Second Cohomology of q-deformed Witt superalgebras

0.21 MB·English

by  Faouzi Ammar

Second Cohomology of q-deformed Witt superalgebras
Faouzi Ammar Abdenacer Makhlouf
Universit´e de Sfax Universit´e de Haute Alsace, LMIA,
3 Facult´e des Sciences, 4, rue des Fr`eres Lumi`ere
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B.P. 1171, Sfax 3000,Tunisia F-68093 Mulhouse, France
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2 [email protected] [email protected]

n
Nejib Saadaoui
a
J Universit´e de Sfax

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Facult´e des Sciences,
1
B.P. 1171, Sfax 3000,Tunisia

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A [email protected]
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January 15, 2013
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h
t
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Abstract
[

Thepurposeofthispaperistocomputethesecondadjointcohomologygroupofq-deformed
1
v Witt superalgebras. They are Hom-Lie superalgebras obtained by q-deformation of Witt Lie
4 superalgebra, that is one considers σ-derivations instead of classical derivations.
0
7
2 Introduction
.
1
0
The theory of Hom-Lie superalgebras was introduced in [3]. Representations and a cohomology
3
1 theories of Hom-Lie superalgebraswas providedin[4]. Moreover wehave studiedcentral extensions
:
v and provide as application computations of the derivations and scalar second cohomology group of
i
X q-deformed Witt superalgebras. In this paper, we aim to provide computation of second adjoint
r cohomology group of q-deformed Witt superalgebras.
a
The Witt algebra is one of the simplest infinite dimensional Lie algebra. This Lie algebra of
vector fields, was defined by E. Cartan. It is established that it admits one central extension,
that is Virasoro algebras. The computation of the cohomology and the formal rigidity of Witt
and Virasoro algebras was established by Fialowski [7], see also [8, 10, 11, 13, 14, 35]. For Lie
superalgebras we refer to [30, 32, 33, 34].
In the first Section we review some preliminaries, the cohomology of Hom-Lie superalgebras
and deformation theory. In Section 2, we describe q-Witt superalgebras. The main result, about
second adjoint cohomology of q-deformed Witt superalgebras, is stated in Section 3. Its proof is
given by computing even and odd adjoint second cohomology groups.
1
1 Preliminaries
Let G be a linear superspace over a field K that is a Z -graded linear space with a direct sum
2
G = G ⊕G . The elements of G , j ∈ Z , are said to be homogenous of parity j. The parity of
0 1 j 2
a homogeneous element x is denoted by |x|. The space End(G) is Z -graded with a direct sum
2
End(G) = End (G)⊕End (G), where End (G) = {f ∈ End(G) : f(G ) ⊂ G }. The elements of
0 1 j i i+j
Endj(G) are said to be homogenous of parity j. Let E = ⊕n∈ZEn be a Z-graded linear space, a
linear map f ∈ End(E) is called of degree s if f(E ) ⊂ E , for all n ∈ Z.
n n+s
Definition 1.1. [3]AHom-Lie superalgebrais atriple(G, [.,.], α)consistingofasuperspaceG, an
even bilinear map [.,.] :G×G → G and an even superspace homomorphism α :G → G satisfying
[x,y] =−(−1)|x||y|[y,x], (1.1)
(−1)|x||z|[α(x),[y,z]]+(−1)|z||y|[α(z),[x,y]]+(−1)|y||x|[α(y),[z,x]] = 0, (1.2)
for all homogeneous element x,y,z in G.
1.1 Cohomology of Hom-Lie Superalgebras
Let (G,[.,.],α) be a Hom-Lie superalgebra and V = V ⊕V an arbitrary vector superspace. Let
0 1
G ×V → V
β ∈ Gl(V) be an arbitrary even linear self-map on V and [.,.] : be a bilinear
V
(g,v) 7→ [g,v]
V
map satisfying [G ,V ] ⊂ V where i,j ∈ Z .
i j V i+j 2
Definition 1.2. The triple (V,[.,.] ,β) is called a Hom-module on the Hom-Lie superalgebra
V
G = G ⊕G or G-module V if the even bilinear map [.,.] satisfies
0 1 V
[α(x),β(v)] = β([x,v] ) (1.3)
V V
[[x,y],β(v)] = [α(x),[y,v]] −(−1)|x||y|[α(y),[x,v]] , (1.4)
V V V
for all homogeneous elements x,y in G and v ∈ V.
