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COHOMOLOGY SPACE OF aff(n|1) IN THE LIE
SUPERALGEBRA OF
SUPERPSEUDODIFFERENTIAL OPERATORS
Othmen Ncib
To cite this version:
SUPERALGEBRA OF SUPERPSEUDODIFFERENTIAL OPERATORS
OTHMEN NCIB
Abstract. We investigate the second cohomology space of the Lie superalge-bras aff(n|1) on the superline R1|n, for n = 1, 2 in the superspace of weighted densities and in the lie superalgebra of superpseudodifferential operators. We explicitly give cocycles spanning this cohomology spaces.
1. Introduction
The theory of cohomology and deformations of homomorphisms has been applied in several mathematical and physical domains specially for classical mechanics. For this we find many recent papers on this theory [2, 3, 4, 9, 10, 14, 15, 16, 17, 19]. According to Nijenhuis-Richardson, the space H1(g, End(V )), where g is a Lie (su-per) algebra and V is a g-module, classifies the infinitesimal deformations of the g-module V , while the obstructions are elements of H2(g, End(V )). In this paper, we studied the computation of the second cohomology space of the affine Lie su-peralgebra aff(n|1) on the (1, n)-dimensional superspace R1|n with coefficients in the Lie superalgebra SΨDO(n) of superpseudodifferential operators with smooth coefficients.
Let Fn
λ be the aff(n|1)-module of weighted densities on R
1|n of weight λ, where the action is given by the Lie derivative. The Lie superalgebra SΨDO(n) is also an aff(n|1)-module. The action is given by the natural embedding of aff(n|1) in SΨDO(n).
In [6], O. Basdouri calculated the spaces Hn(aff(0|1), Fλ) and Hn(aff(0|1), ΨDO), n = 1, 2. We computed the second cohomology spaces H2(aff(n|1), Fn
λ) and
H2(aff(n|1), SΨDO(n)), n = 1, 2, that we interested to H2(aff(n|1), Grp(SΨDO(n))), where Grp(SΨDO(n)) is the graded module isomorphic, as aff(n − 1|1)-module, to a direct sum of modules of weighted densities. In this context, we apply the theory of spectral sequences to a filtered module over a Lie algebra given by Ovsienko and Roger.
2. Definitions and Notations 2.1. Pseudodifferential operators on R.
Key words and phrases. superpseudodifferential operators, Lie superalgebra, cohomology,
Weighted densities, symbols.
To every pseudodifferential operator F onR one associates its symbol ([11, 12, 13]). Let ΨDO(1) be the space of pseudodifferential symbols on R ([16, 17]). The order of ΨDO(1) is defined to be
ord(F ) ={sup k ∈ Z|fk(x)̸= 0} for any F (x, ξ, ξ−1) =∑ k∈Z
ξkfk(x)∈ ΨDO(1),
where fk(x) ∈ C∞(R) with fk = 0 for k sufficiently large. By means of the two natural derivations on ΨDO(1)
(2.1) ∂ξ :∑ k fkξk 7−→∑ k kfkξk−1 and ∂x:∑ k fkξk 7−→∑ k fk′ξk one defines a natural Poisson bracket:
(2.2) {F, G} = ∂F ∂ξ ∂G ∂x − ∂F ∂x ∂G ∂ξ f or any F, G∈ ΨDO(1).
On the space of pseudodifferential symbols an associative algebra structure is de-fined by the rule ([1])
(2.3) F◦ G =∑ k≥0 1 k! : ∂kF ∂ξk ∂kG ∂xk : where : . : stands for the “normal ordering” defined as (2.4) : f (x)ξkg(x)ξℓ:= f (x)g(x)ξk+ℓ.
This is a natural generalization of the Wick product. As usual, we can also define the Lie bracket associated with the◦–product [F, G] = F ◦ G − G ◦ F ; we denote the Lie algebra with this structure by ΨDO(1)L to distinguish from the Poisson Lie algebra structure.
2.2. The Lie superalgebra of contact vector fields onR1|n. Let R1|n be the superspace with local coordinates (x; θ
1, . . . , θn), where θ = (θ1, . . . , θn) are the odd variables. Any contact structure onR1|n can be given by the following 1-form:
(2.5) αn = dx +
n ∑ i=1
θidθi.
On the space C∞(R1|n), we consider the contact bracket (2.6) {F, G} = F G′− F′G−1
2(−1) |F |∑n
i=1
ηi(F ).ηi(G),
where the superscript ’ stands for ∂
∂x, ηi = ∂θ∂i − θi
∂
∂x and|F | is the parity of F . Note that the derivations ηi are the generators of n-extended supersymmetry and generate the kernel of the form (2.5) as a module over the ring of smooth functions. Let Vect(R1|n) be the superspace of vector fields onR1|n:
and consider the superspaceK(n) of contact vector fields on R1|n. That is, K(n) is the superspace of vector fields onR1|npreserving the distribution singled out by the 1-form αn:
K(n) ={X ∈ Vect(R1|n)| there exists F ∈ C∞(R1|n) such that LX(αn) = F αn }
. where LX is the Lie derivative along the vector field X.
The Lie superalgebraK(n) is spanned by the fields of the form:
XF = F ∂x− 1 2 n ∑ i=1 (−1)|F |ηi(F )ηi, where F∈ C∞(R1|n).
The bracket inK(n) can be written as:
[XF, XG] = X{F, G}.
For every contact vector field XF, one define a one-parameter family of first-order differential operators on C∞(R1|n):
(2.7) LλXF = XF+ λF′, λ∈ R. We easily check that
(2.8) [LλXF, L
λ XG] = L
λ X{F,G}.
