supertransvectants, cohomology, and deformations

20
Supertransvectants, cohomology, and deformations Nizar Ben Fraj, Ismail Laraiedh, and Salem Omri Citation: J. Math. Phys. 54, 023501 (2013); doi: 10.1063/1.4789539 View online: http://dx.doi.org/10.1063/1.4789539 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i2 Published by the AIP Publishing LLC. Additional information on J. Math. Phys. Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Upload: salem

Post on 15-Dec-2016

215 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: Supertransvectants, cohomology, and deformations

Supertransvectants, cohomology, and deformationsNizar Ben Fraj, Ismail Laraiedh, and Salem Omri Citation: J. Math. Phys. 54, 023501 (2013); doi: 10.1063/1.4789539 View online: http://dx.doi.org/10.1063/1.4789539 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i2 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 2: Supertransvectants, cohomology, and deformations

JOURNAL OF MATHEMATICAL PHYSICS 54, 023501 (2013)

Supertransvectants, cohomology, and deformationsNizar Ben Fraj,1,a) Ismail Laraiedh,2,b) and Salem Omri3,c)

1Institut Preparatoire aux Etudes D’ingenieur de Nabeul, University of Carthage, Tunisie2Departement de Mathematiques, Faculte des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie3Departement de Mathematiques, Faculte des Sciences de Gafsa, Zarroug 2112,Gafsa, Tunisie

(Received 9 November 2012; accepted 10 January 2013; published online 1 February 2013)

Over the (1, N)-dimensional real superspace, N = 2, 3, we classify osp(N |2)-invariantbinary differential operators acting on the superspaces of weighted densities, whereosp(N |2) is the orthosymplectic Lie superalgebra. This result allows us to computethe first differential osp(N |2)-relative cohomology of the Lie superalgebra K(N ) ofcontact vector fields with coefficients in the superspace of linear differential operatorsacting on the superspaces of weighted densities. We classify generic formal osp(3|2)-trivial deformations of the K(3)-module structure on the superspaces of symbols ofdifferential operators. We prove that any generic formal osp(3|2)-trivial deformationof this K(3)-module is equivalent to its infinitesimal part. This work is the simplestgeneralization of a result by the first author et al. [Basdouri, I., Ben Ammar, M., BenFraj, N., Boujelbene, M., and Kammoun, K., “Cohomology of the Lie superalgebraof contact vector fields on K1|1 and deformations of the superspace of symbols,”J. Nonlinear Math. Phys. 16, 373 (2009)]. C© 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4789539]

I. INTRODUCTION

For motivations, see Bouarroudj’s paper7 (see also Refs. 3, 6, and 12) of which this work is themost natural superization, other possibilities being cohomology of polynomial versions of variousinfinite dimensional “stringy” Lie superalgebras (for their list, see Ref. 16). This list contains severalinfinite series and several exceptional superalgebras, but to consider cohomology relative a “middle”subsuperalgebra similar, in a sense, to sl(2) is only possible when such a subsuperalgebra exists,which only happens in a few cases. Here, we consider the simplest of such cases.

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]:

X ddx

( f ) = X f ′ + λX ′ f,

where X, f ∈ R[x], and X ′ := d Xdx . Denote by Fλ the vect(1)-module structure on R[x] defined

by Lλ for a fixed λ. Geometrically, Fλ = {f dxλ | f ∈ R[x]

}is the space of polynomial weighted

densities of weight λ ∈ R. The space Fλ coincides with the space of vector fields, functions, anddifferential 1-forms for λ = − 1, 0 and 1, respectively.

Denote by Dλ,μ := Homdiff(Fλ,Fμ) the vect(1)-module of linear differential operators with thenatural vect(1)-action denoted Lλ,μ

X (A). Each module Dλ,μ has a natural filtration by the order ofdifferential operators; the graded moduleSλ,μ := grDλ,μ is called the space of symbols. The quotient-module Dk

λ,μ/Dk−1λ,μ is isomorphic to the module of weighted densities Fμ−λ−k ; the isomorphism is

a)E-mail: [email protected])E-mail: [email protected])E-mail: [email protected].

