 2 3 1 3
Then dγ = 0 gives a = 0. Set ϕ (M) = b. Then (dϕ) (L,M) = bλ = γ (L,M), namely,
λ1 λ1,λ2 2 λ1,λ2
γ (L,M) is also a coboundary. Thus H˜2(HV,C)= 0.
λ1,λ2
5
For q = 3, we need to consider k = 1,2,3 by (3.4). Let γ ∈ D˜3(HV,C) be a 3-cocycle. In the
case of k = 1, we can assume that γ (L,M,M) = c(λ −λ ) for some c ∈ C. One can check
λ1,λ2,λ3 2 3
that it satisfies the following equation
(dγ) (L,L,M,M)
λ1,λ2,λ3,λ4
= c(−(λ −λ )φ (L,M,M)+λ φ (L,M,M)
1 2 λ1+λ2,λ3,λ4 3 λ2,λ1+λ3,λ4
−λ φ (L,M,M)−λ φ (L,M,M)+λ φ (L,M,M))
4 λ2,λ1+λ4,λ3 3 λ1,λ2+λ3,λ4 4 λ1,λ2+λ4,λ3
= c(−(λ −λ )(λ −λ )+λ (λ +λ −λ )
1 2 3 4 3 1 3 4
−λ (λ +λ −λ )−λ (λ +λ −λ )+λ (λ +λ −λ ))
4 1 4 3 3 2 3 4 4 2 4 3
= 0. (3.6)
Anditisanotcoboundary,becauseitcan bethecoboundaryofa2-cochain ϕ (M,M) ofdegree
λ1,λ2
0, which must be zero by skew-symmetry of ϕ. In the case when k = 2, we suppose that
γ (L,L,M) = (λ −λ )(a(λ +λ )+bλ ), for some a,b ∈ C. (3.7)
λ1,λ2,λ3 1 2 1 2 3
It satisfies (dγ) (L,L,L,M) = 0. Setting ϕ (L,M) = aλ , we have
λ1,λ2,λ3,λ4 λ1,λ2 1
(dϕ) (L,L,M) = −a(λ −λ )(λ +λ +λ ),
λ1,λ2,λ3 1 2 1 2 3
and
(dϕ) (L,L,M)+γ (L,L,M) = (b−a)(λ −λ )λ .
λ1,λ2,λ3 λ1,λ2,λ3 1 2 3
Thus γ (L,L,M) is equivalent to (λ −λ )λ , which is not a coboundary by (3.5). According
λ1,λ2,λ3 1 2 3
to [2, Theorem 7.1], γ (L,L,L) = (λ −λ )(λ −λ )(λ −λ ) (up to a constant factor) is
λ1,λ2,λ3 1 2 1 3 2 3
also a 3-cocycle, which is not a coboundary. Therefore, dimH3(HV,C) = 3 and H˜3(HV,C) =
Cφ ⊕Cφ ⊕Cφ , where
1 2 3
φ := φ (L,M,M) = λ −λ , (3.8)
1 1λ1,λ2,λ3 2 3
φ := φ (L,L,M) = (λ −λ )λ , (3.9)
2 2λ1,λ2,λ3 1 2 3
φ := φ (L,L,L) = (λ −λ )(λ −λ )(λ −λ ), (3.10)
3 3λ1,λ2,λ3 1 2 1 3 2 3
andwhere,wetakeφ forexample,theskew-symmetricfunctionφ : HV⊗HV⊗HV → C[λ ,λ ,λ ]
1 1 1 2 3
has values λ −λ on L⊗M ⊗M and 0 on others.
2 3
For q = 4, we need to consider k = 2,3. Let γ ∈ D˜4(HV,C). By skew-symmetry of γ with
whose degree as a polynomial taken into account, we assume that
γ (L,L,M,M) = e (λ −λ )(λ −λ ), for some e ∈ C. (3.11)
λ1,λ2,λ3,λ4 1 1 2 3 4 1
Wehave(dγ) (L,L,L,M,M) = 0.Thusψ := ψ (L,L,M,M) = (λ −λ )(λ −
λ1,λ2,λ3,λ4,λ5 1 1λ1,λ2,λ3,λ4 1 2 3
λ ) is a 4-cocycle. If ψ is the coboundary of a 3-cochain φ (L,M,M) of degree 1, then
4 1 λ1,λ2,λ3
φ (L,M,M) must be a constant factor of λ −λ , whose coboundary is zero by (3.6). Hence
λ1,λ2,λ3 2 3
ψ is not a coboundary. Similarly, assume that
1
γ (L,L,L,M) = e (λ −λ )(λ −λ )(λ −λ ), for some e ∈ C. (3.12)
λ1,λ2,λ3,λ4 2 1 2 2 3 1 3 2
One can check that it satisfies dγ = 0. We should point out that ψ := ψ (L,L,M,M) =
2 2λ1,λ2,λ3,λ4
(λ −λ )(λ −λ )(λ −λ ) is not a coboundary. Because it can be the coboundary of a 3-cochain
1 2 2 3 1 3
6
φ (L,L,M)ofdegree2,whichshouldbeofform(λ −λ )(a(λ +λ )+bλ ),whosecoboundary
λ1,λ2,λ3 1 2 1 2 3
is zero by (3.7). Hence H˜4(HV,C) = Cψ ⊕Cψ .