We thus obtain a one-parameter family of K(n)-modules on C∞(R1|n) that we denote Fn
λ, the space of all weighted densities on C∞(R1|n) of weight λ with respect to αn: (2.9) Fnλ= { F αλn | F ∈ C∞(R 1|n ) } .
In particular, we have F0λ=Fλ. Obviously the adjointK(n)-module is isomorphic to the space of weighted densities on C∞(R1|n) of weight −1.
InK(n), there is a subalgebra aff(n|1) of affine transformations aff(n|1) = Span(X1, Xx, Xθi, Xθiθj
)
, 1≤ i, j ≤ n.
called the affine Lie Superalgebra on the (1, n)−dimensional superspace R1|n. The case n = 0 corresponds to the classical setting: K(0) = Vect(R) = {F ∂x|F ∈ C∞(R)} and the corresponding affine Lie algebra aff(0|1) is nothing but the classical Lie algebra aff(1) which is isomorphic to the Lie subalgebra of Vect(R) generated by{∂x, x∂x}. Note that, the Lie superalgebra aff(n − 1|1) is isomorphic to
aff(n− 1|1)i={XF ∈ aff(n|1) | ∂iF = 0}. Therefore, the spaces of weighted densities Fn
λ are also aff(n− 1|1)-modules. In [8] it was proved that, as aff(n− 1|1)-module, we have
(2.10) Fnλ ≃ Fnλ−1⊕ Π(Fnλ+−11 2
2.3. Superpseudodifferential operators onR1|n.
Let T∗R1|nbe the cotangent bundle onR1|nwith local coordinates (x, θ1,· · · , θn, ξ, ¯θ1,· · · , ¯θn), where |¯θi| = 1. The superspace of the supercommutative alge-bra SP(n) of pseudodifferential symbols on R1|n with its natural multiplication is spanned by the series
SP(n) = ∞ ∑ k=−M ∑ ϵ=(ϵ1,··· ,ϵn) ak, ϵ(x, θ)ξ−kθ¯ϵ1 1 ...¯θ ϵn n | ak, ϵ∈ R[x, θ]; ϵi = 0, 1; M ∈ N , where θ = (θ1, . . . , θn). This space has a Lie superalgebra structure given by the following Poisson bracket:
{A, B} = ∂ξA∂xB− ∂xA∂ξB− (−1)|A| 2 ∑ i=1 ( ∂iA∂θ¯iB + ∂θ¯iA∂iB ) , where ∂x=∂x∂ , ∂ξ= ∂ξ∂, ∂i= ∂θ∂i and ∂¯θ=∂ ¯∂θi.
Of courseSP(0) is the classical spaces of symbols, usually denoted P = { F (x, ξ)| F (x, ξ) = m ∑ k=−∞ fk(x)ξk } .
The associative superalgebra of pseudodifferential operators SΨD(n) on R1|n has the same underlying vector space asSP(n), but the multiplication is defined by the following rule:
A◦ B = ∑ α≥0, νi=0, 1 (−1)νi(|A|+1) α! (∂ α ξ∂ νi ¯ θiA)(∂ α x∂ νi θiB).
Denote by SΨDO(n) the Lie superalgebra with the same superspace as SΨD(n) and the supercommutator defined on homogeneous elements by:
[A, B] = A◦ B − (−1)|A||B|B◦ A. Of courseSΨDO(0) = ΨDO(1)L.
3. The structure of SP(n) as an aff(n|1)-module The natural embedding of aff(n|1) into SΨDO(n)
(3.1) π : aff(n|1) −→ SΨDO(n)
defined by π(XF) = F ξ−12∑ni=1(−1)|F |ηi(F )¯ζi, where ¯ζi= ¯θi− θiξ, for n = 1, 2, and π(XF) = F ξ for n = 0
induces an aff(n|1)-module structure on SΨDO(n).
Setting deg x = deg θi = 0, deg ξ = deg ¯θi = 1 for all i, we endow the Poisson superalgebraSP(n) with a Z-grading:
(3.2) SP(n) = g⊕
(3.3) SPp(n) =
{
F ξ−p+G1ξ−p−1θ¯1+G2ξ−p−1θ¯2+· · ·+H1,2ξ−p−2θ¯1θ¯2+· · · | F, Gi, Hi,j ∈ C∞(R1|n) } is the homogeneous subspace of degree −p. Each element of SΨDO(n) can be
expressed as A =∑ k∈Z (Fk+ G1kξ−1θ¯1+· · · + Hk1,2ξ−2θ¯1θ¯2+· · · )ξ−k, where Fk, Gik, H i,j k ∈ C∞(R
1|n). We define the order of A to be ord(A) = sup{k | Fk ̸= 0 or Gik̸= 0 or H
i,j k ̸= 0}.
This definition of order equipsSΨDO(n) with a decreasing filtration as follows: set Fp={A ∈ SΨDO(n) | ord(A) ≤ −p},
where p∈ Z. So we have
(3.4) . . .⊂ Fp+1⊂ Fp⊂ . . .
This filtration is compatible with the multiplication and the super Poisson bracket, that is, for A ∈ Fp and B ∈ Fq, one has A◦ B ∈ Fp+q and {A, B} ∈ Fp+q−1. This filtration makesSΨDO(n) an associative filtered superalgebra. Moreover, this filtration is compatible with the natural aff(n|1)-action on SΨDO(n). Indeed,
XF(A) = [XF, A]∈ Fp f or any XF ∈ aff(n|1) and A ∈ Fp.
