The Binary Invariant Differential Operators on Weighted Densities on the superspace R 1|n and Cohomology
aa r X i v : . [ m a t h . R T ] D ec The Binary Invariant Differential Operators on WeightedDensities on the superspace R | n and Cohomology Mabrouk Ben Ammar ∗ Nizar Ben Fraj † Salem Omri ‡ June 25, 2018
Abstract
Over the (1 , n )-dimensional real superspace, n >
1, we classify K ( n )-invariant binarydifferential operators acting on the superspaces of weighted densities, where K ( n ) is the Liesuperalgebra of contact vector fields. This result allows us to compute the first differentialcohomology of K ( n ) with coefficients in the superspace of linear differential operatorsacting on the superspaces of weighted densities–a superisation of a result by Feigin andFuchs. We explicitly give 1-cocycles spanning these cohomology spaces. Mathematics Subject Classification (2000). 53D55
Key words : Cohomology, Superalgebra.
This work is a direct continuation of [9, 10] and [2, 3] listed among other things, binarydifferential operators invariant with respect to a supergroup of diffeomorphisms and computedcohomology of polynomial versions of various infinite dimensional Lie superalgebras.Let vect (1) be the Lie algebra of polynomial vector fields on R . Consider the 1-parameterdeformation of the vect (1)-action on R [ x ]: L λX ddx ( f ) = Xf ′ + λX ′ f, where X, f ∈ R [ x ] and X ′ := dXdx . Denote by F λ the vect (1)-module structure on R [ x ] definedby L λ for a fixed λ . Geometrically, F λ = (cid:8) f dx λ | f ∈ R [ x ] (cid:9) is the space of polynomial weighteddensities of weight λ ∈ R . The space F λ coincides with the space of vector fields, functionsand differential 1-forms for λ = − , λ,µ := Hom diff ( F λ , F µ ) the vect (1)-module of linear differential operators withthe natural vect (1)-action. Feigin and Fuchs [5] computed H ( vect (1); D λ,µ ), where H ∗ diff denotes the differential cohomology; that is, only cochains given by differential operatorsare considered. They showed that non-zero cohomology H ( vect (1); D λ,µ ) only appear forparticular values of weights that we call resonant which satisfy µ − λ ∈ N . These spaces arise ∗ D´epartement de Math´ematiques, Facult´e des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie. E. mail:[email protected], † Institut Sup´erieur de Sciences Appliqu´ees et Technologie, Sousse, Tunisie. E. mail: benfraj [email protected] ‡ D´epartement de Math´ematiques, Facult´e des Sciences de Gafsa, Zarroug 2112 Gafsa, Tunisie. E. mail:omri [email protected],
1n the classification of infinitesimal deformations of the vect (1)-module S µ − λ = L ∞ k =0 F µ − λ − k ,the space of symbols of D λ,µ .On the other hand, Grozman [10] classified all vect (1)-invariant binary differential opera-tors on R acting in the spaces F λ . He showed that all invariant operators are of order ≤ R | n endowed with its standard contact structuredefined by the 1-form α n , and the Lie superalgebra K ( n ) of contact polynomial vector fieldson R | n . We introduce the K ( n )-module F nλ of λ -densities on R | n and the K ( n )-module oflinear differential operators, D nλ,µ := Hom diff ( F nλ , F nµ ), which are super analogues of the spaces F λ and D λ,µ , respectively. The classification of the K (1)-invariant binary differential operatorson R | acting in the spaces F λ is due to Leites et al. [9], while the space H (cid:16) K (1); D λ,µ (cid:17) hasbeen computed by Basdouri et al. [1] (see also [3]) and the space H (cid:16) K (2); D λ,µ (cid:17) has beencomputed by the second author [2]. We also mention that Duval and Michel studied a similarproblem for the group of contactomorphisms of the supercircle S | n instead of K ( n ) relatedto the link between discrete projective invariants of the supercircle, and the cohomology ofthe group of its contactomorphisms [4].In this paper we classify all K ( n )-invariant binary differential operators on R | n actingin the spaces F nλ for n >
1. We use the result to compute H (cid:16) K ( n ); D nλ,µ (cid:17) for n >
2. Weshow that, as in the classical setting, non-zero cohomology H (cid:16) K ( n ); D nλ,µ (cid:17) only appear forresonant values of weights which satisfy µ − λ ∈ N . Moreover, we give explicit basis of thesecohomology spaces. These spaces arise in the classification of infinitesimal deformations ofthe K ( n )-module S nµ − λ = L k ≥ F nµ − λ − k , a super analogue of S µ − λ , see [1]. R | n Let R | n be the superspace with coordinates ( x, θ , . . . , θ n ) , where θ , . . . , θ n are the oddvariables, equipped with the standard contact structure given by the following 1-form: α n = dx + n X i =1 θ i dθ i . (2.1)On the space R [ x, θ ] := R [ x, θ , . . . , θ n ], we consider the contact bracket { F, G } = F G ′ − F ′ G −