0022-2488/2013/54(2)/023501/19/$30.00 C©2013 American Institute of Physics54, 023501-1

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 3: Supertransvectants, cohomology, and deformations

023501-2 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

provided by the principal symbol map σ pr defined by

A =k∑

i=0

ai (x)

(∂

∂x

)i

�→ σpr(A) = ak(x)(dx)μ−λ−k,

(see, e.g., Ref. 11). Therefore, as a vect(1)-module, the space Sλ,μ depends only on the differenceβ = μ − λ, so that Sλ,μ can be written as Sβ , and we have

Sβ =∞⊕

k=0

Fβ−k

as vect(1)-modules. The space of symbols of order ≤n is

Snβ :=

n⊕k=0

Fβ−k .

The space Dλ,μ cannot be isomorphic as a vect(1)-module to the corresponding space of symbols,but is a deformation of this space in the sense of Richardson Neijenhuis.18

If we restrict ourselves to the Lie subalgebra of vect(1) generated by { ddx , x d

dx , x2 ddx }, isomorphic

to sl(2), we get a family of infinite dimensional sl(2)-modules, still denoted Fλ, Dλ,μ, and Sβ .Gordan14 classified all sl(2)-invariant binary differential operators on R acting in the spaces Fλ.Bouarroudj and Ovsienko8 computed H1

diff

(vect(1), sl(2); Dλ,μ

), where Hi

diff denotes the differentialcohomology; that is, only cochains given by differential operators are considered. They showed thatnonzero cohomology H1

diff

(vect(1), sl(2); Dλ,μ

)only appear for particular values of weights that we

call resonant, which satisfy μ − λ ∈ N. Moreover, the formal deformations that become trivial oncethe action is restricted to sl(2) were completely described in Ref. 5.

In this paper, we study the simplest super analog of the problem solved in Ref. 5, 8, and 14,namely, we consider the superspace R1|N equipped with the contact structure determined by a 1-form αN, and the Lie superalgebra K(N ) of contact polynomial vector fields on R1|N . We introducethe K(N )-module FN

λ of λ-densities on R1|N and the K(N )-module of linear differential operators,DN

λ,μ := Homdiff(FNλ ,FN

μ ), which are super analogues of the spaces Fλ and Dλ,μ, respectively. TheLie superalgebra osp(N |2), a super analogue of sl(2), can be realized as a subalgebra of K(N ).We classify all osp(N |2)-invariant binary differential operators on R1|N acting in the spaces FN

λ forN = 2 and 3. We use the result to compute H1

diff

(K(N ), osp(N |2); DN

λ,μ

)for N = 2 and 3. We show

that nonzero cohomology H1diff

(K(N ), osp(N |2); DN

λ,μ

)only appear for resonant values of weights

that satisfy μ − λ ∈ 12N. Moreover, we give explicit basis of these cohomology spaces. These spaces

allows us to study the generic formal osp(N |2)-trivial deformations of the action of K(N ) on thespace SN

μ−λ = ⊕k≥0 FN

μ−λ− k2, a super analogue of Sμ−λ, see Ref. 3.

II. DEFINITIONS AND NOTATIONS

In this section, we recall the main definitions and facts related to the geometry of the superspaceR1|N ; for more details, see Refs. 11 and 15–17.

A. The Lie superalgebra of contact vector fields on R1|N

Let R1|N be the superspace with coordinates (x, θ1, . . . , θN), where θ1, . . . , θN are the oddvariables, equipped with the standard contact structure given by the following 1-form:

αN = dx +N∑

i=1

θi dθi . (2.1)

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 4: Supertransvectants, cohomology, and deformations

023501-3 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

On the space R[x, θ ] := R[x, θ1, . . . , θN ], we consider the contact bracket

{F, G} = FG ′ − F ′G − 1

2(−1)p(F)

N∑i=1

ηi (F) · ηi (G), (2.2)

where ηi = ∂∂θi

− θi∂∂x and p(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 (2.1) as a module over the ring ofpolynomial functions. Let VectPol(R1|N ) be the superspace of polynomial vector fields on R1|N :

VectPol(R1|N ) =

{F0∂x +

N∑i=1

Fi∂i | Fi ∈ R[x, θ ] for all i

},

where ∂i = ∂∂θi

and ∂x = ∂∂x , and consider the superspace K(N ) of contact polynomial vector fields

on R1|N . That is, K(N ) is the superspace of vector fields on R1|N preserving the distribution singledout by the 1-form αN:

K(N ) = {X ∈ VectPol(R

1|N )|there existsF ∈ R[x, θ ]such thatL X (αN ) = FαN}.