The induced aff(n|1)-module structure on the quotient Fp/Fp+1 is isomorphic to that of the aff(n|1)-module SPp(n). Therefore,
SP(n) ≃ g⊕
p∈ZFp/Fp+1. 3.1. Cohomology Spaces .
Let g = g0⊕g1be a Lie superalgebra acting on a super vector space V = V0⊕V1. The space of k-cochaines with values in V is the g-module
Ck(g, V ) := Hom(∧kg; V ).
The coboundary operator δk : Ck(g, V ) −→ Ck+1(g, V ) is a g-map satisfying δk ◦ δk−1= 0. The kernel of δk, denoted Zk(g, V ), is the space of k-cocycles, among them, the elements in the range of δk−1are called k-coboundries. We denote Bk(g, V ) the space of k-coboundries.
By definition, the kthcohomology space is the quotient space Hk(g, V ) = Zk(g, V )/Bk(g, V ).
We will only need the formula of δn (which will be simply denoted δ) in degrees 0, 1 and 2: for v∈ C0(g, V ) = V, δv(x) := (−1)|x||v|x.v, for Υ∈ C1(g, V ),
(3.5) δ(Υ)(x, y) := (−1)|x||Υ|x.Υ(y)− (−1)|y|(|x|+|Υ|)y.Υ(x)− Υ([x, y]), and for Ω∈ C2(g, V ),
(3.6)
δ(Ω)(x, y, z) := (−1)|Ω||x|x.Ω(y, z)− (−1)|y|(|Ω|+|x|)y.Ω(x, z) + (−1)|z|(|Ω|+|x|+|y|)z.Ω(x, y) −Ω([x, y], z) + (−1)|y||z|Ω([x, z], y)− (−1)|x|(|y|+|z|)Ω([y, z], x),
Hom(g, V ) isZ2-graded via
(3.7) Hom(g, V )b=⊕a∈Z2Hom(ga, Va+b); b∈ Z2.
4. The second cohomology space H2(aff(n|1), Fnλ), where n = 0, 1, 2. 4.1. The second cohomology space Hdif f2 (aff(1),Fλ) .
Theorem 4.1. [6]
Hdif f2 (aff(1),Fλ) = {
R if λ = 0 ; 0 otherwise. This space spanned by the following nontrivial 2-cocycle
(4.1) ω0(Xf, Xg) = (f g′− f′g) where Xf, Xg∈ aff(1).
4.2. The second cohomology space H2
dif f(aff(1|1), F1λ) . Theorem 4.2. Hdif f2 (aff(1|1), F1λ) = { R if λ = 0 ; 0 otherwise. This space spanned by the following nontrivial 2-cocycle
(4.2) C(XF, XG) = (F G′− F′G) +1 2(−1)
|F ||G|(F′η
1(G)− η1(F )G′)θ1 where XF, XG∈ aff(1|1).
To prove the theorem 4.2, we need the following results
Proposition 4.3. [7] Any 2-cocycle C∈ Zdif f2 (aff(1|1), F1λ) has the following gen-eral form C(XF, XG) = 2 ∑ i,j=0 αijηi1(F )ηj1(G)αλ1,
where αij are functions depending only on θ1 and the parity of F and G. The dependence on the parity of F and G is given by the followin relation
αij(F, G) = (−1)εij(F,G)αji(G, F ),
where εij(F, G) = ij(p(F ) + 1)(p(G) + 1) + p(F )p(G) + 1.
Lemma 4.4. Any 2-cocycle C ∈ Zdif f2 (aff(1|1), F1λ) is a coboundary if and only if its restriction to aff(1) is a coboundary
Proof. We study separately the even and the odd cases and explain here only the even case. The odd case treated similarly.
to aff(1), is a 2-coboundary.
Reciprocally, if C/aff(1) is a 2-coboundary then
C(Xf, Xg) = δ(b)(Xf, Xg), f orall Xf, Xg∈ aff(1), where b is a map from aff(1) into F1λ.
By replaycing C by C− δ(b), we can suppose that C/aff(1)= 0. But, in this case, by considering the equations:
δ(C)(X, Y, Z) = 0; X, Y, Z∈ {X1, Xx, Xθ1},
we deduce the expression of the coefficients αij. Similarly we study the odd case. We find the following results
C(XF, XG) = { β [( F′η1(G)− η1(F )G′ ) θ1−12(−1)|F ||G|η1(F )η1(G) ] , if λ = 0 0, otherwise.
But, we check that C = δ(b) where b is given by b(XF) = 1
2η1(F )θ1
Proof of theorem 4.2 :
According to proposition 4.3, the isomorphism (2.10), n=1 and the theorem 4.1 we conclude that the nontrivial cohomology space H2(aff(1|1), F1
λ) only can appear if λ∈ {−12, 1}.
Now, we apply the aff(1)-isomorphism H2(aff(1), F1λ)≃ H 2 (aff(1), fλ)⊕ H2 ( aff(1), Π(fλ+1 2) ) ,
and the theorem 4.1 we find up to a coboundary and up to a scalar factor the restriction of any 2-cocycle C to aff(1). As before, we consider separately the even and the odd cases. Even cohomology spaces only can appear if λ = 0 and odd cohomology spaces only can appear if λ =−12. In each case, we have the restriction of C to aff(1), and we complete the expression obtained by the corresponding other terms having the same parity of the last expression then we apply the 2-cocycle condition we get C(XF, XG) = α [ (F G′− F′G) +12(−1)|F ||G| ( F′η1(G)− η1(F )G′ ) θ1 ] + β [( F′η1(G)− η1(F )G′ ) θ1−12(−1)|F ||G|η1(F )η1(G) ] , if λ = 0 0, otherwise. where α, β∈ R.
for any map b from aff(1|1) into F1λ having the general form b(XF) = 2 ∑ i=0 aiηi1(F ),
where ai∈ C∞(R1|1). Indeed the term (f0g′0− f0′g0) appear in the expression of C2 and not appear in the expression of δ(b).