12 ( − | F | n X i =1 η i ( F ) · η i ( G ) , (2.2)where η i = ∂∂θ i − θ i ∂∂x and | F | is the parity of F . Note that the derivations η i are the generatorsof n-extended supersymmetry and generate the kernel of the form (2.1) as a module over thering of polynomial functions. Let Vect Pol ( R | n ) be the superspace of polynomial vector fieldson R | n : Vect Pol ( R | n ) = ( F ∂ x + n X i =1 F i ∂ i | F i ∈ R [ x, θ ] for all i ) , ∂ i = ∂∂θ i and ∂ x = ∂∂x , and consider the superspace K ( n ) of contact polynomial vec-tor fields on R | n . That is, K ( n ) is the superspace of vector fields on R | n preserving thedistribution singled out by the 1-form α n : K ( n ) = (cid:8) X ∈ Vect
Pol ( R | n ) | there exists F ∈ R [ x, θ ] such that L X ( α n ) = F α n (cid:9) . The Lie superalgebra K ( n ) is spanned by the fields of the form: X F = F ∂ x − n X i =1 ( − | F | η i ( F ) η i , where F ∈ R [ x, θ ].In particular, we have K (0) = vect (1). Observe that L X F ( α n ) = X ( F ) α n . The bracket in K ( n ) can be written as: [ X F , X G ] = X { F, G } . We introduce a one-parameter family of modules over the Lie superalgebra K ( n ). As vectorspaces all these modules are isomorphic to R [ x, θ ], but not as K ( n )-modules.For every contact polynomial vector field X F , define a one-parameter family of first-orderdifferential operators on R [ x, θ ]: L λX F = X F + λF ′ , λ ∈ R . (2.3)We easily check that [ L λX F , L λX G ] = L λX { F,G } . (2.4)We thus obtain a one-parameter family of K ( n )-modules on R [ x, θ ] that we denote F nλ , thespace of all polynomial weighted densities on R | n of weight λ with respect to α n : F nλ = n F α λn | F ∈ R [ x, θ ] o . (2.5)In particular, we have F λ = F λ . Obviously the adjoint K ( n )-module is isomorphic to the spaceof weighted densities on R | n of weight − . A differential operator on R | n is an operator on R [ x, θ ] of the form: A = M X k =0 X ε =( ε , ··· ,ε n ) a k,ǫ ( x, θ ) ∂ kx ∂ ε · · · ∂ ε n n ; ε i = 0 , M ∈ N . (2.6)Of course any differential operator defines a linear mapping F α λn ( AF ) α µn from F nλ to F nµ for any λ, µ ∈ R , thus the space of differential operators becomes a family of K ( n )-modules D nλ,µ for the natural action: X F · A = L µX F ◦ A − ( − | A || F | A ◦ L λX F . (2.7)3imilarly, we consider a multi-parameter family of K ( n )-modules on the space D nλ ,...,λ m ; µ ofmulti-linear differential operators: A : F nλ ⊗ · · · ⊗ F nλ m −→ F nµ with the natural K ( n )-action: X F · A = L µX F ◦ A − ( − | A || F | A ◦ L λ ,...,λ m X F , where L λ ,...,λ m X F is defined by the Leibnitz rule. We also consider the K ( n )-module Π (cid:16) D nλ ,...,λ m ; µ (cid:17) with the K ( n )-action (Π is the change of parity operator): X F · Π( A ) = Π (cid:16) L µX F ◦ A − ( − ( | A | +1) | F | A ◦ L λ ,...,λ m X F (cid:17) . Since − η i = ∂ x , and ∂ i = η i − θ i η i , every differential operator A ∈ D nλ,µ can be expressed inthe form A ( F α λn ) = X ℓ =( ℓ ,...,ℓ n ) a ℓ ( x, θ ) η ℓ . . . η ℓ n n ( F ) α µn , (2.8)where the coefficients a ℓ ( x, θ ) are arbitrary polynomial functions.The Lie superalgebra K ( n −
1) can be realized as a subalgebra of K ( n ): K ( n −
1) = n X F ∈ K ( n ) | ∂ n F = 0 o . Therefore, D nλ ,...,λ m ; µ and F nλ are K ( n − i in { , , . . . , n − } , K ( n −
1) is isomorphic to K ( n − i = n X F ∈ K ( n ) | ∂ i F = 0 o . Proposition 2.1.
As a K ( n − -module, we have D nλ,µ ; ν ≃ e D n − λ,µ ; ν := D n − λ,µ ; ν ⊕ D n − λ + ,µ + ; ν ⊕ D n − λ,µ + ; ν + ⊕ D n − λ + ,µ ; ν + ⊕ Π (cid:18) D n − λ,µ ; ν + ⊕ D n − λ,µ + ; ν ⊕ D n − λ + ,µ ; ν ⊕ D n − λ + ,µ + ; ν + (cid:19) . (2.9)Proof. For any F ∈ R [ x, θ ], we write F = F + F θ n where ∂ n F = ∂ n F = 0and we prove that L λX H F = L λX H F + ( L λ + X H F ) θ n . Thus, it is clear that the map ϕ λ : F nλ → F n − λ ⊕ Π( F n − λ + ) F α λn ( F α λn − , Π( F α λ + n − )) , (2.10)is K ( n − K ( n − F nλ ⊗ F nµ to F n − λ ⊗ F n − µ ⊕ F n − λ ⊗ Π( F n − µ + ) ⊕ Π( F n − λ + ) ⊗ F n − µ ⊕ Π( F n − λ + ) ⊗ Π( F n − µ + )4enoted ψ λ,µ . Therefore, we deduce a K ( n − λ,µ,ν : e D n − λ,µ ; ν → D nλ,µ ; ν A ϕ − ν ◦ A ◦ ψ λ,µ . (2.11)Here, we identify the K ( n − (cid:16) D n − λ,µ ; ν ′ (cid:17) → Hom diff (cid:0) F n − λ ⊗ F n − µ , Π( F n − ν ′ ) (cid:1) , Π( A ) Π ◦ A, Π (cid:16) D n − λ,µ ′ ; ν (cid:17) → Hom diff (cid:16) F n − λ ⊗ Π( F n − µ ′ ) , F n − ν (cid:17) , Π( A ) A ◦ (1 ⊗ Π) , Π (cid:16) D n − λ ′ ,µ ; ν (cid:17) → Hom diff (cid:0) Π( F n − λ ′ ) ⊗ F n − µ , F n − ν (cid:1) , Π( A ) A ◦ (Π ⊗ σ ) , Π (cid:16) D n − λ ′ ,µ ′ ; ν ′ (cid:17) → Hom diff (cid:16) Π( F n − λ ′ ) ⊗ Π( F n − µ ′ ) , Π( F n − ν ′ ) (cid:17) , Π( A ) Π ◦ A ◦ (Π ⊗ σ ◦ Π) , D n − λ,µ ′ ; ν ′ → Hom diff (cid:16) F n − λ ⊗ Π( F n − µ ′ ) , Π( F n − ν ′ ) (cid:17) , A Π ◦ A ◦ (1 ⊗ Π) , D n − λ ′ ,µ ′ ; ν → Hom diff (cid:16) Π( F n − λ ′ ) ⊗ Π( F n − µ ′ ) , F n − ν (cid:17) , A A ◦ (Π ⊗ σ ◦ Π) , D n − λ ′ ,µ ; ν ′ → Hom diff (cid:0) Π( F n − λ ′ ) ⊗ F n − µ , Π( F n − ν ′ ) (cid:1) , A Π ◦ A ◦ (Π ⊗ σ ) , where λ ′ = λ + , µ ′ = µ + , ν ′ = ν + and σ ( F ) = ( − | F | F . ✷ K ( n ) -Invariant Binary Differential Operators In this section, we will classify all K ( n )-invariant binary differential operators acting on thespaces of weighted densities on R | n for n ≥
2. As a first step towards these classifications,we shall need the list of binary K (1)-invariant differential operators acting on the spaces ofweighted densities on R | . K (1) -invariant binary differential operators In [9], Leites et al. classified all binary K (1)-invariant differential operators F λ ⊗ F µ → F ν, F α λ ⊗ Gα µ T λ,µ,ν ( F, G ) α ν . ν k = λ + µ + k for k = 0 , , , T λ,µ,ν ( F, G ) =
F G, T a,b , , ( F, G ) = a ( − | F | F η ( G ) + bη ( F ) G, a, b ∈ R , T λ,µ,ν ( F, G ) = µ η ( F ) G − λ ( − | F | F η ( G ) , T λ,µ,ν ( F, G ) = µF ′ G − ( − | F | η ( F ) η ( G ) − λF G ′ , T ,µ,ν ( F, G ) = S ( F, G ) − µη ( F ′ ) G, T λ, ,ν ( F, G ) = S ( F, G ) − λ ( − | F | F η ( G ′ , T , , ( F, G ) = F ′ G ′ + ( − | F | ( η ( F ′ ) η ( G ) − η ( F ) η ( G ′ )) , T − , , ( F, G ) = 3
F G ′′ − ( − | F | M ( F, G ) + 2 F ′ G ′ , T , − , ( F, G ) = 3 F ′′ G + ( − | F | M ( G, F ) + 2 F ′ G ′ , T λ, − λ − , ( F, G ) = λ ( − | F | F η ( G ′ ) + ( λ + 1) η ( F ′ ) G + ( λ + ) S ( F, G ) , (3.12)where M ( F, G ) = 2 η ( F ) η ( G ′ ) + η ( F ′ ) η ( G ) and S ( F, G ) = η ( F ) G ′ + ( − | F | F ′ η ( G ) . Observe that the operation T λ,µ,ν is nothing but the well-known Poisson bracket on R | and the operation T λ,µ,ν is just the Buttin bracket in coordinates θ and p := Π( α ) with x serving as parameter (see, e.g. [6, 7, 8, 9]). K ( n ) -invariant binary differential operators for n ≥ Now, we describe the spaces of K ( n )-invariant binary differential operators F nλ ⊗ F nµ −→ F nν for n ≥
2. We prove that these spaces are nontrivial only if ν = λ + µ or ν = λ + µ + 1 andthey are, in some sense, spanned by the following even operators defined on R [ x, θ ] ⊗ R [ x, θ ]: a ( F, G ) =
F G, b ( F, G ) = µF ′ G − λF G ′ − ( − | F | P ni =1 η i ( F ) η i ( G ) , c ( F, G ) = ( − | F | ( η ( F ) η ( G ) − η ( F ) η ( G )) + 2 µη ( η ( F )) G ) , d ( F, G ) = ( − | F | ( η ( F ) η ( G ) − η ( F ) η ( G )) + 2 λF η η ( G )) , e ( F, G ) = ( − | F | ( λ + ) ( η ( F ) η ( G ) − η ( F ) η ( G )) + λF η η ( G ) + ( λ + 1) η ( η ( F )) G. (3.13)More precisely, we have Theorem 3.1.
Let n ≥ and F nλ ⊗ F nµ −→ F nν , F α λn ⊗ Gα µn T nλ,µ,ν ( F, G ) α νn be a nontrivial K ( n ) -invariant binary differential operator. Then ν = λ + µ or ν = λ + µ + 1 . Moreover,(a) If ν = λ + µ then T nλ,µ,ν = α a . (b) If ν = λ + µ + 1 then, for n > or n = 2 but λµν = 0 we have T nλ,µ,ν = α b and if n = 2 and λµν = 0 then T nλ,µ,ν has the form α b + β c , α b + β d or α b + β e in accordance with λ = 0 , µ = 0 or ν = 0 . Here, α, β ∈ R and a , b , c , d , e are defined by (3.13). n = 2. The K (2)-invariance of any element of D λ,µ ; ν is equiva-lent to invariance with respect just to the vector fields X F ∈ K (2) such that ∂ ∂ F = 0 thatgenerate K (2). That is, an element of D λ,µ ; ν is K (2)-invariant if and only if it is invariant withrespect just to the two subalgebras K (1) and K (1) . Obviously, the K (1)-invariant elements ofΠ( D λ,µ ; ν ) can be deduced from those given in (3.12) by using the following K (1)-isomorphism D λ,µ ; ν → Π( D λ,µ ; ν ) , A Π( A ◦ ( σ ⊗ σ )) (3.14)Now, by isomorphism (2.11) we exhibit the K (1)-invariant elements of D λ,µ ; ν . Of course,these elements are identically zero if 2( ν − µ − λ ) = − , , , , , , , . More precisely, any K (1)-invariant element T of D λ,µ ; ν can be expressed as follows T = X j,ℓ,k =0 , Ω j,ℓ,kλ,µ,ν Ψ λ,µ,ν (cid:16) Π j + ℓ + k (cid:16) T λ + j ,µ + ℓ ,ν + k ◦ ( σ j + ℓ + k ⊗ σ j + ℓ + k ) (cid:17)(cid:17) + X j,ℓ,k =0 , Ω j,ℓ,k,aλ,µ,ν,b Ψ λ,µ,ν (cid:18) Π