The Lie superalgebra K(N ) is spanned by the fields of the form

X F = F∂x − 1

2

N∑i=1

(−1)p(F)ηi (F)ηi , where F ∈ R[x, θ ].

In particular, we have K(0) = vect(1). Observe that L X F (αN ) = X1(F)αN . The bracket in K(N ) canbe written as

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

The Lie superalgebra K(N − 1) can be realized as a subalgebra of K(N ):

K(N − 1) ={

X F ∈ K(N )|∂N F = 0}.

Note also that, for any i in {1, 2, . . . , N − 1}, K(N − 1) is isomorphic to

K(N − 1)i ={

X F ∈ K(N )|∂i F = 0}.

B. The subalgebra osp(N|2)

In K(N ), there is a subalgebra osp(N |2) of projective transformations

osp(N |2) = Span(X1, Xx , Xx2 , Xθi , Xxθi , Xθi θ j

), 1 ≤ i, j ≤ N .

We easily see that osp(N − 1|2) is a subalgebra of osp(N |2):

osp(N − 1|2) ={

X F ∈ osp(N |2)|∂N F = 0}.

Note also that, for any i in {1, 2, . . . , N − 1}, osp(N − 1|2) is isomorphic to

osp(N − 1|2)i ={

X F ∈ osp(N |2)|∂i F = 0}.

C. Modules of weighted densities

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 XF, define a one-parameter family of first-orderdifferential operators on R[x, θ ]:

LλX F

= X F + λF ′, λ ∈ R. (2.3)

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 5: Supertransvectants, cohomology, and deformations

023501-4 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

We easily check that

[LλX F

,LλXG

] = LλX{F,G} . (2.4)

We thus obtain a one-parameter family of K(N )-modules on R[x, θ ] that we denote FNλ , the space

of all polynomial weighted densities on R1|N of weight λ with respect to αN:

FNλ = {

FαλN | F ∈ R[x, θ ]

}. (2.5)

In particular, we have F0λ = Fλ. Obviously, the adjoint K(N )-module is isomorphic to the space of

weighted densities on R1|N of weight − 1.

D. Differential operators on weighted densities

A differential operator on R1|N is an operator on R[x, θ ] of the form

A =M∑

k=0

∑ε=(ε1,···,εN )

ak,ε(x, θ )∂kx ∂

ε11 · · · ∂εN

N ; εi = 0, 1; M ∈ N. (2.6)

Of course, any differential operator defines a linear mapping FαλN �→ (AF)αμ

N from FNλ to FN

μ forany λ, μ ∈ R, thus the space of differential operators becomes a family of K(N )-modules DN

λ,μ forthe natural action

X F · A = Lμ

X F◦ A − (−1)p(A)p(F) A ◦ Lλ

X F. (2.7)

Similarly, we consider a multi-parameter family of K(N )-modules on the space DNλ1,...,λm ;μ of multi-

linear differential operators: A : FNλ1

⊗ · · · ⊗ FNλm

−→ FNμ with the natural K(N )-action

X F · A = Lμ

X F◦ A − (−1)p(A)p(F) A ◦ L

λ1,...,λmX F

,

where Lλ1,...,λmX F

is defined by the Leibnitz rule. Since −η2i = ∂x , and ∂i = ηi − θiη

2i , every differential

operator A ∈ DNλ,μ can be expressed in the form

A(FαλN ) =

∑�=(�1,...,�N )

a�(x, θ )η�11 . . . η

�NN (F)αμ

N , (2.8)

where the coefficients a�(x, θ ) are arbitrary polynomial functions.

Proposition 2.1: As a osp(N − 1|2)-module, we have

DNλ,μ;ν � D

N−1λ,μ;ν := D

N−1λ,μ;ν ⊕ D

N−1λ+ 1

2 ,μ+ 12 ;ν

⊕ DN−1λ,μ+ 1

2 ;ν+ 12⊕ D

N−1λ+ 1

2 ,μ;ν+ 12⊕

(D

N−1λ,μ;ν+ 1

2⊕ D

N−1λ,μ+ 1

2 ;ν⊕ D

N−1λ+ 1

2 ,μ;ν⊕ D

N−1λ+ 1

2 ,μ+ 12 ;ν+ 1

2

), (2.9)

where is the change of parity operator.