4.3. The second cohomology space H2(aff(2|1), F2λ) .
Recall that the Lie superalgebra aff(2|1) has two subsuperalgebras aff(1|1)i for i = 1, 2 isomorphic to aff(1|1) defined by
aff(1|1)i ={XF ∈ aff(1|1)/∂θ3−iF = 0, F ∈ R1|1}
Denote by ℑiλ the aff(1|1)i-module of tensor densities of degree λ on R1i|1, where R1|1
i is the superline with local coordinates (x, θi), i = 1, 2.
Theorem 4.5. 1) The cohomology space
H2(aff(1|1)i, F2λ)0= {
R if λ = 0 ; 0 otherwise. This space spanned by the following nontrivial 2-cocycle
(4.3) Ωi0(XF, XG) = (F G′− F′G) + 1 2(−1) |F ||G|(F′η i(G)− ηi(F )G′)θi, where XF, XG∈ aff(1|1)i.
2) The cohomology space
H2(aff(1|1)i, F2λ)1= {
R if λ = −1 2 ; 0 otherwise. This space spanned by the following nontrivial 2-cocycle (4.4) Ωi−1 2 (XF, XG) = [ (F G′− F′G)θi+1 2(−1) |F ||G|(F′η
i(G)− ηi(F )G′)θiθ3−i ]
α−
1 2
1,i, where XF, XG∈ aff(1|1)i and α−
1 2
1,i = dx + θidθi, i = 1, 2. Proof. The map
Φ : F2 λ −→ ℑiλ⊕ Π(ℑiλ+1 2 ) F αλ 2 7−→ (
(1− θ3−i∂θ3−i)(F )αλ1,i, Π ( (−1)|F |+1∂θ3−i(F )αλ+ 1 2 1,i )) , where i = 1, 2 and Π stands for the parity change map, provides us with an iso-morphism of aff(1|1)i-modules. In fact, we easily check that
This map induces the following isomorphism between cohomology spaces: H2(aff(1|1)i, F2λ)≃ H 2 (aff(1|1)i, ℑiλ)⊕ H 2 (aff(1|1)i, Π(ℑiλ+1 2 )).
From this isomorphism and the theorem 4.2, we conclude the 2-cocycles (4.3–4.4). In [7] we can show the following Lemma.
Lemma 4.6. Up to a coboundary, any 2-cocycle Ω∈ Z2
dif f(aff(n|1), Fnλ) has the following general form
Ω(XF, XG) = 2 ∑ i1,...,in;j1,...,jn=0 ai1,...,in;j1,...,jnη i1 1...η in n (F )η j1 1...η jn n (G)α λ n, where ai1,...,in;j1,...,jn depending only on θi and the parity of F and G.
Proposition 4.7. Up to a scalar factor and a coboundary, there exist only one
nontrivial 2-cocycle eΩ∈ H2(aff(2|1), F2
λ), given by eΩ(XF, XG) = ( η1η2(F )G− F η1η2(G) ) −1 2(−1)|F ||G| [ (η1η2(F )η1(G)− η1(F )η1η2(G))θ1+ (η1η2(F )η2(G)− η2(F )η1η2(G))θ2 ] such that its restriction to aff(1|1)i, i = 1, 2 , is coboundary.
Proof. Let Ω ∈ Z2
dif f(aff(2|1), F 2
λ), such that its restriction Ω/aff(1|1)1 is a
2-coboundary :
Ω(XF, XG) = δ(b)(XF, XG), f or all XF, XG∈ aff(1|1)1.
By replacing Ω by Ω− δ(b), we can suppose that Ω/aff(1|1)1 = 0. But, in this case
by studing separately the even and the odd cases for each subspace aff(2|1), then applying the 2-cocycle condition one gets:
Ω = a1Ω1+ a2Ω2, where Ω1(XF, XG) = ( η1η2(F )G− F η1η2(G) ) − 1 2(−1)|F ||G| ( η1η2(F )η1(G)− η1(F )η1η2(G) ) θ1 −1 2(−1)|F ||G| ( η1η2(F )η2(G)− η2(F )η1η2(G) ) θ2, Ω2(XF, XG) = (−1)|F |+|G| [ (η1η2(F )η2(G)− η2(F )η1η2(G))θ1+ (η1(F )η1η2(G)− η1η2(F )η1(G))θ2 ] , and a1, a2∈ R.
Note that any nonrivial 2-cocycle Ω∈ Z2
dif f(aff(2|1), F 2
λ) is a coboundary if there exist a map b from aff(2|1) into F2
λ having the general form
(4.5) b(XF) = 2 ∑ i,j=0 βijηi1η j 2(F ) ,
where βij ∈ C∞(R1|2), such that Ω = δ(b). By a direct computation, we check that :
Ω2(XF, XG) = δ(b)(XF, XG), where b(XF) =−η1η2(F )θ1θ2.
