Comptes Rendus

Comptes Rendus

Mathématique

Hafedh Khalfoun and Ismail Laraiedh

The linear n

–invariant differential operators and n

–relative cohomology

Volume 358, issue 1 (2020), p. 45-58.

<https://doi.org/10.5802/crmath.22>

© Académie des sciences, Paris and the authors, 2020.

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2020, 358, n1, p. 45-58

https://doi.org/10.5802/crmath.22

Lie Algebras, Mathematical Physics /

Algèbres de Lie, Physique mathématique

The linear n _{(1 |} N )–invariant di ff erential operators and n _{(1 |} N )–relative cohomology

Opérateurs di ff érentiels linéaires n (1 | N )–invariants et cohomology n (1 | N )–relative

Hafedh Khalfoun^a,band Ismail Laraiedh^a,b

aDépartement de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie

bDepartement of Mathematics, College of Sciences and Humanities - Kowaiyia, Shaqra University, Kingdom of Saudi Arabia.

E-mails:[email protected], [email protected].

Abstract.Over the (1,N)-dimensional supercircleS^1|N, we classifyn(1|N)-invariant linear differential oper- ators acting on the superspaces of weighted densities onS^1|N, wheren(1|N) is the Heisenberg Lie superal- gebra. This result allows us to compute the first differentialn(1|N)-relative cohomology of the Lie superal- gebraK(N) of contact vector fields with coefficients in the superspace of weighted densities. ForN=0, 1, 2, we investigate the firstn_(1|N)-relative cohomology space associated with the embedding ofK(N) in the superspace of the supercommutative algebraS P(N) of pseudodifferential symbols onS^1|Nand in the Lie superalgebraSΨDO(S^1|N) of superpseudodifferential operators with smooth coeffcients. We explicity give 1-cocycles spanning these cohomology spaces.

Résumé.Sur le supercercle (1,N)-dimensionnelS^1|N, nous classifions les opérateurs différentiels linéaires n(1|N)-invariant agissant sur les densités tensorielles surS^1|N, oùn(1|N) est la superalgèbre de Lie de Heisenberg. Ce résultat permet de calculer le premier espace de cohomologie différentielsn(1|N)-relative de la superalgèbre de Lie des champs de vecteurs de contactK(N) à coefficients dans le superespace des densités tensorielles. PourN=0, 1, 2, nous etudions le premier espace de cohomologien(1|N)-relative de K(N) dans le superespace de l’algèbre supercommutativeS P(N) des symboles pseudodifférentiels sur S^1|Net dans la superalgèbre de LieSΨDO(S^1|N) des opérateurs superpseudodifférentiels. Nous donnons explicitement les 1-cocycles engendrent ces espaces de cohomologie.

Mathematical subject classification (2010).53D55, 14F10, 17B10, 17B68.

Manuscript received 19th June 2019, revised 7th August 2019 and 29th October 2019, accepted 6th January 2020.

1. Introduction

Let Vect(S¹) is the Lie algebra of smooth vector fields on the circleS¹. Consider the 1-parameter deformation of the Vect(S¹)-action onC_C^∞(S¹):

L^λ

X_dx^d(f)=X f⁰+λX⁰f,

where X,f ∈C_C^∞(S¹) and X⁰:= ^dX_dx. Denote by Fλ the Vect(S¹)-module structure on C_C^∞(S¹) defined byL^λfor a fixedλ. Geometrically,F_λ=©

fdx^λ|f ∈C_C^∞(S¹)ª

is the space of weighted densities of weightλ∈R. The spaceF_λcoincides with the space of vector fields, functions and differential 1-forms forλ= −1, 0 and 1, respectively.

Denote by D_λ,µ:=Hom_diff(F_λ,Fµ) the Vect(S¹)-module of linear differential operators with the natural Vect(S¹)-action denotedL^λ_X^,µ(A). Each module D_λ,µ has a natural filtration by the order of differential operators; the graded moduleS_λ,µ:=gr D_λ,µis called thespace of symbols.