Proof: Any element F ∈ R[x, θ ] can be uniquely written as follows: F = F1 + F2θN, where∂NF1 = ∂NF2 = 0. Therefore, for any X H ∈ osp(N − 1|2), we easily check that

LλX H

F = LλX H

F1 + (Lλ+ 1

2X H

F2)θN .

Thus, the following map is an osp(N − 1|2)-isomorphism:

ϕλ : FNλ → F

N−1λ ⊕ (FN−1

λ+ 12)

FαλN �→ (F1α

λN−1, (F2α

λ+ 12

N−1)),(2.10)

So, we get the natural osp(N − 1|2)-isomorphism from FNλ ⊗ FN

μ to

FN−1λ ⊗ FN−1

μ ⊕ FN−1λ ⊗ (FN−1

μ+ 12) ⊕ (FN−1

λ+ 12) ⊗ FN−1

μ ⊕ (FN−1λ+ 1

2) ⊗ (FN−1

μ+ 12)

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 6: Supertransvectants, cohomology, and deformations

023501-5 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

denoted ψλ,μ. Therefore, we deduce a osp(N − 1|2)-isomorphism

�λ,μ,ν : DN−1λ,μ;ν → DN

λ,μ;ν

A �→ ϕ−1ν ◦ A ◦ ψλ,μ.

(2.11)

Here, we identify the osp(N − 1|2)-modules via the following isomorphisms:

(D

N−1λ,μ;ν ′

)→ Homdiff

(F

N−1λ ⊗ FN−1

μ , (FN−1ν ′ )

), (A) �→ ◦ A,

(D

N−1λ,μ′;ν

)→ Homdiff

(F

N−1λ ⊗ (FN−1

μ′ ), FN−1ν

), (A) �→ A ◦ (1 ⊗ ),

(D

N−1λ′,μ;ν

)→ Homdiff

( (FN−1

λ′ ) ⊗ FN−1μ , FN−1

ν

), (A) �→ A ◦ ( ⊗ σ ),

(D

N−1λ′,μ′;ν ′

)→ Homdiff

( (FN−1

λ′ ) ⊗ (FN−1μ′ ), (FN−1

ν ′ ))

, (A) �→ ◦ A ◦ ( ⊗ σ ◦ ),

DN−1λ,μ′;ν ′ → Homdiff

(F

N−1λ ⊗ (FN−1

μ′ ), (FN−1ν ′ )

), A �→ ◦ A ◦ (1 ⊗ ),

DN−1λ′,μ′;ν → Homdiff

( (FN−1

λ′ ) ⊗ (FN−1μ′ ), FN−1

ν

), A �→ A ◦ ( ⊗ σ ◦ ),

DN−1λ′,μ;ν ′ → Homdiff

( (FN−1

λ′ ) ⊗ FN−1μ , (FN−1

ν ′ )), A �→ ◦ A ◦ ( ⊗ σ ),

where λ′ = λ + 12 , μ′ = μ + 1

2 , ν ′ = ν + 12 and σ (F) = ( − 1)|F|F. �

III. SUPERTRANSVECTANTS: AN EXPLICIT FORMULA

The 1|N-supertransvectants are the bilinear osp(N |2)-invariant maps

Jτ,λ,Nk : FN

τ ⊗ FNλ −→ FN

τ+λ+k

(FατN , Gαλ

N ) �→ Jτ,λ,Nk (F, G)ατ+λ+k

N ,(3.1)

where k = 0, 12 , 1, 3

2 , 2, . . .. For N = 1, Gieres and Theisen13 listed the supertransvectants. Gargoubiand Ovsienko12 gave an interpretation of these operators. In Ref. 13, the supertransvectants areexpressed in terms of supercovariant derivative. In Ref. 3, the supertransvectants appear in thecontext of the osp(1|2)-relative cohomology and their explicit formula, in accordance with k ∈ Nor k ∈ N + 1

2 , is as follows:

(i) The operators Jτ,λ,1k labeled by semi-integer k are odd; they are given by

Jτ,λ,1k (F, G) =

∑i+ j=[k]

�τ,λi, j,k

((−1)p(F)(2τ + [k] − j)F (i)η1(G( j)) − (2λ + [k] − i)η1(F (i))G( j)

).