j + ℓ + k (cid:18) T a,bλ + j ,µ + ℓ ,ν + k ◦ ( σ j + ℓ + k ⊗ σ j + ℓ + k ) (cid:19)(cid:19) where T λ + j ,µ + ℓ ,ν + k , T a,bλ + j ,µ + ℓ ,ν + k are defined by (3.12). The coefficients Ω j,ℓ,kλ,µ,ν and Ω j,ℓ,k,aλ,µ,ν,b are, a priori, arbitrary constants, but the invariance of T with respect K (1) imposes somesupplementary conditions over these coefficients and determines thus completely the space of K (2)-invariant elements of D λ,µ ; ν . By a direct computation, we get:Ω , , λ,µ,λ + µ = Ω , , λ,µ,λ + µ = Ω , , λ,µ,λ + µ , Ω , , λ,µ,λ + µ +1 = 2Ω , , λ,µ,λ + µ +1 = Ω , , λ,µ,λ + µ +1 = Ω , , λ,µ,λ + µ +1 . All other coefficients vanish except for ν = λ + µ + 1 with λµν = 0, in which case we havealso the following non-zero coefficients:Ω , , λ, ,ν = − Ω , , λ, ,ν = λ +12 Ω , , λ, ,ν = Ω , , λ, ,ν for λ = − , Ω , , λ, ,ν = − Ω , , λ, ,ν = − Ω , , , λ, ,ν, = Ω , , λ, ,ν for λ = − , Ω , , ,µ,ν = − µ +12 Ω , , ,µ,ν = Ω , , ,µ,ν = Ω , , ,µ,ν for µ = − , Ω , , ,µ,ν = − Ω , , , ,µ,ν, = Ω , , ,µ,ν = Ω , , ,µ,ν for µ = − , Ω , , λ,µ, = Ω , , λ,µ, = Ω , , λ,µ, = (2 λ + 1)Ω , , λ,µ, for λ = − , Ω , , λ,µ, = Ω , , λ,µ, = Ω , , λ,µ, = − , , , λ,µ, , for λ = − . Thus, we easily check that Theorem 3.1 is proved for n = 2.(ii) Now, we assume that n ≥ n . First notethat the K ( n )-invariance of any element of D nλ,µ ; ν is equivalent to invariance with respectjust to the fields X F ∈ K ( n ) such that ∂ · · · ∂ n F = 0 that generate K ( n ). That is, anelement of D nλ,µ ; ν is K ( n )-invariant if and only if it is invariant with respect to the subalgebras K ( n −
1) and K ( n − i , i = 1 , . . . , n −
1. Thus, as before, we prove that our result holdsfor n = 3 . Assume that it holds for n ≥
3. Then, by recurrence assumption and isomorphism(2.11), we deduce that any nontrivial K ( n )-invariant element T of D n +1 λ,µ ; ν only can appear if2( ν − µ − λ ) = − , , , , , , and it has the general following form: T = X j,ℓ,k =0 , ∆ j,ℓ,kλ,µ,ν Ψ λ,µ,ν (cid:16) Π j + ℓ + k (cid:16) T nλ + j ,µ + ℓ ,ν + k ◦ ( σ j + ℓ + k ⊗ σ j + ℓ + k ) (cid:17)(cid:17) .
7s before, the coefficients ∆ j,ℓ,kλ,µ,ν are, a priori, arbitrary constants, but the invariance of T with respect K ( n ) i , i = 1 , . . . , n , shows that∆ , , λ,µ,λ + µ = ∆ , , λ,µ,λ + µ = ∆ , , λ,µ,λ + µ , ∆ , , λ,µ,λ + µ +1 = 2∆ , , λ,µ,λ + µ +1 = ∆ , , λ,µ,λ + µ +1 = ∆ , , λ,µ,λ + µ +1 and all other coefficients are identically zero. Therefore, we easily check that T is expressedas in Theorem 3.1. ✷ For n ≥
2, the even operation T nλ,µ,λ + µ +1 ( F, G ) = µF ′ G − λF G ′ −
12 ( − | F | n X i =1 η i ( F ) η i ( G ) (3.15)defines a structure of Poisson Lie superalgebra on R | n . Indeed, consider the continuous sum(direct integral) of all spaces F nλ : F n = ∪ λ ∈ R F nλ . The collection of the operations (3.15) defines a bilinear map T n : F n ⊗ F n → F n . The followingstatement can be checked directly. Proposition 3.1.
The operation T n satisfies the Jacobi and Leibniz identities, then it equipsthe space F n with a Poisson superalgebra structure. Note that this Proposition is a simplest generalization of a result by Gargoubi andOvsienko for n = 1 (see [7]). Let us first recall some fundamental concepts from cohomology theory (see, e.g., [3]). Let g = g ¯0 ⊕ g ¯1 be a Lie superalgebra acting on a superspace V = V ¯0 ⊕ V ¯1 and let h be asubalgebra of g . (If h is omitted it assumed to be { } .) The space of h -relative n -cochains of g with values in V is the g -module C n ( g , h ; V ) := Hom h (Λ n ( g / h ); V ) . The coboundary operator δ n : C n ( g , h ; V ) −→ C n +1 ( g , h ; V ) is a g -map satisfying δ n ◦ δ n − = 0.The kernel of δ n , denoted Z n ( g , h ; V ), is the space of h -relative n - cocycles , among them, theelements in the range of δ n − are called h -relative n - coboundaries . We denote B n ( g , h ; V ) thespace of n -coboundaries.By definition, the n th h -relative cohomolgy space is the quotient spaceH n ( g , h ; V ) = Z n ( g , h ; V ) /B n ( g , h ; V ) . We will only need the formula of δ n (which will be simply denoted δ ) in degrees 0 and 1: for v ∈ C ( g , h ; V ) = V h , δv ( g ) := ( − p ( g ) p ( v ) g · v , where V h = { v ∈ V | h · v = 0 for all h ∈ h } , and for Υ ∈ C ( g , h ; V ), δ (Υ)( g, h ) := ( − | g || Υ | g · Υ( h ) − ( − | h | ( | g | + | Υ | ) h · Υ( g ) − Υ([ g, h ]) for any g, h ∈ g . .1 The space H ( K ( n ); D nλ,µ ) In this subsection, we will compute the first differential cohomology spaces H ( K ( n ); D nλ,µ )for n ≥
3. Our main result is the following:
Theorem 4.1.