Hence, we say that (V,[.,.] ,β) is a representation of G.
V
Remark 1.3. When β is the zero-map, we say that the module V is trivial.
Let x ,··· ,x bek homogeneous elements of G. We denote by |(x ,··· ,x )| = |x |+···+|x | (
1 k 1 k 1 k
mod 2) the parity of an element (x ,...,x ) in Gk.
1 k
ThesetCk(G,V)of k-cochains on spaceG withvalues in V,is thesetof k-linear mapsf : ⊗kG → V
satisfying
f(x ,...,x ,x ,...,x ) =−(−1)|xi||xi+1|f(x ,...,x ,x ,...,x ) for 1 ≤ i ≤ k−1.
1 i i+1 k 1 i+1 i k
For k = 0 we have C0(G,V) = V.
The map f is called even (resp. odd) when we have f(x ,...,x ) ∈ V (resp. f(x ,...,x ) ∈ V )
1 k 0 1 k 1
for all even (resp odd ) element (x ,...,x )∈ Gk.
1 k
A k-cochain on G with values in V is defined to be a k-hom-cochain f ∈ Ck(G, V) such that it is
compatible with α andβ in thesensethat β◦f = f◦α, i.e. β◦f(x ,...,x )= f(α(x ),...,α(x )).
1 k 1 k
Denote Ck (G, V) the set of k-hom-cochains:
α,β
Ck (G, V) = {f ∈ Ck(G, V): β ◦f = f ◦α}. (1.5)
α,β
2
Define δk : Ck(G, V)→ Ck+1(G, V) by setting
δk(f)(x ,...,x )
0 k
= (−1)t+|xt|(|xs+1|+···+|xt−1|)f α(x ),...,α(x ),[x ,x ],α(x ),...,x ,...,α(x )
0 s−1 s t s+1 t k
0≤Xs<t≤k (cid:16) (cid:17)
k b
+ (−1)s+|xs|(|f|+|x0|+···+|xs−1|) αk−1(x ),f x ,...,x ,...,x ,
s 0 s k
" #
Xs=0 (cid:16) (cid:17) V
where f ∈ Ck(G, V), |f| is the parity of f, x ,....,x ∈ G abnd x means that x is omitted.
0 k i i
b
Theorem 1.4. [4] Let (G,[.,.],α) be a multiplicative Hom-Lie superalgebra and (V,[.,.] ,β) be a
V
G-Hom-module.
The pair (⊕ Ck (G, V),{δk} ) defines a cohomology complex, that is δk ◦δk−1 = 0.
k>0 α,β k>0
Let (G,[.,.],α) be a multiplicative Hom-Lie superalgebra and (V,[.,.] ,β) be a G-Hom-module.
V
We have with respect the cohomology defined by the coboundary operators
δk : Ck (G,V) −→ Ck+1(G,V).
α,β α,β
• The k-cocycles space is defined as Zk(G,V) = ker δk.
The even (resp. odd ) k-cocycles space is defined as Zk(G,V) = Zk(G,V)∩ (Ck (G,V))
0 α,β 0
(resp. Zk(G,V) = Zk(G,V)∩(Ck (G,V)) .
1 α,β 1
• The k-coboundary space is defined as Bk(G,V)= Im δk−1.
The even (resp. odd ) k-coboundaries space is Bk(G,V) = Bk(G,V)∩(Ck (G,V)) (resp.
0 α,β 0
Bk(G,V) =Bk(G,V)∩(Ck (G,V)) .
1 α,β 1
• The kth cohomology space is the quotient Hk(G,V) = Zk(G,V)/Bk(G,V). It decomposes as
well as even and odd kth cohomology spaces.
Finally, we denote by Hk(G,V) = Hk(G,V) ⊕ Hk(G,V) the set kth cohomology space and by
0 1
⊕ Hk(G,V) the cohomology group of the Hom-Lie superalgebra G with values in V.
k≥0
In the general case, let (G,[.,.],α) be a Hom-Lie superalgebra. We have a 1-coboundary (resp.