Thus, Ω1 is not a coboundary and its restrictions to aff(1|1)i are identically zeros. The main result in this part is the following :
Theorem 4.8. The cohomology space
Hdif f2 (aff(2|1), F2λ) = {
R2 if λ = 0 ; 0 otherwise. The non trivial 2-cocycles spanned this space are :
Ω0(XF, XG) = (F G′− F′G)−12(−1)|F |+|G| [( η1(F )G′− F′η1(G) ) θ1+ ( η2(F )G′− F′η2(G) ) θ2 ] , eΩ0(XF, XG) = ( η1η2(F )G− F η1η2(G) ) −1 2(−1)|F ||G| [( η1η2(F )η1(G)− η1(F )η1η2(G) ) θ1+ ( η1η2(F )η2(G)− η2(F )η1η2(G) ) θ2 ] where XF, XG∈ aff(2|1). Proof. Let Ω∈ Z2 dif f(aff(2|1), F 2
λ), such that up to a scalar factor and a coboundary its restrictions to aff(1|1)i, i = 1, 2. As before we separately the even and the odd cases. By the theorem 4.5, the even cohomology spaces only can appear if λ = 0 and the odd cohomology spaces only can appear if λ =−1
2. In each case, we compute the restriction of Ω to both aff(1|1)1and aff(1|1)2, then we deduce the general form of its expression by adding to the obtained expression the corresponding other terms with the same homogeneity and parity. The 2-cocycle condition leads to
Ω(XF, XG) = α [ (F G′− F′G)−12(−1)|F |+|G| ( (η1(F )G′− F′η1(G))θ1 +(η2(F )G′− F′η2(G))θ2 )] +βΩ2(XF, XG) if λ = 0, 0 otherwise ,
where α, β∈ R and Ω2is defined in the proposition 4.7.
We proved that, the 2-cocycle Ω2(XF, XG) = δ(b)(XF, XG), where b(XF) =−η1η2(F )θ1θ2. According to the expression 4.5, and by a direct computation we deduce that there is not exist a map b from aff(2|1) into F2λ, such that the 2-cocycle
(F G′− F′G)−12(−1)|F |+|G| (
(η1(F )G′− F′η1(G))θ1+ (η2(F )G′− F′η2(G))θ2 )
= δ(b)(XF, XG), for all XF, XG ∈ aff(2|1), indeed, the term f0g0′ exist in the expres-sion of this 2-cocycle but not exist in the expresexpres-sion of δ(b).
In the proposition 4.7, we proved that, any nontrivial 2-cocycle from aff(2|1) with coefficients in F2
λsuch that its restriction to aff(1|1)i, i = 1, 2, is trivial if and only if it coincides, up o a scalar factor and coboundary, to eΩ0, where eΩ0 is given in
proposition 4.7. Theorem 4.8 is proved.
5. The second cohomology space H2(aff(n|1), SP(n)), where n = 0, 1, 2. 5.1. The second cohomology space H2(aff(1),P) .
Theorem 5.1. [6]
Hdif f2 (aff(1),Pn) = {
R if n = 0 ; 0 otherwise. The nontrivial 2-cocycle spanned this space is
(5.1) ω(Xf, Xg) = (f g′− f′g) where Xf, Xg∈ aff(1).
5.2. The second cohomology space H2(aff(1|1), SP(1)) . The main result in this subsection is the following
Theorem 5.2. The cohomology space
Hdif f2 (aff(1|1), SPn(1)) = {
R if n = 0 ; 0 otherwise. The nontrivial 2-cocycle spanned this space is
(5.2) Λ(XF, XG) = ( 1 + (−1)|F |+|G| )[ (F G′− F′G) +12(−1)|F ||G| ( F′η1(G)− η1(F )G′ ) θ1 ] + ( 1 + 12(−1)|F |+|G| )[ η1(F )G′− F′η1(G) ] ζ1ξ−1. where XF, XG∈ aff(1|1).
To prove the theorem 5.2, we need the following results :
Proposition 5.3. [4] The aff(1|1)-module SPn(1) has a direct sum decomposition of the two aff(1|1)-modules, SP1
n(1) andSP 2 n(1), defined by SP1 n(1) ={(1 + (−1)|F |)F ξ−n+ η1(F )ξ−n−1ζ1/F ∈ C∞(R1|2)} and SP2 n(1) ={F ξ−n−1ζ1/F ∈ C∞(R1|2)} and the maps
(5.3) φ1: F1n −→ SPn1(1) and φ2: π(F1n+1 2 ) −→ SP2 n(1) F αn 1 7−→ ( 1 + (−1)|F |)F ξ−n+ η1(F )ξ−n−1ζ1, π(F α n+1 2 1 ) 7−→ F ξ−n−1ζ1, provides us with isomorphisms of aff(1|1)-modules.
Proof of theorem 5.2 :
5.3. The second cohomology space Hdif f2 (aff(2|1), SP(2)) . In this subsection we need the following results :
Proposition 5.4. [5] 1) As an aff(1|1)i-module, i = 1, 2, we have
SPn(2)≃ F2n⊕ Π(F2n+1 2 ⊕ F 2 n+1 2 )⊕ F2n+1 for n = 0,−1. 2) For n̸= 0, −1 :
a) The following subspace of SPn(2) :
(5.4) SPn, i(2) =
BF(n,i) = F θ3−iξ−n−1ζ¯3−i+ θ3−iη¯3−i(F )ξ−n−2ζi¯ζ¯3−i − 1 2θ3−iη¯i(F )ξ−n−2ζ¯iζ¯3−i| F ∈ C∞(R 1|2) , where ¯ζi= ¯θi− θiξ, is an aff(1|1)i- module, i = 1, 2, isomorphic to F2
n+1. b) As an aff(1|1)i-module we have SPn(2)/SPn, i(2)≃ F2n⊕ Π(F2n+1 2 ⊕ F 2 n+1 2 ), i = 1, 2.