(3.2)

(ii) The operators Jτ,λ,1k , where k ∈ N, are even; set J

τ,λ,10 (F, G) = FG and

Jτ,λ,1k (F, G) =

∑i+ j=k−1

(−1)p(F)�τ,λi, j,k−1η1(F (i))η1(G( j)) −

∑i+ j=k

�τ,λi, j,k−1 F (i)G( j), (3.3)

where(x

i

) = x(x−1)···(x−i+1)i! and [k] denotes the integer part of k, k > 0, and

�τ,λi, j,k = (−1) j

(2τ + [k]

j

)(2λ + [k]

i

). (3.4)

(iii) If τ, λ ∈ Ik = {0, − 12 , −1, . . . , − [k]

2 }, then Jτ,λ,1k is the unique (up to a scalar factor) bilinear

osp(1|2)-invariant bilinear differential operator F1τ ⊗ F1

λ −→ F1τ+λ+k .

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 7: Supertransvectants, cohomology, and deformations

023501-6 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

Gargoubi and Ovsienko12 further studied the exceptional cases, they showed that:(iv) If one of the weights τ or λ belongs to Ik but the second one does not, then J

τ,λ,1k is the unique

(up to a scalar factor). If, say, τ = 1−m4 for some odd m, then the corresponding bilinear

osp(1|2)-invariant operators is given by

J1−m

4 ,λ,1k (F, G) = J

1+m4 ,λ,1

k− m2

(ηm1 (F), G). (3.5)

(v) If τ = 1−m4 for some odd m and λ = 1−�

4 for some odd �, and if �+m2 > k, then J

τ,λ,1k is still

unique (up to a scalar factor) and is of the form

J1−m

4 , 1−�4 ,1

k (F, G) = J1+m

4 , 1−�4 ,1

k− m2

(ηm1 (F), G) = J

1−m4 , 1+�

4 ,1

k− �2

(F, η�1(G)). (3.6)

(vi) If τ = 1−m4 for some odd m and λ = 1−�

4 for some odd �, and if �+m2 ≤ k, then the space of

osp(1|2)-invariant bilinear differential operator F1τ ⊗ F1

λ −→ F1τ+λ+k is two dimensional and

spanned by

J1−m

4 , 1−�4 ,1

k (F, G) = J1+m

4 , 1−�4 ,1

k− m2

(ηm1 (F), G) = J

1−m4 , 1+�

4 ,1

k− �2

(F, η�1(G)), (3.7)

J1−m

4 , 1−�4 ,1

k (F, G) = J1+m

4 , 1+�4 ,1

k− m+�2

(ηm1 (F), η�

1(G)). (3.8)

A. osp(2|2)-invariant bilinear differential operators

The third author computed the generic osp(2|2)-invariant bilinear differential operators

Jτ,λ,2k : F2

τ ⊗ F2λ −→ F2

τ+λ+k

(Fατ2 , Gαλ

2 ) �→ Jτ,λ,2k (F, G)ατ+λ+k

2 ,

(3.9)

where k = 0, 12 , 1, 3

2 , 2, . . .. Its explicit formula is

Jτ,λ,20 (F, G) = FG (3.10)

and for k ∈ N\{0}, it is

Jτ,λ,2k (F, G) =

∑1≤p≤2

∑i+ j=k−1

(−1)p(F)�τ,λi, j,k−1ηp(F (i))ηp(G( j)) −

∑i+ j=k

�τ,λi, j,k−1 F (i)G( j)

+∑i+ j=k−2 �

τ,λi, j,k−1η1η2(F (i))η1η2(G( j))

(3.11)

or

Jτ,λ,2k (F, G) =

∑i+ j=k−1

(−1)p(F)�τ,λi+1, j+1,k(i + 1)( j + 1)

(η1(F (i))η2(G( j)) − η2(F (i))η1(G( j))

)+

∑i+ j=k �

τ,λi, j,k

(i(2λ + k − i)η1η2(F (i−1))G( j) − j(2τ + k − j)F (i)η1η2(G( j−1))

).

(3.12)For the sake of completeness, Proposition 3.1 gives the uniqueness of supertansvectants ((3.11)and (3.12)) for the generic (τ , λ), and a description of osp(2|2)-invariant bidifferential operators inexceptional cases.

Proposition 3.1: (i) If τ, λ ∈ Ik+1 = {0, − 12 , −1, . . . , − [k]+1

2 }, then the space of osp(2|2)-invariant bilinear differential operators is spanned by the operator (3.10) for k = 0, and by theoperators (3.11) and (3.12) for k ∈ N\{0}.