1 2
By [2, Proposition 2.1], themap γ 7→ ∂γ gives an isomorphism H˜q(HV,C) ∼= Hq(∂C˜•) for q ≥ 1.
Therefore,
C(∂φ )⊕C(∂φ )⊕C(∂φ ) if q = 3,
1 2 3
Hq(∂C˜•)=  C(∂ψ )⊕C(∂ψ ) if q = 4, (3.13)
 1 2
0 otherwise.

The computation of H•(HV,C) is based on the short exact sequence of complexes
0 −−−−→ ∂C˜• −−−ι−→ C˜• −−−π−→ C• −−−−→ 0 (3.14)
where ι and π are the embedding and the natural projection, respectively. The exact sequence
(3.14) gives the following long exact sequence of cohomology groups (cf. [2]):
··· −−−−→ Hq(∂C˜•) −−−ιq−→ H˜q(HV,C) −−−πq−→ Hq(HV,C) −−−ωq−→
(3.15)
−−−−→ Hq+1(∂C˜•) −−ι−q+−1→ H˜q+1(HV,C) −−π−q+−→1 Hq+1(HV,C) −−−−→ ···
where ι ,π are induced by ι,π respectively and w is the q−th connecting hommorphism. Given
q q q
∂γ ∈ Hq(∂C˜•) with a nonzero element γ ∈ H˜q(HV,C) of degree k, we have ι (∂γ) = ∂γ ∈
q
H˜q(HV,C). Since deg(∂γ) = deg(γ)+1 = k+1, ∂γ = 0 ∈ H˜q(HV,C). Thus the image im(ι ) of
q
ι is zero for any q ∈ Z . Because ker(π ) = im(ι ) = {0} and im(ω ) = ker(ι ) = Hq+1(∂C˜•),
q + q q q q+1
we obtain the following short exact sequence
0 −−−−→ H˜q(HV,C) −−−πq−→ Hq(HV,C) −−−ωq−→ Hq+1(∂C˜•) −−−−→ 0. (3.16)
Therefore,
dimHq(HV,C) = dimH˜q(HV,C)+dimHq+1(∂C˜•), for all q ≥ 0. (3.17)
Consequently, weobtain (2.8). By (3.16), thebasis of Hq(HV,C)can beobtained by combiningthe
images of a basis of H˜q(HV,C) with the pre-images of a basis of H˜q+1(HV,C). Let ϕ be a nonzero
(q+1)-cocycle of degree k such that ∂ϕ ∈ Hq+1(∂C˜•). By (3.3), we have
d(τ(∂ϕ) = (dτ +τd)(∂ϕ) = (deg(∂ϕ)−k)(∂ϕ) = ((k+1)−k)(∂ϕ) = ∂ϕ. (3.18)
Thus the pre-image ω−1(∂ϕ) of ∂ϕ under the connecting homorphism ω is τ(∂ϕ), i.e., ω−1(∂ϕ) =
q p q
τ(∂ϕ). Finally, let us finish the proof by giving the basis of Hq(HV,C) for q = 2,3,4. For q = 2, we
have known that H˜2(HV,C)= 0 and H3(∂C˜•)= C(∂φ )⊕C(∂φ )⊕C(∂φ ). By (3.1), (3.8)–(3.10),
1 2 3
φ¯ : = (τ(∂φ )) (M,M)
1 1 λ1,λ2
∂
= (−1)2 (∂φ )) (M,M,L)|
∂λ 1 λ1,λ2,λ λ=0
∂
= (λ +λ +λ)(λ −λ )|
1 2 1 2 λ=0
∂λ
= λ −λ , (3.19)
1 2
φ¯ : = (τ(∂φ )) (L,M)
2 1 λ1,λ2
∂
= (−1)2 (∂φ )) (L,M,L)|
∂λ 1 λ1,λ2,λ λ=0
∂
= − (λ +λ +λ)(λ −λ)λ |
1 2 1 2 λ=0
∂λ
= λ2, (3.20)
2
7
φ¯ : = (τ(∂Λ )) (L,L)
3 3 λ1,λ2
∂
= (−1)2 (∂Λ )) (L,L,L)|
∂λ 3 λ1,λ2,λ λ=0
∂
= (λ +λ +λ)(λ −λ )(λ −λ)(λ −λ)|
1 2 1 2 2 1 λ=0
∂λ
= −λ3+λ3. (3.21)
1 2
Thus H2(HV,C)= Cφ¯ ⊕Cφ¯ ⊕Cφ¯ . For q = 3, we have
1 2 3
ψ¯ : = (τ(∂ψ )) (L,M,M)
1 1 λ1,λ2,λ3
∂
= (−1)3 (∂ψ )) (L,M,M,L)|
∂λ 1 λ1,λ2,λ3,λ λ=0
∂
= (λ +λ +λ +λ)(λ −λ)(λ −λ )|
1 2 3 1 3 2 λ=0
∂λ
= λ2−λ2,
2 3
ψ¯ : = (τ(∂ψ )) (L,L,M)
2 2 λ1,λ2,λ3
∂
= (−1)3 (∂ψ )) (L,L,M,L)|
∂λ 2 λ1,λ2,λ3,λ λ=0
∂
= (λ +λ +λ +λ)(λ −λ )(λ −λ)(λ −λ)|
1 2 3 1 2 1 2 λ=0
∂λ
= −λ3−λ2λ +λ3+λ2λ .