Note that for the each of the cases n = 0, and n =−1, the aff(1|1)i-moduleSPn(2) with the grading (3.2) is the direct sum of four aff(1|1)i-modules, i = 1, 2.
For n = 0, the four aff(1|1)i-modules are SP(0, 0, i)(2) = { A(0,0,i)F = F | F ∈ C∞(R1|2) } , SP(0, 1 2, i)(2) = A(0, 1 2, i)
F = θiF + F ¯θiξ−1− 2∂θ3−i(θ3−iF )¯θiξ−1
+ ∂θi(θ3−iF )¯θ3−iξ−1+ F′θ3−iθi¯θ¯3−iξ−2| F ∈ C∞(R1|2) , g SP(0, 1 2, i)(2) = e A(0, 12, i)
F = 2∂θi(θiF )¯θ3−iξ−1+ ∂θ3−i(F θi)¯θ3−iξ−1
− ∂θ3−i∂θi(F θiθ3−i)¯θ3−iξ−1+ θiθ3−i∂θ3−i∂θiF ¯θ3−iξ−1
+ (−1)|F |+1(∂θiF ξ−2θ¯iθ¯3−i− ∂θ3−iF ξ−2θ¯iθ¯3−i) + F′θiξ−2θi¯θ¯3−i| F ∈ C∞(R1|2)
, SP(0, 1, i)(2) =
A(0, 1, i)F = F θ3−iξ−1ζ¯3−i+ θ3−iη¯3−i(F )ξ−2ζi¯ζ¯3−i − 1
2θ3−iηi¯(F )ξ−2ζi¯ζ¯3−i| F ∈ C∞(R 1|2)
For n =−1, the four aff(1|1)i-modules are SP(−1, 0, i)(2) = { A(F−1, 0, i)= F ξ +(−1)2|F |+1 ( ¯ η1(F )¯ζ1+ ¯η2(F )¯ζ2 ) | F ∈ C∞(R1|2)} , SP(−1,1 2, i)(2) = A(−1,12, i)
F = F ¯ζi+ ¯ηi(θ3−iF )¯θ3−i+ ∂θ3−i(θiF )¯θ3−i + (−1)|F |+1∂θ3−iF ¯θiθ¯3−iξ−1 | F ∈ C∞(R1|2)
, g SP(−1, 1 2, i)(2) = { e A(−1, 1 2, i)
F = F ¯ζi+ F ¯θ3−i+ ¯ηi(θ3−iF )¯θ3−i| F ∈ C∞(R1|2) }
,
SP(−1, 1, i)(2) = {
A(F−1, 1, i) = F θ3−iζ¯3−i+ θ3−iη¯3−i(F )ξ−1ζi¯ζ¯3−i − 1
2θ3−iη¯i(F )ξ−1ζ¯iζ¯3−i| F ∈ C∞(R1|2) }
.
The action of aff(1|1)i-module onSP(n, j, i)(2) (resp. gSP(n, 1
2, i)(2)) is given by XF.A(n, j, i)G ={πi(XF), A (n, j, i) G } = A (n, j, i) Ln+j XF(G) for j = 0, 1 2, 1 and n = 0,−1, where, G = g0+g1θ1+g2θ2+g12θ1θ2∈ C∞(R1|2) and πiis the canonical embedding of aff(1|1)i intoSP(2) defined by
πi(XF) = F ξ + (−1)|F |+1 2 ( ¯ η1(F ) ¯ζ1+ ¯η2(F ) ¯ζ2 ) , where F = f0+ fiθi. For j = 0,12, 1 and n = 0,−1, let F αn+j2 = (f0+ f1θ1+ f2θ2+ f12θ1θ2)αn+j2 ∈ F
2 n+j. Then, The natural maps
(5.5)
ψi
n, j: F2n+j −→ SP(n, j, i)(2) (resp. gSP(n, 1 2, i)(2))
F αn+j2 7−→ A(n, j, i)F ,
provide us with isomorphisms of aff(1|1)i-modules, i = 1, 2. The main result in this subsection is the following.
Theorem 5.5. The cohomology space
Hdif f2 (aff(2|1), SPn(2)) = {
R2 if n =−1, 0 ; 0 otherwise. For n =−1, the nontrivial 2-cocycles are:
For n = 0, the nontrivial 2-cocycles are: Υ3(XF, XG) = (F G′− F′G) +12(−1)|F |+|G| [( η1(F )G′− F′η1(G) ) θ1+ ( η2(F )G′− F′η2(G) ) θ2 ] , Υ4(XF, XG) = ( η1η2(F )G− F η1η2(G) ) +12(−1)|F |+|G| [( η1η2(F )η1(G)− η1(F )η1η2(G) ) θ1 + ( η1η2(F )η2(G)− η2(F )η1η2(G) ) θ2 ] where XF, XG∈ aff(2|1).
To prove the theorem 5.5, we need the following Proposition:
Proposition 5.6. There exist, up to a scalar factor and a coboundary, two
non-trivial 2-cocycles Φ1 and Φ2 from aff(2|1) to SP(2), given by
Φ1(XF, XG) = ( η1η2(F )G− F η1η2(G) ) +12(−1)|F |+|G| [( η1η2(F )η1(G)− η1(F )η1η2(G) ) θ1 + ( η1η2(F )η2(G)− η2(F )η1η2(G) ) θ2 ] , Φ2(XF, XG) = ( η1η2(F )G′− F′η1η2(G) ) ζ1ζ2ξ−1
such that any nonzero linear combination is a nontrivial 2-cocycle and their restric-tions to aff(1|1)i, i = 1, 2, are coboundaries.