(ii) If one of the weights τ or λ belongs to Ik + 1 but the second one does not, then the space ofosp(2|2)-invariant bilinear differential operators is trivial if k is semi-integer and two-dimensional

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 8: Supertransvectants, cohomology, and deformations

023501-7 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

if k is integer. If, say, τ = 1−m4 for some odd m, then the corresponding space of osp(2|2)-invariant

bilinear differential operators is spanned by

J1−m

4 ,λ,2k (F, G) = J

m−14 ,λ,2

k− m2 + 1

2(ηm−2

1 η2(F), G), (3.13)

J1−m

4 ,λ,2k (F, G) = J

m−14 ,λ,2

k− m2 + 1

2(ηm−2

1 η2(F), G). (3.14)

If, say, λ = 1−�4 for some odd �, then the corresponding space of osp(2|2)-invariant bilinear differ-

ential operators is spanned by

Jτ, 1−�

4 ,2k (F, G) = J

τ, �−14 ,2

k− l2 + 1

2(F, η�−2

1 η2(G)), (3.15)

Jτ, 1−�

4 ,2k (F, G) = J

τ, �−14 ,2

k− �2 + 1

2(F, η�−2

1 η2(G)). (3.16)

(iii) If τ = 1−m4 for some odd m and λ = 1−�

4 for some odd � and if �+m2 > k + 1 then the

corresponding space of osp(2|2)-invariant bilinear differential operators is still two-dimensionaland spanned by

Jm−1

4 , 1−�4 ,2

k− m2 + 1

2(ηm−2

1 η2(F), G) = J1−m

4 , �−14 ,2

k− l2 + 1

2(F, η�−2

1 η2(G)), (3.17)

Jm−1

4 , 1−�4 ,2

k− m2 + 1

2(ηm−2

1 η2(F), G) = J1−m

4 , �−14 ,2

k− �2 + 1

2(F, η�−2

1 η2(G)). (3.18)

(iv) If τ = 1−m4 for some odd m and λ = 1−�

4 for some odd � and if l+m2 ≤ k + 1 then the

corresponding space of osp(2|2)-invariant bilinear differential operators is four-dimensional andspanned by

Jm−1

4 , 1−�4 ,2

k− m2 + 1

2(ηm−2

1 η2(F), G) = J1−m

4 , �−14 ,2

k− l2 + 1

2(F, η�−2

1 η2(G)), (3.19)

Jm−1

4 , 1−�4 ,2

k− m2 + 1

2(ηm−2

1 η2(F), G) = J1−m

4 , �−14 ,2

k− l2 + 1

2(F, η�−2

1 η2(G)), (3.20)

(−1)p(F)Jm−1

4 , �−14 ,2

k− m+�2 +1

(ηm−21 η2(F), η�−2

1 η2(G)), (3.21)

(−1)p(F )Jm−1

4 , �−14 ,2

k− m+�2 +1

(ηm−21 η2(F), η�−2

1 η2(G)). (3.22)

Proof: Let T : F2τ ⊗ F2

λ −→ F2μ be an osp(2|2)-invariant bilinear differential operator. Observe

that the osp(2|2)-invariance of T is equivalent to invariance with respect just to the subalgebrasosp(1|2) and osp(1|2)1. Note that, the osp(1|2)-invariant elements of (D2

τ,λ;μ) can be deducedfrom those given in (3.2) and (3.3) by using the following osp(1|2)-isomorphism:

D1τ,λ;μ → (D1

τ,λ;μ), A �→ (A ◦ (σ ⊗ σ )). (3.23)

Now, by isomorphism (2.11), we exhibit the osp(1|2)-invariant elements of D2τ,λ;μ. Of course, these

elements are identically zero if 2(μ − τ − λ) /∈ N. So, the parameters τ , λ and μ must satisfy

μ = τ + λ + k, where k ∈ 1

2N.

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 9: Supertransvectants, cohomology, and deformations

023501-8 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

The corresponding operators will be denoted Jτ,λ,2k . Obviously, if k is integer, then the operator J

τ,λ,2k

is even and if k is semi-integer, then the operator Jτ,λ,2k is odd.

(i) The uniqueness of 1|2-supertansvectants follows from the uniqueness of 1|1-supertansvectants.