2-coboundary operator) defined on G-valued cochains Ck(G,G) such as for x,y,z ∈ G
δ1(f)(x,y) = −f([x,y])+(−1)|x||f|[x,f(y)]−(−1)|y|(|f|+|x|[y,f(x)], (1.6)
δ2(f)(x,y,z) = −f([x,y],α(z))+(−1)|z||y|f([x,z],α(y))+f(α(x),[y,z])
+(−1)|x||f|[α(x),f(y,z)]−(−1)|y|(|f|+|x|)[α(y),f(x,z)]
+(−1)|z|(|f|+|x|+|y|)[α(z),f(x,y)]. (1.7)
A straightforward calculation shows that δ2 ◦δ2 = 0. We denote by H1(G,G) (resp. H2(G,G)) the
corresponding 1st and 2nd cohomology groups.
3
1.2 Deformations of Hom-Lie superalgebras.
In this section we extend to Hom-Lie superalgebras the one-parameter formal deformation theory
introduced by Gerstenhaber [16] for associative algebras. It was extended to Hom-Lie algebras in
[28, 2].
Definition 1.5. Let (G,[.,.],α) be a Hom-Lie superalgebra. A one -parameter formal deformation
of G is given by the K[[t]]-bilinear map [.,.] : G[[t]]×G[[t]] −→ G[[t]] of the form [.,.] = ti[.,.] ,
t t i
i≥0
where each [.,.] is an even bilinear map [.,.] : G ×G −→ G (extended to be K[[t]]-biliXnear) and
i i
[.,.] = [.,.] satisfying the following conditions
0
[x,y] = −(−1)|x||y|[y,x] , (1.8)
t t
(−1)|x||z|[α(x),[y,z] ] +(−1)|z||y|[α(z),[x,y] ] +(−1)|y||x|[α(y),[z,x] ] = 0. (1.9)
t t t t t t
k
The deformation is said of order k if [.,.] = ti[.,.] .
t i
i=0
X
Given two deformations G = (G,[.,.] ,α) and (G′ = G,[.,.]′,α) of G where [.,.] = ti[.,.] and
t t t t t i
i≥0
X
[.,.]′ = ti[.,.]′ with [.,.] = [.,.]′ = [.,.]. We say that G and G′ are equivalent if there exists a
t i 0 0 t t
i≥0
X
formal automorphism φ = φ ti where φ ∈ End(G) and φ = id , such that
t i i 0 G
i≥0
X
φ ([x,y] ) = [φ (x),φ (y)]′.
t t t t t
A deformation G is said to be trivial if and only if G is equivalent to G (viewed as a superalgebra
t t
on G[[t]].)
The identity (1.9) is called deformation equation and it is equivalent to
(cid:9) (−1)|x||z|ti+j[α(x),[y,z] ] = 0,
x,y,z i j
i≥0,j≥0
X
i.e.
(cid:9) (−1)|x||z|ts[α(x),[y,z] ] = 0,
x,y,z i s−i
i≥0,s≥0
X
or
ts (cid:9) (−1)|x||z|ts[α(x),[y,z] ] = 0.
x,y,z i s−i
s≥0 i≥0
X X
The deformation equation is equivalent to the following infinite system
s
(cid:9) (−1)|x||z|[α(x),[y,z] ] = 0, for s =0,1,2,··· (1.10)
x,y,z i s−i
i=0
X
In particular, For s = 0 we have (cid:9) (−1)|x||z|[α(x),[y,z] ] = 0 which is the super Hom-Jacobi
x,y,z 0 0
identity of G.
4
The equation for s = 1, is equivalent to δ2([.,.] ) = 0. Then [.,.] is a 2-cocycle ([.,.] ∈ Z2(G,G).