The quotient-module D^k_λ,µ/D^k−1_λ,µ is isomorphic to the module of weighted densitiesF_µ−λ−k; the isomorphism is provided by the principal symbol mapσrdefined by:

A=

k

X

i=0

a_i(x) µ ∂

∂x

¶i

7→σpr(A)=a_k(x)(dx)^µ−λ−k,

We study the classification of n(1|N)-invariant linear differential operators onS^1|N acting in the spacesF_λ^N. Ovsienko and Roger [11] calculated the space H¹(Vect(S¹),ΨDO(S¹)), where Vect(S¹) is the Lie algebra of smooth vector fields on the circleS¹ andΨDO(S¹) is the space of pseudodifferntial operators. The action is given by the natural embedding of Vect(S¹) in ΨDO(S¹). They used the results of D. B. Fuks [5] on the cohomology of Vect(S¹) with coeffi- cients in tensor densities to determine the cohomology with coefficients in the graded mod- ule Grad(ΨDO(S¹)), namely H¹(Vect(S¹), Grad^p(ΨDO(S¹))); here Grad^p(ΨDO(S¹)) is isomor- phic, as Vect(S¹)-module, to the space of tensor densitiesFp of degreep on S¹. To compute H¹(Vect(S¹),ΨDO(S¹)), V. Ovsienko and C. Roger applied the theory of spectral sequences to a filtered module over a Lie algebra.

In this paper we consider the superspaceS^1|Nequipped with the contact structure determined by a 1-formαN, and the Lie superalgebraK(N) of contact vector fields onS^1|N. We introduce the K(N)-moduleF^N_λ ofλ-densities onS^1|Nand theK(N)-module of linear differential operators, D^N_λ,µ:=Hom_diff(F_λ^N,F^N_µ), which are super analogues of the spacesFλand D_λ,µ, respectively. We classify alln(1|N)-invariant linear differential operators onS^1|Nacting in the spacesF^N_λ. We use the result to compute H¹_diff(K(N),n(1|N),F^N_λ). We show that, the non-zero cohomology only appear for resonant values of weights. Moreover, we give explicit bases of these cohomology spaces. ForN =0, 1, 2, we follow again the same methods by V. Ovsienko and C. Roger [11] to compute then(1|N)-relative cohomology H¹(K(N),n(1|N),SΨDO(S^1|N)), wheren(1|N) is the Heisenberg Lie superalgebra, andSΨDO(S^1|N) is the space of superpseudodifferential operators onS^1|N. Moreover, we give explicit bases of these cohomology spaces.

2. Definitions and notations

In this section, we recall the main definitions and facts related to the geometry of the superspace S^1|N; for more details, see [6, 7, 8, 9, 10].

2.1.

The Lie superalgebra of contact vector fields on S

We define the supercircleS^1|Nin terms of its superalgebra of functions, denoted byC_C^∞(S^1|N) and consisting of elements of the form:

F=

N

X

s=0

X

1≤i1<i2<···<is≤N

fi₁i₂...i_s(x)θi₁. . .θi_s,

wheref_i1i2...is∈C_C^∞(S¹), and wherexis the even indeterminate,θ1, . . . ,θNare the odd indetermi- nates, i.e.,θiθj= −θjθi. Consider the standard contact structure given by the following 1-form:

αN=dx+

N

X

i=1

θidθi. On the spaceC_C^∞(S^1|N), we consider the contact bracket

{F,G}=F G⁰−F⁰G−1

2(−1)^p(F)

N

X

i=1

ηi(F)·ηi(G), whereηi=_∂θ^∂_i−θi ∂

∂xandp(F) is the parity ofF. Let Vect_C(S^1|N) be the superspace of vector fields onS^1|N:

Vect_C(S^1|N)= (

F₀∂x+

N

X

i=1

F_i∂i

¯

¯

¯

¯

¯

F_i∈C_C^∞(S^1|N) )

,

where∂i=_∂θ^∂_i and∂x=_∂^∂_x, and consider the superspaceK(N) of contact vector fields onS^1|N: K(N)=©

X∈Vect_C(S^1|N)¯

¯there existsF∈C_C^∞(S^1|N) such thatL_X(αN)=FαNª , The Lie superalgebraK(N) is spanned by the fields of the form:

X_F=F∂x−1

2(−1)^p(F)

N

X

i=1

ηi(F)ηi, whereF∈C_C^∞(S^1|N).