Proof. According to the lemma 4.6, any differential 2-cocycle in Z2(aff(2|1), SP(2)) has the following general form
Λ(XF, XG) = +∞ ∑ n=−M,1≤i,j≤2 ai,j;k,lηi1ηj2(F )ηk1ηl2(G)ζ1ζ2ξ−n + +∞ ∑ n=−M,1≤i,j≤2 bi,j;k,lηi1η j 2(F )η k 1η l 2(G)ζ1ξ−n+1 + +∞ ∑ n=−M,1≤i,j≤2 ci,j;k,lηi1ηj2(F )ηk1ηl2(G)ζ2ξ−n+1 + +∞ ∑ n=−M,1≤i,j≤2 di,j;k,lηi1η j 2(F )η k 1η l 2(G)ξ−n+2
where M ∈ N and ai,j;k,l, bi,j;k,l, ci,j;k,l, di,j;k,lare functions depending only on θi and the parity of F and G. Assume that Λ/aff(1|1)1 is a 2-coboundary:
Λ(XF, XG) = δ(b)(XF, XG), f or all Xf, XG∈ aff(1|1)1.
By replacing Λ by Λ− δ(b), we can suppose that Λ/aff(1|1)1 = 0. But, in this case
by studing separately the even and the odd cases for each subspaceSPn(2), then applying the 2-cocycle condition one gets:
Φ1(XF, XG) = ( η1η2(F )G− F η1η2(G) ) +12(−1)|F |+|G| [( η1η2(F )η1(G)− η1(F )η1η2(G) ) θ1 + ( η1η2(F )η2(G)− η2(F )η1η2(G) ) θ2 ] , Φ2(XF, XG) = ( η1η2(F )G′− F′η1η2(G) ) ζ1ζ2ξ−1, Φ3(XF, XG) = 12 [( η1η2(F )G′− F′η1η2(G) ) θ1ζ1+ (−1)|F |+|G| ( η1(F )η1η2(G)− η1η2(F )η1(G) ) ζ1ξ−1 ] , Φ4(XF, XG) = 12 [( η1η2(F )G′− F′η1η2(G) ) θ2ζ2+ (−1)|F |+|G| ( η2(F )η1η2(G)− η1η2(F )η2(G) ) ζ2ξ−1 ] , and a1, a2, a3, a4∈ R.
Note that any differential 2-cocycle Ω ∈ Z2(aff(2|1), SP(2)) is a coboundary if there exist a map b from aff(2|1) into SP(2) having the general form
b = M ∑ k=−∞ 0≤i,j≤2 0≤ε≤3 αi,jk,εξkηi1ηj2+ M ∑ k=−∞ 0≤i,j≤2 0≤ε≤3 βk,εi,jθ¯1ξk−1ηi1η j 2+ M ∑ k=−∞ 0≤i,j≤2 0≤ε≤3 γk,εi,jθ¯2ξk−1ηi1η j 2+ M ∑ k=−∞ 0≤i,j≤2 0≤ε≤3 µi,jk,εθ¯1θ¯2ξk−2ηi1η j 2,
where αi,jk,ε, βk,εi,j, γk,εi,j, µi,jk,ε ∈ C∞(R1|2), such that Ω = δ(b). So, by a direct computation we check that :
(i) Φ3= δ(−η1η2(F )θ1θ1ξ−1) and Φ4= δ(−η1η2(F )θ2θ2ξ−1).
(ii) Φ1and Φ2are not coboundaries. Indeed the terms f0g12and f0′g12are exists respectively in Φ1 and Φ2 but not in δ(b). Moreover, it’s easy to see that Φ1 and Φ2are not cohomologous. Thus we complete the proof.
Proof of theorem 5.5
By the Proposition 5.6, any nontrivial 2-cocycle from aff(2|1) with coefficients in SP(2) such that its restriction to aff(1|1)i, i = 1, 2, is trivial if and only if it coincides, up to a scalar factor and a coboundary, to Φ1and Φ2, denoted respectively Υ4and Υ1.
Now, according to Theorem 4.5 and Proposition 5.4, one get, up to a scalar factor and a coboundary, cocycles such that their restriction to aff(1|1)i, i = 1, 2, are nontrivials. Therefore, the space H2
dif f(aff(1|1)i,SPn(2)), i = 1, 2, has the following structure dimension:
For n = −1, the space Hdif f2 (aff(1|1)i,SP−1(2)) is spanned by the following nontrivvial 2-cocycles: T1−1,i(XF, XG) = ψi−1,1 ( Ωi0(XF, XG) ) , T2−1,i(XF, XG) = ψi−1,1 2 ( Π(Ωi−1 2 (XF, XG)) ) , e T2−1,i(XF, XG) = eψi−1,1 2 ( Π(Ωi−1 2 (XF, XG)) ) . For n = 0, the space H2
dif f(aff(1|1)i,SP0(2)) is spanned by the following non-trivvial 2-cocycles: T10,i(XF, XG) = ψi0,0 ( Ωi0(XF, XG) ) , where Ωi 0 and Ωi−1 2
, are defined in (4.3)-(4.4) and ψi
n,j, eψn,ji are as in (5.5). We prove by direct computation that:
For n =−1: Only the 2-cocycle T1−1,i that extend to a nontrivial 2-cocycle of aff(2|1) with coefficients in SP(2). Its extention is the 2-cocycle Υ2of Theorem 5.5. For n = 0: The 2-cocycle T10,igives a nontrivial extention in H2(aff(2|1), SP(2)), defined by the nontrivial 2-cocycle Υ3of Theorem 5.5.