1 1 1
We deal here with G-valued cohomology. For s ≥ 2, the identities (1.10) are equivalent to:
s−1
δ2([.,.] )(x,y,z) = − (cid:9) (−1)|x||z|[α(x),[y,z] ] .
s x,y,z i s−i
i=1
X
Let (G,[.,.],α) be a Hom-Lie superalgebra and [.,.] be an element of Z2(G,G). The 2-cocycle
1
[.,.] is said to be integrable if there exists a family ([.,.] ) such that [.,.] = ti[.,.] defines a
1 p p≥0 t i
i≥0
X
formal deformation G = (G,[.,.] ,α) of G.
t t
One may also prove:
Theorem 1.6. Let (G,[.,.],α) be a Hom-Lie superalgebra and G = (G,[.,.] ,α) be a one-parameter
t t
formal deformation of G, where [.,.] = ti[.,.] . Then there exists an equivalent deformation
t i
i≥0
X
(G′ = G,[.,.]′,α), where [.,.]′ = ti[.,.]′ such that [.,.]′ ∈Z2(G,G) and doesn’t belong to B2(G,G).
t t t i 1
i≥0
X
Hence, if H2(G,G) = 0 then every formal deformation is equivalent to a trivial deformation.
The Hom-Lie superalgebra is called rigid.
2 The q-Witt superalgebras
Let A = A ⊕A be an associative superalgebra. We assume that A is super-commutative, that is
0 1
for homogeneous elements a,b, the identity ab = (−1)|a||b|ba holds.
Definition 2.1. A σ-derivation D (i ∈ Z ) on A is an endomorphism satisfying:
i 2
D (ab) = D (a)b+(−1)i|a|σ(a)D (b),
i i i
where a,b ∈ A are homogeneous element and |a| is the parity of a.
A σ-derivation D is said to be an even σ-derivation and D is an odd σ-derivation. The set of
0 1
all σ-derivations is denoted by Der (A). Therefore, Der (A) = Der (A) ⊕ Der (A) , where
σ σ σ 0 σ 1
Der (A) (resp Der (A) ) is the space of even (resp. odd) σ-derivations.
σ 0 σ 1
Let A = A ⊕A be a super-commutative associative superalgebra, such A = C[t,t−1] and
0 1 0
A = θA where θ is the Grassman variable (θ2 = 0). We set {n} = 1−qn, a q-number, where
1 0 1−q
q ∈ C\{0,1} and n ∈ N. Let σ be the algebra endomorphism on A defined by
σ(tn)= qntn and σ(θ)= qθ.
Let ∂ and ∂ be two linear maps on A defined by
t θ
∂ (tn) = {n}tn, ∂ (θtn)= {n}θtn,
t t
∂ (tn)= 0, ∂ (θtn) = qntn.
θ θ
5
Lemma 2.2. [3] The linear map ∆ = ∂ +θ∂ on A is an even σ-derivation.
t θ
Hence, ∆(tn)= {n}tn and ∆(θtn)= {n+1}θtn.
Let Wq = A·∆ be a superspace generated by elements L = tn ·∆ of parity 0 and elements
n
G = θtn·∆ of parity 1.
n
Let [−,−] be a bracket on the superspace Wq defined by
[L ,L ]= ({m}−{n})L , (2.1)
n m n+m
[L ,G ] = ({m+1}−{n})G . (2.2)
n m n+m
The others brackets are obtained by supersymmetry or equals 0.
It is easy to see that Wq is a Z−graded algebra Wq = ⊕n∈ZWnq, where Wnq = spanC{Ln,Gn}. The
elements L and G are said of degree n.
n n
Let α be an even linear map on Wq defined on the generators by
α(L ) = (1+qn)L , (2.3)
n n
α(G )= (1+qn+1)G . (2.4)
n n
Proposition 2.3. [3] The triple (Wq,[−,−],α) is a Hom-Lie superalgebra.
In the sequel we refer to this Hom-Lie superalgebras as Wq. We call it q-deformed Witt
superalgebra.