In particular, we haveK(0)=Vect_C(S¹). The bracket inK(N) can be written as:

[XF,XG]=X{F,G}.

The Lie superalgebraK(N−1) can be realized as a subalgebra ofK(N):

K(N−1)={X_F∈K(N)|∂NF=0} . Note also that, for anyiin {1, 2, . . . ,N},K(N−1) is isomorphic to

K(N−1)ⁱ={X_F∈K(N)|∂iF=0} .

2.2.

The Heisenberg subalgebra n

The Heisenberg Lie superalgebran(1|N) can be realized as a subalgebra ofK(N):

n(1|N)=Span¡ X₁,X_θi¢

, 1≤i≤N. We easily see thatn(1|N−1) is a subalgebra ofn(1|N):

n(1|N−1)={X_F∈n(1|N−1)|∂NF=0} . Note also that, for anyiin {1, 2, . . . ,N−1},n(1|N−1) is isomorphic to

n(1|N−1)ⁱ={XF∈n(1|N−1)|∂iF=0} .

2.3.

Modules of weighted densities

For every contact vector fieldX_F, define a one-parameter family of first-order differential opera- tors onC^∞_C(S^1|N):

L^λ_X

F=X_F+λF⁰, λ∈C. We easily check that

h L^λ_X

F,L^λ_X

G

i

=L^λ_X

{F,G}.

We thus obtain a one-parameter family ofK(N)-modules onC_C^∞(S^1|N) that we denoteF^N_λ, the space of all weighted densities onS^1|Nof weightλwith respect toαN:

F_λ^N₌ⁿFα^λ_N¯

¯

¯F∈C^∞(S^1|N)o .

2.4.

Di

erential operators on weighted densities

A differential operator onS^1|Nis an operator onC_C^∞(S^1|N) of the form:

A=

M

X

k=0

X

ε=(ε1,...,εN)

a_k,²(x,θ)∂^kx∂^ε₁¹. . .∂^ε_N^N;εi=0, 1;M∈N.

Of course any differential operator defines a linear mappingFα^λ_N7→(AF)α^µ_N fromF^N_λ toF_µ^Nfor anyλ,µ∈R, thus the space of differential operators becomes a family ofK(N)-modulesD_λ^N_,µfor the natural action:

XF·A=L^µ_X

F◦A−(−1)^p(A)p(F)A◦L^λ_X

F. Every differential operatorA∈D^N_λ,µcan be expressed in the form

A(Fα^λ_N)= X

`=(`1,...,`N)

a_{`</sub>(x,θ)η<sup>`</sup>₁¹. . .η<sup>`</sup><sub>N</sub><sup>N</sup>(F)α<sup>µ</sup><sub>N</sub>, where the coefficientsa<sub>`}(x,θ) are arbitrary functions.

Lemma 1 ([2]). As aK(N−1)-module, we have D^N_λ,µ'D^N_λ,µ⁻¹_⊕D^N−1

λ+¹2,µ+¹2⊕Π µ

D^N−1

λ,µ+¹2⊕D^N−1

λ+¹2,µ

¶

, (1)

whereΠis the change of parity operator.

2.5.

Pseudodi

erential operators on S

LetT^∗S^1|N be the cotangent bundle onS^1|N with local coordinates (x,θ1, . . . ,θN,ξ, ¯θ1, . . . , ¯θN), wherep( ¯θi)=1. The superspace of the supercommutative algebraS P(N) of pseudodifferential symbols onS^1|Nwith its natural multiplication is spanned by the series

S P(N)= ( ∞

X

k=−M

X

²=(²1,...,²N)

a_k,²(x,θ)ξ^−kθ¯^²₁¹. . . ¯θ^²_N^N

¯

¯

¯

¯

¯

a_k,²∈C_C^∞(S^1|N);²i=0, 1;M∈N )

. This space has a structure of the Poisson Lie superalgebra given by the following bracket:

{A,B}=∂_ξA∂xB−∂xA∂_ξB−(−1)^p(A)

N

X

i=1

³∂iA∂θ¯iB+∂θ¯iA∂iB´ , where∂x=_∂^∂_x,∂_ξ=_∂^∂_ξ,∂i=_∂^∂_θi and∂_θ

i=_∂^∂

θi. Of courseS P(0) is the classical space of symbols, usually denotedP.