The space H ( K ( n ); D nλ,µ ) has the following structure: H ( K ( n ); D nλ,µ ) ≃ R if n = 3 and µ − λ = 0 , , ,n = 4 and µ − λ = 0 , ,n ≥ and µ − λ = 0 , otherwise . (4.16) A base for the nontrivial H ( K ( n ); D nλ,µ ) is given by the cohomology classes of the 1-cocycles: Υ nλ,λ ( X G ) = G ′ Υ λ,λ + ( X G ) = η η η ( G ) if λ = − ∂ ( G ) η η − η η ( ∂ ( G )) ζ − ( − | G | θ M η ( G ) η if λ = − Υ λ,λ + ( X G ) = Ξ G ′ + 2 λη η η ( G ′ ) + η η η ( G ) η if λ = − G ′ + P ≤ i As a K ( n − -module, we have D nλ,µ ≃ D n − λ,µ ⊕ D n − λ + ,µ + ⊕ Π (cid:18) D n − λ,µ + ⊕ D n − λ + ,µ (cid:19) . (4.20)Proof. By isomorphism (2.10), we deduce a K ( n − λ,µ : D n − λ,µ ⊕ D n − λ + ,µ + ⊕ Π (cid:18) D n − λ,µ + ⊕ D n − λ + ,µ (cid:19) → D nλ,µ A ϕ − µ ◦ A ◦ ϕ λ . (4.21)Here, we identify the K ( n − (cid:18) D n − λ,µ + (cid:19) → Hom diff (cid:18) F n − λ , Π( F n − µ + ) (cid:19) Π( A ) Π ◦ A, Π (cid:18) D n − λ + ,µ (cid:19) → Hom diff (cid:18) Π( F n − λ + ) , F n − µ (cid:19) Π( A ) A ◦ Π , D n − λ + ,µ + → Hom diff (cid:18) Π( F n − λ + ) , Π( F n − µ + ) (cid:19) A Π ◦ A ◦ Π . . Corollary 4.2. The space H ( K (2); D λ,µ ) has the following structure: H ( K (2); D λ,µ ) ≃ R if µ − λ = 0 , , R if µ − λ = , , R if µ − λ = − , , , otherwise . (4.22) The corresponding spaces H ( K (2); D λ,λ + k ) are spanned by the cohomology classes of the1-cocycles Θ ,j,ℓλ,λ + k and e Θ ,j,ℓλ,λ + k , defined by Θ ,j,ℓλ,λ + k ( X G ) = Φ λ,λ + k (cid:16) Π j + ℓ (cid:16) σ j + ℓ ◦ Υ λ + j ,λ + k + ℓ ( X G ) (cid:17)(cid:17) (4.23)10 nd e Θ ,j,ℓλ,λ + k ( X G ) = Φ λ,λ + k (cid:16) Π j + ℓ (cid:16) σ j + ℓ ◦ e Υ λ + j ,λ + k + ℓ ( X G ) (cid:17)(cid:17) , (4.24) where j, ℓ = 0 , , k ∈ {− , . . . , } , Υ λ,µ , e Υ λ,µ are as in (4.19), Φ λ,µ is as in (4.21). Further-more, the space H ( K (2); D λ,λ + k ) has the same parity as the integer k . Proof. First, it is easy to see that the map χ : D nλ,µ → Π (cid:16) D nλ,µ (cid:17) defined by χ ( A ) = Π( σ ◦ A )satisfies L λ,µX G ◦ χ = ( − | G | χ ◦ L λ,µX G for all X G ∈ K ( n ) . Thus, we deduce the structure of H ( K ( n ); Π( D nλ,µ )) from H ( K ( n ); D nλ,µ ). Indeed, to anyΥ ∈ Z ( K ( n ); D nλ,µ ) corresponds χ ◦ Υ ∈ Z ( K ( n ); Π( D nλ,µ )). Obviously, Υ is a couboundaryif and only if χ ◦ Υ is a couboundary.Second, according to Proposition 4.1, we obtain the following isomorphism between coho-mology spaces:H (cid:16) K ( n − D nλ,µ (cid:17) ≃ H (cid:16) K ( n − D n − λ,µ (cid:17) ⊕ H (cid:18) K ( n − D n − λ + ,µ + (cid:19) ⊕ H (cid:18) K ( n − D n − λ + ,µ ) (cid:19) ⊕ H (cid:18) K ( n − D n − λ,µ + ) (cid:19) . Thus, we deduce the structure of H ( K (2); D λ,µ ). ✷ H ( K ( n ) , K ( n − i ; D nλ,µ ) As a first step towards the proof of Theorem 4.1, we shall need to study the K ( n − i -relativecohomology H ( K ( n ) , K ( n − i ; D nλ,µ ). Hereafter all ǫ ’s are constants and we will use thesuperscript i when we consider the superalgebra K ( n ) i instead of K ( n ).Let g = h ⊕ p be a Lie superalgebra, where h is a subalgebra and p is a h -module such that[ p , p ] = h . Consider a 1-cocycle Υ ∈ Z ( g ; V ), where V is a g -module. The cocycle relationreads ( − | g || Υ | g · Υ( h ) − ( − | h | ( | g | + | Υ | ) h · Υ( g ) − Υ([ g, h ]) = 0 for any g, h ∈ g . Denote Υ h = Υ | h and Υ p = Υ | p . Obviously, if Υ h = 0 then Υ is h -invariant, therefore, the h -relative cohomology space H ( g , h ; V ) is nothing but the space of cohomology classes of1-cocycles vanishing on h . In our situation, g = K ( n ) , h = K ( n − i , p = Π( F n − ,i − ) and V = D nλ,µ . Furthermore, in this case, the 1-cocycle relation yields the following equations:( − | g || Υ | X g · Υ p ( X hθ i ) − ( − | e h | ( | g | + | Υ | ) X hθ i · Υ h ( X g ) − Υ p ([ X g , X hθ i ]) = 0 , (4.25)( − | e g || Υ | X gθ i · Υ p ( X hθ i ) − ( − | e h | ( | e g | + | Υ | ) X hθ i · Υ p ( X gθ i ) − Υ h ([ X gθ i , X hθ i ]) = 0 , (4.26)where g, h ∈ R [ x, θ , . . . , ˘ θ i , . . . , θ n ] and | e h | = | h | + 1. Theorem 4.3. For all n ≥ and for all i = 1 , . . . , n , we have H ( K ( n ) , K ( n − i ; D nλ,µ ) ≃ R if (cid:26) n = 2 and λ = µ = 0 ,n = 3 and ( λ, µ ) = ( − , , otherwise . (4.27)11 oreover, for fixed n = 2 or 3, non-zero relative cohomology H ( K ( n ) , K ( n − i ; D nλ,µ ) arespanned by classes of some K ( n − i -relative cocycles which are cohomologous. Proof. For n = 1, the result holds from [1], (Lemma 4.1). For n = 2, we deduce the resultfrom [2] (Proposition 4.2). Moreover, the space H ( K (2) , K (1); D λ,λ ), for λ = 0, is spannedby the cohomology class of the K (1)-relative 1-cocycle e Υ λ,λ defined by (4.19). Note that e Υ λ,λ | K (1) is a coboundary, namely, 2 δ ( θ ∂ + θ ∂ ). Therefore, e Υ λ,λ is cohomologous to the K (1) -relative 1-cocycle e Υ λ,λ − δ ( θ ∂ + θ ∂ ) generating the space H ( K (2) , K (1) ; D λ,λ ).Now, we deduce the result for n ≥ Proposition 4.2. 1) a) For ( λ, µ ) = ( − , , any element of Z ( K (3); D λ,µ ) is a coboundary over K (3) ifand only if at least one of its restrictions to the subalgebras K (2) i is a coboundary.b) For ( λ, µ ) = ( − , , there exists a unique, up to a scalar factor and a coboundary,nontrivial 1-cocycle Υ − , ∈ Z ( K (3); D − , ) such that its restrictions to K (2) , K (2) and to K (2) are coboundaries. This 1-cocycle is odd and it is given by: Υ − , ( X G ) = ∂ ( G ) η η − η η ( ∂ ( G )) (1 − θ ∂ ) − ( − | G | θ M η ( G ) η , (4.28) where, for G ∈ R [ x, θ ] , M G is as (4.17).2) For n > , any element of Z ( K ( n ); D nλ,µ ) is a coboundary over K ( n ) if and only ifat least one of its restrictions to the subalgebras K ( n − i is a coboundary. Proof. Let Υ ∈ Z ( K ( n ); D nλ,µ ) and assume that the restriction of Υ to some K ( n − i is a coboundary, that is, there exists b ∈ D nλ,µ such thatΥ( X F ) = δ ( b )( X F ) = ( − | F || b | X F · b for all X F ∈ K ( n − i . By replacing Υ by Υ − δb , we can suppose that Υ | K ( n − i = 0. Thus, the map Υ is K ( n − i -invariant and therefore the equation (4.26) becomes:( − | e g || Υ | X gθ i · Υ( X hθ i ) − ( − | e h | ( | e g | + | Υ | ) X hθ i · Υ( X gθ i ) = 0 . (4.29)According to the isomorphism (2.10), the map Υ is decomposed into four componentsΠ( F n − ,i − ) ⊗ F n − ,iλ → F n − ,iµ , Π( F n − ,i − ) ⊗ Π( F n − ,iλ + ) → Π( F n − ,iµ + ) , Π( F n − ,i − ) ⊗ F n − ,iλ → Π( F n − ,iµ + ) , Π( F n − ,i − ) ⊗ Π( F n − ,iλ + ) → F n − ,iµ . (4.30)So, each of these bilinear maps is K ( n − i -invariant. Therefore, their expressions are given byTheorem 3.1 with the help of isomorphisms (2.11) and (3.14). More precisely, using equation(4.29), we get up to a scalar factor: • For n ≥ λ, µ ) = ( − , 0) if n = 3 , Υ = δ ( θ i ) if µ = λ − ,δ (1 − θ i ∂ i ) if µ = λ,δ ( ∂ i ) if µ = λ + , For n = 3 and µ = λ + = 0 , Υ = ǫ Υ ,i − , + ǫ δ ( ∂ i ) , (4.32)where Υ ,i − , is the 1-cocycle on K (3) with coefficients in D − , defined byΥ ,i − , ( X G ) = ∂ i ( G ) η ℓ η k − η ℓ η k ( ∂ i ( G )) (1 − θ i ∂ i ) ++ θ i ( η ℓ η i ( G ) η k − η k η i ( G ) η ℓ ) η i (4.33)with ℓ, k = i and ℓ < k. Obviously, Υ , − , = Υ − , with Υ − , is as in (4.28), and a directcomputation shows that for j = 1 , − , + ( − j Υ ,j − , = 2( − j δ (( θ η j + θ j η ) η − j ) . (4.34)Thus, up to a scalar factor and a coboundary, Υ = Υ − , . Therefore, in order to completethe proof of Proposition 4.2, we have to study the cohomology class of the 1-cocycle Υ − , inH ( K (3) , D − , ) . Lemma 4.3. The 1-cocycle Υ − , defines a nontrivial cohomology class over K (3) . Its re-strictions to K (2) , K (2) and to K (2) are coboundaries. Proof. It follows from equation (4.34) that the restriction of Υ − , to K (2) vanishes andto K (2) and to K (2) are coboundaries. Now, assume that there exists an odd operator A ∈ D − , such that Υ − , is equal to δA. By isomorphism (4.21), the operator A can beexpressed as A = Φ − , ( A , A , Π( A ) , Π( A )) , where A ∈ D − , , A ∈ D , , A ∈ D − , and A ∈ D , . Thus, since the map D λ,µ → Π( D λ,µ ) , B Π( B ◦ σ )is a K (2)-isomorphism, the condition Υ − , |K (2) = 0 tell us that A , A , A ◦ σ and A ◦ σ are K (2)-invariant linear maps. Therefore, up to a scalar factor, each of A , A , A ◦ σ and A ◦ σ is the identity map [8]: F λ → F λ , F α λ F α λ . Thus, we obtain A ( F α − ) = ǫ∂ ( F ) . Finally, it is easy to check that the equation Υ − , = δ ( A ) has no