3 Second cohomology H2(Wq,Wq)
In this section, we aim to compute the second cohomology group of Wq with values in itself. For
all q-deformed 1-cocycle (resp 2-cocycle) on Wq we have with respect to (1.6),(1.7), respectively
0 = δ1(f)(x ,x )
0 1
= −f([x ,x ])+(−1)|x0||f|[x ,f(x )]−(−1)|x1|(|f|+|x0|[x ,f(x )], (3.1)
0 1 0 1 1 0
0 = δ2(f)(x ,x ,x )
0 1 2
= −f([x ,x ],α(x ))+(−1)|x2||x1|f([x ,x ],α(x ))+f(α(x ),[x ,x ])
0 1 2 0 2 1 0 1 2
+(−1)|x0||f|[α(x ),f(x ,x )]−(−1)|x1|(|f|+|x0|)[α(x ),f(x ,x )]
0 1 2 1 0 2
+(−1)|x2|(|f|+|x0|+|x1|)[α(x ),f(x ,x )]. (3.2)
2 0 1
Taking the pair (x,y) to be respectively (L ,L ) and (L ,G ) in (3.1), we obtain
n p n p
δ1(f)(L ,L )= −f([L ,L ])+[L ,f(L )]−[L ,f(L ]= 0.
n p n p n p p n
(3.3)
(δ1(f)(Ln,Gp) = −f([Ln,Gp])+[Ln,f(Gp)]−(−1)|f|[Gp,f(Ln)]= 0.
6
Taking the triple (x,y,z) to be (L ,L ,L ), (L ,L ,G ), and (L ,G ,G ) in (3.2), respec-
n m p n m p n m p
tively, we obtain
0 = −f([L ,L ],α(L ))+f([L ,L ],α(L ))+f(α(L ),[L ,L ])
n m p n p m n m p
+[α(L ),f(L ,L )]−[α(L ),f(L ,L )]+[α(L ),f(L ,L )]. (3.4)
n m p m n p p n m
0 = −f([L ,L ],α(G ))+f([L ,G ],α(L ))+f(α(L ),[L ,G ])
n m p n p m n m p
+[α(L ),f(L ,G )]−[α(L ),f(L ,G )]+(−1)|f|[α(G ),f(L ,L )]. (3.5)
n m p m n p p n m
0 = −f([L ,G ],α(G ))−f([L ,G ],α(G ))+f(α(L ),[G ,G ])
n m p n p m n m p
+[α(L ),f(G ,G )]−(−1)|f|[α(G ),f(L ,G )]−(−1)|f|[α(G ),f(L ,G )]. (3.6)
n m p m n p p n m
Our main theorem is
Theorem 3.1. The second cohomology group of q-deformed Witt superalgebras Wq with values in
the adjoint module vanishes, i.e.
H2(Wq,Wq) = {0}.
Hence, every formal deformation is equivalent to a trivial deformation.
Inthesequelweproceedto provethisresultby computingthesecondeven andoddcohomology
groups. We have
H2(Wq,Wq) = H2(Wq,Wq)⊕H2(Wq,Wq)
0 1
where H2(Wq,Wq) (resp. H2(Wq,Wq)) is the even (resp. odd) subspace.
0 1
3.1 Second even cohomology H2(Wq,Wq)
0
We denote by H2 (Wq,Wq) the even subspace of degree s. That is given by even 2-cochains of
0,s
degree s, i.e. for all homogeneous elements x ,x ∈ Wq of degree respectively m,n, f(x ,x ) is of
1 2 1 2
degree m+n+s.
Assume now that f is an even 2-cocycle of degree s. We set
f(L ,L )= a L , f(L ,G ) = b G andf(G ,G )= c L .
n p s,n,p s+n+p n p s,n,p s+n+p n p s,n,p s+n+p
When there is no ambiguity with the degree s, the coefficients a , b , c are denoted by
s,n,p s,n,p s,n,p
a , b , c .
n,p n,p n,p
By (3.4), we have
0 = −(1+qp)({m}−{n})a +(1+qm)({p}−{n})a
n+m,p n+p,m
+(1+qn)({p}−{m})a +(1+qn)({m+p+s}−{n})a (3.7)
n,m+p m,p
−(1+qm)({p+n+s}−{m})a +(1+qp)({n+m+s}−{p})a .
n,p n,m
Therefore by (3.5), we obtain
0 = −(1+qp+1)({m}−{n})b −(1+qm)({p+1}−{n})b
n+m,p m,n+p
+(1+qn)({p+1}−{m})b +(1+qn)({m+p+1+s}−{n})b (3.8)
n,m+p m,p
−(1+qm)({p+n+1+s}−{m})b −(1+qp+1)({p+1}−{n+m+s})a .
n,p n,m
7
Since [G ,G ]= 0, by (3.6), we obtain
n m
0 = −(1+qp+1)({m+1}−{n})c −(1+qm+1)({p+1}−{n})c
n+m,p n+p,m
+(1+qn)({m+p+s}−{n})c . (3.9)
m,p
Proposition 3.2. If s 6= 0, 2, the subspaces H2 (Wq,Wq) vanishe.