The associative superalgebra of pseudodifferential operators SΨDO(S^1|N) on S^1|N has the same underlying vector space as S P(N), but the multiplication is now defined by the following rule:

A◦B= X

α≥0,νi=0, 1

(−1)^p(A)+1 α!

³∂^α_ξ∂^ν_θ¯ⁱ

iA´³

∂^αx∂^ν_iⁱB´ .

Denote bySΨDO(S^1|N)SLthe Lie superalgebra with the same superspace asSΨDO(S^1|N) and the supercommutator defined on homogeneous elements by:

[A,B]=A◦B−(−1)^p(A)p(B)B◦A.

In particular, we haveSΨDO(S^1|0)=ΨDO(S¹).

3. The structure ofS P(N)as aK(N)–module The natural embedding ofK(N) intoS P(N) defined by

π(X_F)=Fξ−(−1)^p(F) 2

N

X

i=1

ηi(F)ζi, where ζi=θ¯i−θiξ, induces aK(N)-module structure onS P(N).

Setting degx =degθi =0, degξ=deg ¯θi =1 for all i, we endow the Poisson superalgebra S P(N) with aZ-grading:

S P(N)= f M

n∈ZS Pn(N), where e

Ln∈Z=(L

n<0)L Q

n≥0and S Pn(N)=©

Fξ⁻ⁿ+G₁ξ⁻ⁿ⁻¹θ¯1+G₂ξ⁻ⁿ⁻¹θ¯2+ · · · +H_1,2ξ⁻ⁿ⁻²θ¯1θ¯2+. . .¯

¯F,G_i,H_i,j∈C_C^∞(S^1|N)ª is the homogeneous subspace of degree−n.

Note that each element ofSΨDO(S^1|N) can be expressed as A=X

k∈Z

(F_k+G¹_kξ⁻¹θ¯1+ · · · +H^1,2

k ξ⁻²θ¯1θ¯2+. . . )ξ^−k, whereF_k,Gⁱ_k,H_k^i,j∈C_C^∞(S^1|N). We define theorderofAto be

ord(A)=supn k¯

¯

¯F_k6=0 orGⁱ_k6=0 orH_k^i,j6=0o .

This definition of order equipsSΨDO(S^1|N) with a decreasing filtration as follows: set F_n=©

A∈SΨDO(S^1|N)¯

¯ord(A)≤ −nª , wheren∈Z. So we have

· · · ⊂Fn+1⊂Fn⊂. . . .

This filtration is compatible with the multiplication and the super Poisson bracket, that is, for A∈FnandB∈Fp, one hasA◦B∈Fn+pand {A,B}∈Fn+p−1. This filtration makesSΨDO(S^1|N) an associative filtered superalgebra. Moreover, this filtration is compatible with the naturalK(N)- action onSΨDO(S^1|N). Indeed,

X_F(A)=[X_F,A]∈Fnfor anyX_F∈K(N) andA∈Fn.

The inducedK(N)-module structure on the quotientFn/Fn+1is isomorphic to that of theK(N)- moduleS Pn(N). Therefore,

S P(N)' f M

n∈Z

F_n/F_n+1.

4.

n

N

Now, we describe the spaces ofn(1|N)-invariant linear differential operatorsF^N_λ→F^N_µ forN∈N. Our main result of this section is the following:

Theorem 2.LetN_λ^N_,µ:F^N_{λ →}F^N_µ, (Fα^λ_N)7→N_λ^N_,µ(F)α^µ_N be a non-zeroN(1|N)-invariant linear differential operator. Then, up to a scalar factor, the mapN_λ,µ^N is given by:

N_λ,µ^N(F)= (P

k≥0γkF^(k), for N∈N P

k≥0γkη1η2. . .ηN(F^(k)), for N≥1, (2) whereγk∈R.

Proof. (i). ForN=0, the generic form of any such a differential operator is N_λ⁰_,µ:F⁰_λ→Fµ⁰,Fdx^λ7→

m

X

i=0

γiF⁽ⁱ⁾dx^µ,

whereγi∈C^∞(S¹) are arbitrary functions andF⁽ⁱ⁾stands for ^diF

dxⁱ. The invariance property with respect to the vector fieldX=_dx^d implies that^d_dx^γi =0.