1 1 3 2 2 3
Hence, H3(HV,C)= Cφ ⊕Cφ ⊕Cφ ⊕Cψ¯ ⊕Cψ¯ and H4(HV,C) = Cψ ⊕Cψ by (3.17). (cid:3)
1 2 3 1 2 1 2
Lemma 3.3 Theorem 2.6 (2) holds.
Proof. For a 6= 0, define an operator τ : C˜q(HV,C ) → C˜q−1(HV,C ) by
2 a a
(τ γ) (X ,··· ,X )= (−1)q−1γ (X ,··· ,X ,L)| , (3.22)
2 λ1,···,λq−1 1 q−1 λ1,···,λq−1,λ 1 q−1 λ=0
for X ,··· ,X ∈ {L,M}. By the fact that ∂C˜q(HV,C ) = (a+ q λ )C˜q(HV,C ), we have
1 q−1 a i=1 i a
P
q
((dτ +τ d)γ) (X ,··· ,X ) = λ γ (X ,··· ,X )
2 2 λ1,···,λq 1 q (cid:0)Pi=1 i(cid:1) λ1,···,λq 1 q
≡ −aγ (X ,··· ,X ) (mod ∂C˜q(HV,C ). (3.23)
λ1,···,λq 1 q a
Letγ ∈C˜q(W,C )beaq-cochain suchthatdγ ∈ ∂C˜q+1(HV,C ), namely, thereisa(q+1)-cochain
a a
φ such that dγ = (a+ q+1λ )φ. By (3.22), we have τ dγ = (a+ q λ )τ φ∈ ∂C˜q(HV,C ). It
i=1 i 2 i=1 i 2 a
P P
follows from (3.23) that γ ≡ −d(a−1τ γ) is a reduced coboundary. (cid:3)
2
Lemma 3.4 Theorem 2.6(3) holds.
Proof. In this case, ∂C˜q(HV,M ) = (∂ + q λ )C˜q(HV,M ). Define an operator τ :
∆,α i=1 i ∆,α 3
P
Cq(HV,M )→ Cq−1(HV,M ) by
∆,α ∆,α
(τ γ) (X ,··· ,X )= (−1)q−1γ (X ,··· ,X ,L)| ,
3 λ1,···,λq−1 1 q−1 λ1,···,λq−1,λ 1 q−1 λ=0
for X ,··· ,X ∈ {L,M}. We have
1 q−1
q
((dτ +τ d)γ) (X ,··· ,X ) = L γ (X ,··· ,X )| + λ γ (X ,··· ,X )
3 3 λ1,···,λq 1 q λ λ1,···,λq 1 q λ=0 (cid:0)iP=1 i(cid:1) λ1,···,λq 1 q
q
= ∂+α+ λ γ (X ,··· ,X )
(cid:0) iP=1 i(cid:1) λ1,···,λq 1 q
≡ αγ (X ,··· ,X ) (mod ∂C˜q(HV,M )). (3.24)
λ1,···,λq 1 q ∆,α
8
If γ is a reduced q-cocycle, then, by (3.24), γ ≡ d(α−1τ γ) is a reduced coboundary, since α 6= 0.
3
Thus Hq(HV,M )= 0 for all q ≥ 0. (cid:3)
∆,α
This completes the proof of Theorem 2.6.
Acknowledgements. This work was supported by National Natural Science Foundation grants of China
(11301109,11526125)andtheResearchFundfortheDoctoralProgramofHigherEducation(20132302120042).
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