0,s
Proof. We define an endomorphism g of Wq by
1 1
g(L ) = f(L ,L ) and g(G ) = f(L ,G ).
p qp{s} 0 p p qp+1{s} 0 p
By (3.3) and (2.1) we have
δ1(g)(L ,L ) = −{p}g(L )+{p+s}g(L ).
0 p p p
So
δ1(g)(L ,L )= qp{s}g(L ).
0 p p
We define a 2-cocycle h by
h= f −δ1(g).
Therefore
h(L ,L ) = f(L ,L )−δ1(g)(L ,L ) = qp{s}g(L )−qp{s}g(L )= 0. (3.10)
0 p 0 p 0 p p p
Taking m = 0 in (3.4), with (2.1) and (2.3) we obtain
0 = (1+qp+1){n}f(L ,L )+2({p}−{n})f(L ,L )+(1+qn){p}f(L ,L )
n p n+p 0 n p
+(1+qn)[L ,f(L ,L )]−2[L ,f(L ,L )]+(1+qp)[L ,f(L ,L )]. (3.11)
n 0 p 0 n p p n 0
Since h is a 2-cocycle, we can replace f by h in (3.11), and using (3.10) we obtain
0 = (1+qp){n}h(L ,L )+(1+qn){p}h(L ,L )−2{s+n+p}h(L ,L ).
n p n p n p
From this, using the fact that s is not vanishing, we obtain
h(L ,L )= 0 ∀ n,p ∈ Z. (3.12)
n p
Since g(G )= 1 f(L ,G ), by (3.3) and (2.2) we have
p qp+1{s} 0 p
δ1(g)(L ,G )= −{p+1}g(G )+{p+s+1}g(G )= qp+1{s}g(G ).
0 p p p p
Then
h(L ,G ) = f(L ,G )−δ1(g)(L ,G )= 0.
0 p 0 p 0 p
Taking m = 0 in (3.5), by (2.1), (2.2), (2.3) and (2.4), we obtain
(1+qp){n}f(L ,G )+2({p+1}−{n})f(G ,L )+(1+qn){p+1}f(L ,G )
n p p+n 0 n p
+(1+qn)[L ,f(L ,G )]−2[L ,f(L ,G )]+(−1)|f|(1+qp+1)[G ,f(L ,L )] = 0.
n 0 p 0 n p p n 0
8
Since h is a 2-cocycle and h(L ,G ) = h(L ,L ) = 0 we can deduce
0 p 0 n
(1+qp+1){n}h(L ,G )+(1+qn){p+1}h(L ,G )−2{p+n+s+1}h(L ,G ) =0,
n p n p n p
which implies, under the condition s 6= 0, that
h(L ,G )= 0. (3.13)
n p
Taking n = 0 in (3.6), since [G ,G ]= h(L ,G ) = h(L ,L ) = 0 and h is a 2-cocycle we have
m p n p n p
−h([L ,G ],α(G ))−h([L ,G ],α(G ))+[α(L ),h(G ,G )] = 0.
0 m p 0 p m 0 m p
Then
−(1+qp+1){m+1}h(G ,G )−(1+qm+1){p+1}h(G ,G )+2{m+p+s}h(G ,G ) = 0,
m p p m m p
which implies, under the condition s 6= 2, that h(G ,G ) = 0.
m p
It follows that h ≡ 0. Hence f is a coboundary.
Lemma 3.3. Let f be an even 2-cocycle of degree zero (s = 0) and g be an even endomorphism of
Wq. If h = f −δ1(g) then
h(L ,L )= 0, h(L ,G )= 0, h(L ,L ) = 0 and h(L ,G )= 0 ∀n∈ Z, m ∈ Z∗.