(ii). By induction onN. ForN =1, letN_λ¹_,µ:F¹_λ→F¹_µbe ann(1|1)-invariant linear differential operator. The n(1|1)-invariance of N_λ¹_,µ is equivalent to invariance with respect just to the subalgebran(1|0) and the vector fieldsX_θ1. Using the fact that, asvect(S¹)-modules,

F¹_λ'F⁰_λ⊕Π µ

F_λ+⁰ 1 2

¶

, (3)

we can deduce, by induction hypothesis, the restriction ofN_λ,µ¹ to each component of the right- hand side of (3). The invariance ofN_λ¹_,µwith respectX_θ1determine thus completely the space of n(1|1)-invariant linear differential operatorF¹_λ→F¹_µ.

Now, assume that the result holds for N >1. Observe that the n(1|N)-invariance of any linear differential operatorsN_λ^N_,µ:F^N_{λ →}F^N_µ is equivalent to invariance with respect just to the subalgebrasn(1|N−1) andn(1|N−1)ⁱ,i =1, . . . ,N−1, and thatN_λ^N_,µis decomposed into four n(1|N−1)-invariant maps:

Π^ı³ F^N−1_λ+ı

2

´

−→Π^ µ

F^N−1

µ+2^

¶

, ı,=0, 1. (4)

Thus, by induction assumption, we exhibit then(1|N−1)-invariant linear differential operators F^N_{λ →}F^N_µ. More precisely, anyn(1|N−1)-invariant binary differential operatorsN_λ^N_,µ:F^N_{λ →}F^N_µ can be expressed as:

N_λ^N_,µ(F)=Ξλ,µ(1−θN∂θN)(N_λ^N_,µ⁻¹)−Θλ,µ(−1)^p(F)∂θN(N_λ^N_,µ⁻¹)θN, Nf_λ^N_,µ(F)=(−1)^p(F)Ω_λ,µ(1−θi∂_θi)(Nf_λ^N_,µ⁻¹)θN+Γ_λ,µ(∂_θi(Nf_λ^N_,µ⁻¹),

where the coefficientsΩλ,µ,Γλ,µ,Ξλ,µandΘλ,µare, a priori, arbitrary constants, but the invari- ance ofN_λ,µ^N with respectn(1|N−1)ⁱ,i=1, . . . ,N−1, shows that

Γ_λ−^N

2,λ+k= −Ω_λ−^N

2,λ+k, Ξ_λ,λ+k=Θ_λ,λ+k.

Therefore, we easily check thatN_λ^N_,µis expressed as in Theorem 2. This completes the proof of

Theorem 2.

5. Cohomology

Let us first recall some fundamental concepts from cohomology theory (see, e.g., [4]). Letg₌ g_¯0⊕g_¯1be a Lie superalgebra acting on a superspaceV=V_¯0⊕V_¯1and lethbe a subalgebra ofg. (If his omitted it assumed to be {0}). The space ofh-relativen-cochains ofgwith values inV is the g-module

Cⁿ(g,h;V) :=Hom_h(Λⁿ(g/h);V).

Thecoboundary operatorδn:Cⁿ(g,h;V)−→Cⁿ⁺¹(g,h;V) is ag-map satisfyingδn◦δn−1=0. The kernel ofδn, denotedZⁿ(g,h;V), is the space ofh-relativen-cocycles, among them, the elements in the range ofδn−1are calledh-relativen-coboundaries. We denoteBⁿ(g,h;V) the space ofn- coboundaries.

By definition, then^{t h}h-relative cohomolgy space is the quotient space Hⁿ(g,h;V)=Zⁿ(g,h;V)/Bⁿ(g,h;V).

5.1.

The spaces

n

F

In this subsection, we will compute the first differential cohomology spaces H¹_di

ff(K(N),n(1|N),F^N_λ). Our main result is the following:

Theorem 3.The spaceH¹_diff(K(N),n(1|N),F^N_λ)has the following structure:

H¹_diff(K(N),n(1|N),F^N_λ)=































R² if N=2 and λ=0

R if



















N=0 and λ=0, 1, 2 N=1 and λ=0,¹₂,³₂ N=2 and λ=1 N=3 and λ=0,¹₂ N≥4 and λ=0 0 otherwise.