6. The space H2
dif f(aff(n|1), SΨDO(n)) for n = 0, 1, and 2.
6.1. The spectral sequence for a filtered module over a Lie (super)algebra.
The reader should refer to [18], for the details of the homological algebra used to construct spectral sequences. We will merely quote the results for a filtered module M with decreasing filtration{Mn}n∈Z over a Lie (super)algebra g so that
Mn+1⊂ Mn, ∪n∈ZMn= M and gMn⊂ Mn .
Consider the natural filtration induced on the space of cochains by setting: Fn(C∗(g, M )) = C∗(g, Mn),
then we have:
dFn(C∗(g, M ))⊂ Fn(C∗(g, M )) (i.e., the filtration is preserved by d); Fn+1(C∗(g, M ))⊂ Fn(C∗(g, M )) (i.e. the filtration is decreasing). Then there is a spectral sequence (Er∗,∗, dr) for r∈ N with dr of degree (r, 1− r) and
E0p,q= Fp(Cp+q(g, M ))/Fp+1(Cp+q(g, M )) and E1p,q= H p+q
(g, Grp(M )). To simplify the notations, we have to replace Fn(C∗(g, M )) by FnC∗. We define
Zrp,q= FpCp+q∩d−1(Fp+rCp+q+1), Brp,q= FpCp+q∩d(Fp−rCp+q−1),
Erp,q= Zrp,q/(Zrp+1,q−1 −1+ Brp,q−1). The differential d maps Zp,q
r into Zrp+r,q−r+1, and hence includes a homomorphism dr: Erp,q−→ E
The spectral sequence converges to H∗(C, d), that is
E∞p,q ∼= FpHp+q(C, d)/Fp+1Hp+q(C, d),
where FpH∗(C, d) is the image of the map H∗(FpC, d)→ H∗(C, d) induced by the inclusion FpC→ C.
6.2. Computing H2
dif f(aff(2|1), SΨDO(n)). Since the cohomology space H2
dif f(aff(2|1), SΨDO(n)) is upper bounded by H2(aff(2|1), SP(n)), we can check that the behavior of the cocycles with values inSΨDO(n) under the successive differentials of the spectral sequence. More pre-cisely, we consider a cocycle with values in SP(n) but we compute its boundary as it was inSΨDO(n) and keep the symbolic part of the result. This gives a new cocycle of degree equal to the degree of the previous one plus one. We iterate this procedure by computing supplementary higher order terms until we find cocycles given in the following results.
Theorem 6.1. [6]
Hdif f2 (aff(1), ΨDO) ≃ R
It is spanned by the classes of the following nontrivial 2-cocycles
(6.1) ∆(Xf, Xg) = f g′− f′g where Xf, Xg∈ aff(1).
Theorem 6.2. The spaces H2
dif f(aff(n|1), SΨDO(n)), where n = 1, 2, are purely even. An explicit description of these spaces is the following:
1.
Hdif f2 (aff(1|1), SΨDO(1)) ≃ R
It is spanned by the classes of the following nontrivial 2-cocycles (6.2) Ξ(XF, XG) = ( 1 + (−1)|F |+|G| )[ (F G′− F′G) +12(−1)|F ||G| ( F′η1(G)− η1(F )G′ ) θ1 ] + ( 1 +12(−1)|F |+|G| )[ η1(F )G′− F′η1(G) ] ζ1ξ−1. 2.
(6.3) Ψ1(XF, XG) = ( η1η2(F )G′− F′η1η2(G) ) ζ1ζ2ξ−1 Ψ2(XF, XG) = (F G′− F′G)ξ +14(−1)|F |+|G| [( F′η1(G)− η1(F )G′ ) ζ1+ ( F′η2(G)− η2(F )G′ ) ζ2 ] +14 [( F′η1η2(G)− η1η2(F )G′ ) θ2ζ1+ ( η1η2(F )G′− F′η1η2(G) ) θ1ζ2 ] +12 [( η1(F )G′− F′η1(G) ) θ1ξ + ( η2(F )G′− F′η2(G) ) θ2ξ ] Ψ3(XF, XG) = (F G′− F′G) +12(−1)|F |+|G| [( η1(F )G′− F′η1(G) ) θ1+ ( η2(F )G′− F′η2(G) ) θ2 ] Ψ4(XF, XG) = ( η1η2(F )G− F η1η2(G) ) +1 2(−1)|F |+|G| [( η1η2(F )η1(G)− η1(F )η1η2(G) ) θ1 + ( η1η2(F )η2(G)− η2(F )η1η2(G) ) θ2 ] + (−1)|F |+|G| [( η1(F )η1η2(G)− η1η2(F )η1(G) ) ζ1 + ( η2(F )η1η2(G)− η1η2(F )η2(G) ) θ2 ] + ( η1η2(F )G′− F′η1η2(G) )( θ1ζ1+ θ2ζ2 ) ACKNOWLEDGEMENTS
It is our duty to thank the professors Mabrouk Ben Ammar, Salem Omri and Claude Roger for their helpful discussions.
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Faculty of Sciences of Gafsa, Zarroug 2112; Gafsa-Tunisia