(ii) For k ∈ N and for fixed τ = 1−m4 for some odd m, any osp(2|2)-invariant element J

τ,λ,2k of

D2τ,λ;μ with μ = τ + λ + k, can be expressed as follows:

Jτ,λ,2k = �τ,λ,μ

(�0

τ,λ,kJτ,λ,1k + �1

τ,λ,kJτ+ 1

2 ,λ+ 12 ,1

k−1 + �2τ,λ,kJ

τ,λ+ 12 ,1

k + �3τ,λ,kJ

τ+ 12 ,λ,1

k

)+

�τ,λ,μ

(

((�0

τ,λ,kJτ,λ,1k+ 1

2+ �1

τ,λ,kJτ,λ+ 1

2 ,1

k− 12

+ �2τ,λ,kJ

τ+ 12 ,λ,1

k− 12

+ �3τ,λ,kJ

τ+ 12 ,λ+ 1

2 ,1

k− 12

)◦ (σ ⊗ σ )

)),

(3.24)where J

τ,λ,1k are defined by (3.5) and the coefficients ��

τ,λ,k and ��τ,λ,k with � = 0, 1, 2, 3 are,

a priori, arbitrary constants, but the invariance of Jτ,λ,2k with respect osp(1|2)1 imposes some

supplementary conditions over these coefficients and determines thus completely the even space ofosp(2|2)-invariant elements of D2

τ,λ;μ. By a direct computation, we get

(2λ + k − m−12 )�1

τ,λ,k = −k�0τ,λ,k,

(2λ + k − m−12 )�2

τ,λ,k = (2λ + k)�0τ,λ,k,

(2λ + k − m−12 )�3

τ,λ,k = −(k − m−12 )�0

τ,λ,k,

�0τ,λ,k = �1

τ,λ,k = �2τ,λ,k = �3

τ,λ,k .

Therefore, we easily check that Jτ,λ,2k is expressed as in (3.13) and (3.14).

Similarly, for k ∈ N and for fixed λ = 1−�4 for some odd �, we get

(2τ + k − �−12 )�1

τ,λ,k = −k�0τ,λ,k,

(2τ + k − �−12 )�2

τ,λ,k = (k − �−12 )�0

τ,λ,k,

(2τ + k − �−12 )�3

τ,λ,k = −(2τ + k)�0τ,λ,k

�0τ,λ,k = �1

τ,λ,k = �2τ,λ,k = �3

τ,λ,k .

Therefore, we easily check that Jτ,λ,2k is expressed as in (3.15) and (3.16).

Now, for k ∈ N + 12 and τ = 1−m

4 for some odd m or λ = 1−�4 for some odd �, then the same

arguments as before, show that the space of osp(2|2)-invariant elements of D2τ,λ;τ+λ+k vanish.

(iii) For τ = 1−m4 for some odd m and λ = 1−�

4 for some odd � and if �+m2 > k + 1, then by

substituting expression (3.6) of Jτ,λ,1k in (3.24) and using the same arguments as before, we get

(k + 1 − m+�2 )�1

τ,λ,k = −k�0τ,λ,k,

(k + 1 − m+�2 )�2

τ,λ,k = (k − �−12 )�0

τ,λ,k,

(k + 1 − m+�2 )�3

τ,λ,k = −(k − m−12 )�0

τ,λ,k,

�0τ,λ,k = �1

τ,λ,k = �2τ,λ,k = �3

τ,λ,k .

Therefore, we easily check that Jτ,λ,2k is expressed as in (3.17) and (3.18).

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 10: Supertransvectants, cohomology, and deformations

023501-9 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

(iv) For τ = 1−m4 for some odd m and λ = 1−�

4 for some odd � and if �+m2 ≤ k + 1, any

osp(2|2)-invariant element Jτ,λ,2k of D2

τ,λ;μ with μ = τ + λ + k, can be expressed as follows:

Jτ,λ,2k = �τ,λ,μ

(�0

τ,λ,kJτ,λ,1k + �1

τ,λ,kJτ+ 1

2 ,λ+ 12 ,1

k−1 + �2τ,λ,kJ

τ,λ+ 12 ,1

k + �3τ,λ,kJ

τ+ 12 ,λ,1

k

)+

�τ,λ,μ

(�0

τ,λ,k Jτ,λ,1k + �1

τ,λ,k Jτ+ 1

2 ,λ+ 12 ,1

k−1 + �2τ,λ,k J

τ,λ+ 12 ,1

k + �3τ,λ,k J

τ+ 12 ,λ,1

k

)+

�τλ,μ

(

((�0

τ,λ,kJτ,λ,1k+ 1

2+ �1

τ,λ,kJτ,λ+ 1

2 ,1

k− 12

+ �2τ,λ,kJ

τ+ 12 ,λ,1

k− 12

+ �3τ,λ,kJ

τ+ 12 ,λ+ 1

2 ,1

k− 12

)◦ (σ ⊗ σ )