The following1-cocyclesΥ_λ^Nspan the corresponding cohomology spaces:

Υ^N0(X_F)=F⁰;N∈N, Υ¹3 2

(X_F)=η¯1(F⁰⁰)α₁³², Υ⁰1(X_F)=F⁰⁰dx¹, Υ²0(X_F)=η¯1η¯2(F)α2, Υ⁰2(X_F)=F⁰⁰⁰dx², Υ²1(X_F)=η¯1η¯2(F⁰)α2, Υ¹1

2

(X_F)=η1(F⁰)α₁¹², Υ³1 2

(X_F)=η¯1η¯2η¯3(F)α₃¹².

(5)

The proof of Theorem 3 will be the subject of subsection 5.2. In fact, we need first the following classical fact:

Lemma 4 ([3]).Any 1-cocycle ΥonK(N) vanishing onn(1|N), with values in F^N_λ, the linear differential operatorN :K(N)→F^N_λ defined by

N(X)=Υ(X), isn(1|N)-invariant.

5.2.

Proof of the Theorem 3

LetΥ_−1,^N _µbe a 1-cocycle onK(N) vanishing onn(1|N), with values inF_µ^N. By Lemma 4, up to a scalar factor,Υ^N_−1,µis a linear differential operatorn(1|N)-invariantN₋^N_1,µ: F^N_−1→F^N_µ. Thus, by Theorem 2, we get the explicit formulae forN₋^N_1,µ:

ForN=0,

nN_−1,⁰_µ(XF)=P

k≥0γkF^(k)dx^µ ForN≥1,

(N_−1,^N_µ(X_F)=P

k≥0γkF^(k)α^µ_N N_−1,^N_µ(X_F)=P

k≥0γkη1η2. . .ηN(F^(k))α^µ_N.

Now let us check if each of the mapsN_−1,^N_µare 1-cocycles. If the mapsN_−1,^N_µare 1-cocycles one has to check the 1−cocycles one has to check the 1-cocycle relation. It reads as follows:

δ(N₋^N1,µ)=(−1)^{p(X)p(N−1,µN )}L^µ_X(N₋^N1,µ(Y))−(−1)^{p(Y)(p(X)+p(N−1,µN ))}L^µ_Y(N₋^N1,µ(X))−N₋^N1,µ([X,Y])

=0,

where X,Y ∈K(N). By direct computation, we can see that only the operatorsN_−1,µ^N =Υ^N_µ expressed as in (5) are 1-cocycles vanishing onn(1|N).

Finally, we study the non-triviality of these 1-cocyclesN₋^N_1,λ. For instance, assume that the 1-cocycleN_−1,2⁰ is trivial, then there exists a densityϕ(x)dx²∈F⁰₂such that

N_−1,2⁰ (X_F)=L²_X

Fϕ(x)dx². (6)

The coefficient ofF⁰⁰⁰is zero in the expression of the coboundary and the coefficient ofF⁰⁰⁰is 1 in the expression of 1-cocycleN_−1,2⁰ . Thus, the relation (6) implies 1=0 which is absurd. With the same arguments, we prove the non-triviality of 1-cocyclesN_−1,0^N ,N_−1,1⁰ ,N_−1,2⁰ ,N_−1,¹ 1

2

,N_−1,¹ 3 2

, N₋²_1,0,N₋²_1,1andN³

−1,¹₂. Therefore, we easily check thatΥ_λ^Nis expressed as in (5). This completes the proof of Theorem 3.

6. H¹_di

ff(K(N),

n

N

ff(K(N),

n

N

The space

ff(K(N),

n

N

ff(K(N),n(1|N);S Pn(N)) inherits the grading (3) ofS Pn(N), so it suffices to compute it in each degree. The main result of this section forN=0, 1, 2, is the following.

Theorem 5.The spaceH¹_diff(K(N),n(1|N);S Pn(N))has the following structure:

H¹_di

ff(K(N),n(1|N);S Pn(N))'





























 R if











N=2 and n=1 N=0 and n=0, 1, 2 N=1 and n=1 R² if

(N=2 and n= −1 N=1 and n=0 R⁵ if N=2 and n=0 0 otherwise.