solutions contradicting ourassumption. Lemma 4.3 is proved. Thus we have completed the proof of Proposition 4.2. ✷ Corollary 4.4. Up to a coboundary, any 1-cocycle Υ ∈ Z ( K (3); D λ,µ ) has the followinggeneral form: Υ( X F ) = X a ℓ ℓ ℓ m m m η ℓ η ℓ η ℓ ( F ) η k η k η k , (4.35) where the coefficients a ℓ ℓ ℓ k k k are functions of θ i , not depending on x . a ℓ ℓ ℓ k k k are some functions of x and θ i , but we shall now prove that ∂ x a ℓ ℓ ℓ k k k = 0. To do that, we shall simply show that X · Υ = 0.We have ( X · Υ)( X F ) := X · Υ( X F ) − Υ([ X , X F ]) for all F ∈ R [ x, θ ] . (4.36)But, from Proposition 4.2 and Corollary 4.2, it follows that, up to a coboundary, Υ( X ) = 0,and therefore the equation (4.36) becomes( X · Υ)( X F ) = X · Υ( X F ) − ( − | F || Υ | X F · Υ( X ) − Υ([ X , X F ]) . (4.37)The right-hand side of (4.37) vanishes because Υ is a 1-cocycle. Thus, X · Υ = 0. ✷ The following lemma gives a description of all coboundaries over K (2) i vanishing on thesubalgebra K (1) m i , where m i ∈ { , , } \ { i } . This description will be useful in the proof ofTheorem 4.1. Lemma 4.5. (see [2]) Any coboundary B i,m i λ,µ ∈ B ( K (2) i ; D ,iλ,µ ) vanishing on K (1) m i is, upto a scalar factor, given by B i,m i λ,µ = δ ( ǫ ∂ m i + ǫ η − i − m i ( θ m i η m i − if ( λ, µ ) = (0 , ) δ ( ǫ ∂ m i + ǫ θ m i η − i − m i η m i ) if ( λ, µ ) = ( − , δ ( ǫ θ m i η − i − m i + ǫ θ m i η m i ) if λ = µ = 0 δ ( ∂ m i η − i − m i ) if ( λ, µ ) = ( − , ) δ ( θ m i η m i ) if λ = µ = 0 δ ( ∂ m i ) if µ = λ + and λ = 0 , − δ ( θ m i ) if µ = λ − otherwise. (4.38) (i) According to Proposition 4.2, the restriction of any nontrivial differential 1-cocycle Υ of K (3) with coefficients in D λ,µ to K (2) i , for i =1, 2, 3, is a nontrivial 1-cocycle except forΥ = ǫ Υ − , + δA, (4.39)where Υ − , is as (4.28), ǫ = 0 and A ∈ D − , . So, if 2( µ − λ ) = − , , , , , , 5, then, byCorollary 4.2, the corresponding cohomology spaces H ( K (3); D λ,µ ) vanish.For 2( µ − λ ) = − , , , , , , 5, let Υ be a 1-cocycle from K (3) to D λ,µ . The map Υ |K (2) i is a 1-cocycle of K (2) i . Therefore, using Corollary 4.2 together with Lemma 4.5 and Theorem4.3 with the help of isomorphism (4.21), we deduce that, up to a coboundary, the non-zerorestrictions of the cocycle Υ on K (2) i can be expressed as (here τ = µ − λ ):For τ = − , , Υ | K (2) i = a ( − i e Θ i, , λ,µ + i (3 − i )2 b (cid:16) Γ i, , , λ,µ + Γ i, , , λ,µ (cid:17) if τ = − , λ = 0 a (cid:16) Λ i , δ ( e A ) − ( − i e Θ i, , λ,µ (cid:17) + b ( Λ i , − Λ i , ) (cid:16) Γ i, , , λ,µ + Γ i, , , λ,µ (cid:17) if τ = − , λ = 0 a ( − i ( e Θ i, , λ,µ − Θ i, , λ,µ ) if τ = a (cid:16) (2 λ + 2)Θ i, , λ,µ + (2 λ + 3)Θ i, , λ,µ (cid:17) if τ = 2 τ = 0, Υ | K (2) i = a (cid:16) Θ i, , λ,µ + Θ i, , λ,µ (cid:17) + b i (cid:16) e Θ i, , λ,µ + e Θ i, , λ,µ (cid:17) +Λ i , (cid:16) b (cid:16) Γ , , , λ,µ − Γ , , , λ,µ (cid:17) − t (cid:16) Γ , , , λ,µ + Γ , , , λ,µ (cid:17)(cid:17) +Λ i , (cid:16) b δ ( e A ) + b (Γ , , , λ,µ − Γ , , , λ,µ ) + t (cid:16) Γ , , , λ,µ + Γ , , , λ,µ (cid:17)(cid:17) if µ = 0 , − a (cid:16) Θ i, , λ,µ + Θ i, , λ,µ (cid:17) + b i e Θ i, , λ,µ − i , (cid:16) b Γ , , , λ,µ, , + b Γ , , , λ,µ, , + b (Γ , , , λ,µ, , − Γ , , , λ,µ ) + t (Γ , , , λ,µ + Γ , , , λ,µ, , ) (cid:17) +Λ i , (cid:16) b ( δ ( e A ) − Γ , , , λ,µ, , ) + b (Γ , , , λ,µ, , − Γ , , , λ,µ )+ b Γ , , , λ,µ, , + t (Γ , , , λ,µ + Γ , , , λ,µ, , ) (cid:17) if µ = 0 a (cid:16) Θ i, , λ,µ + Θ i, , λ,µ (cid:17) + b i e Θ i, , λ,µ − i , (cid:16) b Γ , , , λ,µ, , + b Γ , , , λ,µ, , + b (Γ , , , λ,µ, , − Γ , , , λ,µ ) + t (Γ , , , λ,µ + Γ , , , λ,µ, , ) (cid:17) +Λ i , (cid:16) b ( δ ( e A ) − Γ , , , λ,µ, , ) + b (Γ , , , λ,µ, , − Γ , , , λ,µ )+ b Γ , , , λ,µ, , + t (Γ , , , λ,µ + Γ , , , λ,µ, , ) (cid:17) if µ = − For τ = , Υ | K (2) i = b ( − i (Θ i, , λ,µ − e Θ i, , λ,µ ) + ( Λ i , − Λ i , ) ×× (cid:16) a (Γ i, , , λ,µ, , + Γ i, , , λ,µ, , ) + ( − i t (Γ i, , , λ,µ, , − Γ i, , , λ,µ ) (cid:17) if µ = 0( − i ( a Θ i, , λ,µ + b e Θ i, , λ,µ ) + t ( Λ i , − Λ i , ) ×× (Γ i, , , λ,µ, , + Γ i, , , λ,µ ) − b Λ i , δ ( e A ) if µ = ( − i ( a Θ i, , λ,µ + b e Θ i, , λ,µ ) + t ( Λ i , − Λ i , ) ×× (Γ i, , , λ,µ + Γ i, , , λ,µ, , ) − b Λ i , δ ( e A ) if µ = − ( − i ( a Θ i, , λ,µ + b e Θ i, , λ,µ ) + t ( Λ i , − Λ i , ) ×× (Γ i, , , λ,µ + Γ i, , , λ,µ ) − b Λ i , δ ( e A ) if µ = ± , , where (recall that B i,m i λ,µ depend on ǫ , ǫ ) Γ i,m i ,j,ℓλ,µ,ǫ ,ǫ ( X G ) = Φ iλ,µ (cid:16) Π j + ℓ (cid:16) σ j + ℓ ◦ B i,m i λ + j ,µ + ℓ ( X G ) (cid:17)(cid:17) , e A jℓ = Φ λ,µ (cid:16) Π j + ℓ (cid:16) σ j + ℓ ◦ A λ + j ,µ + ℓ (cid:17)(cid:17) with A λ + j ,µ + ℓ = θ ∂ + θ ∂ ∈ D λ + j ,µ + ℓ , e A = e A + e A , Λ ir,s = ( i − r )( i − s ) , Θ i,j,ℓλ,µ and e Θ i,j,ℓλ,µ are defined by (4.23)–(4.24) and the coefficients a, b, b i and t are constants.So, by Proposition 4.2, H ( K (3); D λ,µ ) = 0 for µ − λ = 1 , . Now, by Corollary 4.4, we canwrite Υ( X hθ θ θ ) = X m,k, ε =( ε ,ε ,ε ) a ,m,k,ε + X j =1 X ≤ i < ···
1. For each case, we solve the equations (4.25) and (4.26) for a, b, b i , t, a ,m,k,ε ,a i ··· i j ,m,k,ε . We obtain1) For 2( µ − λ ) = − , , the coefficient a vanishes; so, by Proposition 4.2, Υ is a coboundary.Hence H ( K (3); D λ,µ ) = 0.2) For µ = λ, the coefficients b i vanish and, up to a coboundary, Υ is a multiple of Υ λ,λ , see Theorem 4.1. Hence dimH ( K (3); D λ,λ ) = 1.3) For 2( µ − λ ) = 1 , the coefficient b vanishes and, up to a coboundary, Υ is a multiple ofΥ λ,λ + , see Theorem 4.1. Hence dimH ( K (3); D λ,λ + ) = 1.4) For 2( µ − λ ) = 3, Υ is a multiple of Υ λ,λ + . Hence dimH ( K (3); D λ,λ + ) = 1.(ii) Note that, by Proposition 4.2, the restriction of any nontrivial differential 1-cocycle Υof K (4) with coefficients in D λ,µ to K (3) i , for i = 1 , . . . , 4, is a nontrivial 1-cocycle. Further-more, using arguments similar to those of the proof of Corollary 4.2 together with the aboveresult, we deduce that H ( K (3) i ; D λ,µ ) = 0 if 2( µ − λ ) = − , , , , , . Then, we consideronly the cases where 2( µ − λ ) = − , , , , , n = 4.(iii) We proceed by recurrence over n . In a similar way as in (ii), we get the result for n = 5. Now, we assume that it holds for some n ≥ 5. Again, the same arguments as in theproof of Corollary 4.2 together with recurrence assumption show that H ( K ( n ) i ; D n +1 λ,µ ) = 0if 2( µ − λ ) = − , , . So, we consider only the cases where 2( µ − λ ) = − , , 1, we proceedas in (i) and we get the result for n + 1. ✷ Acknowledgements We are pleased to thank Dimitry Leites and Valentin Ovsienko formany stimulating discussions and valuable correspondences. References [1] Basdouri, I., Ben Ammar, M., Ben Fraj, N., Boujelbene, M., Kammoun, K.: Cohomologyof the Lie superalgebra of contact vector fields on K | and deformations of the superspaceof symbols. J. Nonlinear Math. Phys. (2009, to appear).[2] Ben Fraj, N.: Cohomology of K (2) acting on linear differential operators on the superspace R | , Lett. Math. Phys. , 159-175 (2008).[3] Conley, C. H.: Conformal symbols and the action of contact vector fields over the super-line, arXiv: 0712.1780v2 [math.RT] .[4] Duval, C; Michel, J. P.: On the projective geometry of the supercircle: a unifiedconstruction of the super cross-ratio and Schwarzian derivative, arXiv: 0710.1544v3[math-ph]. [5] Feigin, B. L., Fuchs, D. B.: Homology of the Lie algebras of vector fields on the line.Func. Anal. Appl. , 201–212 (1980).[6] Gargoubi, H., Mellouli, N., Ovsienko, V.: Differential operators on supercircle: confor-mally equivariant quantization and symbol calculus. Lett. Math. Phys. , 51-65 (2007).167] Gargoubi, H., Ovsienko, V.: Supertransvectants and symplectic geometry, arXiv:0705.1411v1 [math-ph] .[8] Leites, D.: Lie superalgebras. In: Modern Problems of Mathematics. Recent develope-ments, vol. 25, pp. 3–49. VINITI, Moscow (1984, in Russian) [English translation in:JOSMAR v. 30(6), 2481–2512 (1985)].[9] Leites, D., Kochetkov, Yu; Weintrob, A.: New invariant differential operators on super-manifolds and pseudo-(co)homology. In: Lecture Notes in Pure and Applied Mathematics,134, pp. 217–238. Dekker, New York (1991).[10] Grozman, P.: Classification of bilinear invariant operators over tensor fields. FunctionalAnal. Appl.,14:2,(1980), 127–128; for details and proofs, see arXiv: math/0509562arXiv: math/0509562