−1 2 −1 1 n 1 1 m
′
Proof. Let f be an even 2-cocycle. Then f(L ,L ) = f L and f(L ,G )= f G .
n m n,m n+m n m n,m n+m
Let (an)n∈Z be the sequence given recursively by
a = f ,
0 0,1
1
a = a + f ∀n< 0,
n n+1 n,1
{1}−{n}
a = 0,
1
1
a = f −a ,
2 −1,2 −1
{2}−{−1}
1
a = a + f ∀n≥ 2.
n+1 n n,1
{n}−{1}
Let (bm)m∈Z be the sequence given recursively by
b = 0,
0
1 ′
b = b − f ∀m< 0,
m m+1 {1}−{m+1} 1,m
1 ′
b = −a − f ,
1 −1 {−1}−{2} −1,1
1
b = b + f′ ∀m≥ 1.
m+1 m {1}−{m+1} 1,m
Let g be an even endomorphism of Wq given by g(L ) = a L and g(G ) = b G .
n n n n n n
By (3.3) and (3.3) we have recursively
δ1(g)(L ,L ) = ({n}−{m})(a −a −a )L ,
n m n+m m n n+m
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and
δ1(g)(L ,G ) = ({n}−{m+1})(b −b −a )G .
n m n+m m n n+m
If h =f −δ1(g) we have
h = f −({n}−{1})(a −a )= 0 ∀n∈ Z,
n,1 n,1 n+1 n
h = f −({−1}−{2})(a −a −a ) =0,
−1,2 −1,2 1 2 −1
h′ = f′ −({1}−{m+1})(b −b ) = 0 ∀m∈ Z∗,
1,m 1,m m+1 m
′ ′
h = f +({−1}−{2})(b +a ) = 0.
−1,1 −1,1 1 −1
This proves the lemma.
Lemma 3.4. Let f be an even 2-cocycle of degree zero such that f(L ,L ) = 0 and f(L ,L ) = 0.
n 1 −1 2
q q
Then the cohomology class of f is trivial on the space W ×W .
0 0
Proof. Since f is an even 2-cocycle of degree zero, by (3.7) we have
−(1+qp)({m}−{n})a +(1+qm)({p}−{n})a +(1+qn)({p}−{m})a
n+m,p n+p,m n,m+p
+(1+qn)({m+p}−{n})a −(1+qm)({n+p}−{m})a +(1+qp)({n+m}−{p})a
m,p n,p n,m
= 0. (3.14)
Taking m = −1 in (3.14), we have
−(1+qp)({−1}−{n})a +(1+q−1)({p}−{n})a +(1+qn)({p}−{−1})a
n−1,p n+p,−1 n,−1+p
+(1+qn)({−1+p}−{n})a −(1+q−1)({n+p}−{−1})a +(1+qp)({n−1}−{p})a
−1,p n,p n,−1
= 0. (3.15)
Setting m = 1 in (3.14), since a = −a = 0 ∀k ∈Z, we have
1,k k,1
−(1+qp)({1}−{n})a +(1+qn)({p}−{1})a −(1+q)({n+p}−{1})a = 0.(3.16)
n+1,p n,1+p n,p
We investigate the following cases:
Case 1: k = 0
In (3.16) we consider p = 0 this gives
−2({1}−{n})a −(1+q)({n}−{1})a = 0. (3.17)
n+1,0 n,0
That is,
2 1+q
a = a , a = a , ∀n6= 1.
n,0 n+1,0 n+1,0 n,0
1+q 2
Starting from a = −a = 0 this implies for n ≤ 0 that a = 0 and for n ≥ 3 that
1,0 0,1 n,0
a = 1+q n−2a .
n,0 2 2,0
Next we consider (3.15) for n = 0, p = 2. It follows
(cid:0) (cid:1)
−(1+q2){−1}a +(1+q−1){2}a +2({2}−{−1})a
−1,2 2,−1 0,1
+2{1}a −(1+q−1)({2}−{−1})a +(1+q2)({−1}−{2})a
−1,2 0,2 0,−1
= 0.
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