(7)

The following1-cocyclesχn^Nspan the corresponding cohomology spaces:

χ^N₀ =F⁰,for N=0, 2, χ²₋₁=η₁η₂(F)ξ⁻¹ζ1ζ1, χ⁰₁=F⁰⁰ξ⁻¹, χe²₋₁=F⁰ξ⁻¹ζ1ζ1, χ⁰₂=F⁰ξ⁻², χe²₀=η₁η₂(F), χ¹0=(1+(−1)^p(F))F⁰+η1(F⁰)ξ⁻¹ζ1, χb²0(X_F)=(−1)^p(F)¡

η1(F⁰)ζ1+η2(F⁰)ζ2¢ ξ⁻¹, χe¹₀=η₁(F⁰)ξ⁻¹ζ1−2θ1η₁(F⁰), χ²₀(X_F)=F⁰⁰ξ⁻²ζ1ζ2+ −(1)^p(F)¡

η₂(F⁰)ζ1−η₁(F⁰)ζ2

¢ξ⁻¹, χ¹₁=η₁(F⁰⁰)ξ⁻²ζ1−2θ1η₁(F⁰⁰)ξ⁻¹, χ²

0(X_F)=η₁η₂(F⁰)ξ⁻²ζ1ζ2, χ²1(X_F)=2

3F⁽³⁾ξ⁻³ζ1ζ2+ −(1)^p(F)¡

η2(F⁰⁰)ζ1−η1(F⁰⁰)ζ2

¢ξ⁻²+2η1η2(F⁰)ξ⁻¹.

(8)

Proof. The case whereN=0. In this case, we can see that the mapφ:Fn −→Pn defined by φ(F d xⁿ)=Fξ⁻ⁿ provide us with an isomorphism of Vect(S¹)-modules. So, we can deduce the structure of H¹_diff(Vect(S¹),n(1|0);Pn) from H¹_diff(Vect(S¹),n(1|0);Fn) given in Theorem 3.

The case whereN=1. In this case, as aK(1)-module, we have S Pn(1)=S P¹n⊕S P²n, where

S P¹n={(1+(−1)^p(F))Fξ⁻ⁿ+η₁(F)ξ⁻ⁿ⁻¹ζ1,F∈C_C^∞(S^1|1)}, S P²n={Fξ⁻ⁿ⁻¹ζ1−2θ1Fξ⁻ⁿ,F∈C_C^∞(S^1|1)}.

The natural maps

ϕ1: F¹_{n −→S P}¹_n

Fαⁿ₁ 7−→(1+(−1)^p(F))Fξ⁻ⁿ+η1(F)ξ⁻ⁿ⁻¹ζ1, ϕ2: Π

µ F¹

n+¹₂

¶

−→S P¹n

Π µ

Fαⁿ⁺₁ ¹²

¶

7−→Fξ⁻ⁿ⁻¹ζ1−2θ1Fξ⁻ⁿ,

provide us with isomorphisms ofK(1)-modules. Hence, asK(1)-modules, we haveS Pn(1)' F¹_n⊕Π(F¹

n+¹₂). This isomorphism induces the following isomorphism between cohomology spaces:

H¹_diff(K(1),n(1|1);S Pn(1))'H¹_diff(K(1),n(1|1);F¹_n)⊕H¹_diff µ

K(1),n(1|1);Π µ

F¹

n+¹₂

¶¶

. We deduce from this isomorphism and Theorem 3, the 1-cocycles (8).

The case whereN=2. To prove Theorem 5 in this case, we need first the following:

Proposition 6.The spaceH¹_di

ff(K(1)ⁱ,n(1|1)ⁱ,F²_λ)has the following structure:

H¹_diff(K(1)ⁱ,n(1|1)ⁱ,F²_λ)=











R² if λ=0

R if λ= −¹₂,¹₂, 1,³₂ 0 otherwise.

The following 1-cocyclesγⁱ_λspan the corresponding cohomology spaces:

γⁱ0=F⁰, γⁱ3

2 =η1(F⁰⁰), γⁱ1

2 =η1(F⁰), γeⁱ0=(−1)^p(F)η3−i(F⁰)θi, γⁱ1=(−1)^p(F)η3−i(F⁰⁰)θi, γⁱ₋1

2 =F⁰θi. (9)

Proof of Proposition 6. LetFα^λ₂=(f₀+f₁θ1+f₂θ2+f₁₂θ1θ2)α^λ₂∈F²_λ. The map Φ: F²_{λ −→}F^1,i_{λ ⊕Π}

µ F^1,i

λ+¹₂

¶

Fα^λ₂ 7−→

µ

(1−θi∂_θi)(F)α^λ₁,Π µ

(−1)^p(F)+1∂_θi(F)α^λ+₁ ¹²

¶¶

,

provides us with an isomorphism ofK(1)ⁱ-modules. This map induces the following isomor- phism between cohomology spaces:

H¹_diff³

K(1)ⁱ,n(1|1)ⁱ;F²_λ^´_'H¹_diff³

K(1)ⁱ,n(1|1)ⁱ;F^1,i_λ ^´_⊕H¹_diff µ

K(1)ⁱ,n(1|1)ⁱ;Π µ

F^1,i

λ+¹2

¶¶

. (10) Of course, we can deduce the structure of

H¹_di

ff

³K(1)ⁱ,n(1|1)ⁱ;Π³

F^1,i_λ ^´´ from H¹_di

ff

³K(1)ⁱ,n(1|1)ⁱ;F^1,i_λ ^´. Indeed, to anyΥ∈H¹_diff³

K(1)ⁱ,n(1|1)ⁱ;F^1,i_λ ^´correspondsΥe∈H¹_diff³

K(1)ⁱ,n(1|1)ⁱ;Π³

F^1,i_λ ^´´where Υe(X_F)=Π(σ◦Υ(X_F)) withσ(F)=(−1)^p(F)F. Obviously,Υis a coboundary if and anly ifΥe is a coboundary. We deduce from isomorphism (10) and formula (5), the 1-cocycles (9).

Lemma 7.For n∈Z, any element of Z¹(K(2),n(1|2);S Pn(2))is an(1|2)-relative coboundary over K(2)if and only if its restriction to the subalgebraK(1)ⁱ isn(1|1)ⁱ-relative coboundary for i = 1 and 2.

Proof of Lemma 7. It is easy to see that if C is an(1|2)-relative coboundary overK(2), then C_|K(1)ⁱ is an(1|1)ⁱ-relative coboundary ofK(1)ⁱ. Now, assume thatC_|K(1)ⁱ is an(1|1)ⁱ-relative coboundary ofK(1)ⁱfor i = 1 and 2. Using the condition of a 1-cocycle, we prove that there exists an elementn(1|1)ⁱ-invariantG∈S Pn(2) such that

C¡

X_f0+fiθi¢

=© ρ0

¡X_f0+fiθi¢ ,Gª

for anyf₀,f_i∈C_C^∞¡ S¹¢

, i=1, 2 C¡

X_f12θ1θ2¢

=© ρ0

¡X_f12θ1θ2¢ ,Gª

for anyf₁₂∈C_C^∞¡ S¹¢

. We deduce thatC(XF)=©

ρ0(XF) ,Gª

, for anyF∈C_C^∞¡ S^1|2¢

, and thereforeC is an(1|2) -relative

coboundary ofK(2).

We also need the following:

Proposition 8 ([1]).

(1) As aK(1)ⁱ-module, i=1, 2,we have S Pn(2)'F²_n⊕Π

µ F²

n+¹₂⊕F²

n+¹₂

¶

⊕F²_n+1,for n=0,−1. (11) (2) For n6=0,−1 :

(a) The following subspace ofS Pn(2) : S Pn,i=

½

B^(n,i)_F =Fθiθ¯iξ⁻ⁿ⁻¹+θi

µ ηi−1

2η3−i

¶

(F)ζ3−iζiξ⁻ⁿ⁻²

¯

¯

¯

¯

F∈C^∞_C(S^1|2)

¾

(12) is aK(1)ⁱ- module, i=1, 2,isomorphic toF²_n+1.

(b) As aK(1)ⁱ-module we have S Pn(2)/S Pn,i'F²_n⊕Π

µ F²

n+¹₂⊕F²

n+¹₂

¶

, i=1, 2. (13)