))+

�τ,λ,μ

(

((�0

τ,λ,k Jτ,λ,1k+ 1

2+ �1

τ,λ,k Jτ,λ+ 1

2 ,1

k− 12

+ �2τ,λ,k J

τ+ 12 ,λ,1

k− 12

+ �3τ,λ,k J

τ+ 12 ,λ+ 1

2 ,1

k− 12

)◦ (σ ⊗ σ )

)),

where Jτ,λ,1k , J

τ,λ,1k are defined by (3.7) and (3.8) and the coefficients ��

τ,λ,k, ��τ,λ,k, �

�τ,λ,k and ��

τ,λ,k

with � = 0, 1, 2, 3 are, a priori, arbitrary constants. As before, the invariance of Jτ,λ,2k with respect

osp(1|2)1, show that

( 1−m2 − �−1

2 + k)�1τ,λ,k = −k�0

τ,λ,k, k�1τ,λ,k = (k + 1−m

2 )�0τ,λ,k,

( 1−m2 − �−1

2 + k)�2τ,λ,k = (k − �−1

2 )�0τ,λ,k, −k�2

τ,λ,k = ( 1−�2 + k)�0

τ,λ,k,

( 1−m2 − �−1

2 + k)�3τ,λ,k = −( 1−m

2 + k)�0τ,λ,k, −k�3

τ,λ,k = (k − m−12 − �−1

2 )�0τ,λ,k,

�0τ,λ,k = �1

τ,λ,k = �2τ,λ,k = �3

τ,λ,k, �0τ,λ,k = �1

τ,λ,k = �2τ,λ,k = �3

τ,λ,k .

Therefore, we easily check that Jτ,λ,2k is expressed as in (3.19)–(3.22). �

Remark 3.2: It can be checked, by the same arguments as in the proof of part (ii) of Proposition3.1, that the operators (3.11) and (3.12) are, indeed, osp(2|2)-invariant.

IV. osp(3|2)-INVARIANT BILINEAR DIFFERENTIAL OPERATORS

Now, we describe the spaces of osp(3|2)-invariant bilinear differential operators F3τ ⊗ F3

λ

−→ F3μ. Our main result of this section is the following.

Theorem 4.1: (i) There are only the following osp(3|2)-invariant bilinear differential operatorsacting in the spaces F3

τ :

Jτ,λ,3k : F3

τ ⊗ F3λ −→ F3

τ+λ+k

(Fατ3 , Gαλ

3 ) �→ Jτ,λ,3k (F, G)ατ+λ+k

3 ,

where k ∈ 12N. The operators J

τ,λ,3k labeled by semi-integer k, where 2k ∈ N\{0, 1}, are odd; they

are given by

Jτ,λ,3k (F, G) = −

∑1≤p<�≤3

∑i+ j=[k]−1

�τ,λi+1, j+2,k+1�

ijη6−�−p(F (i))ηpη�(G( j))

−∑

1≤p<�≤3

∑i+ j=[k]−1

(−1)p(F)�τ,λi+2, j+1,k+1�

ji ηpη�(F (i))η6−�−p(G( j))

+∑

i+ j=[k]−1

(−1)p(F)�τ,λi, j+3,k+1�

jj+1 F (i)η1η2η3(G( j))

+∑

i+ j=[k]−1

�τ,λi+3, j,k+1�

ii+1η1η2η3(F (i))G( j).

(4.1)

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 11: Supertransvectants, cohomology, and deformations

023501-10 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

The operators Jτ,λ,3k , where k ∈ N, are even; and

Jτ,λ,3k (F, G) = −

∑i+ j=k

�τ,λi, j,k−1 F (i)G( j) −

∑1≤p≤3

∑i+ j=k−1

(−1)p(F)�τ,λi, j,k−1ηp(F (i))ηp(G( j))

−∑

1≤p<�≤3

∑i+ j=k−2

�τ,λi, j,k−1ηpη�(F (i))ηpηl(G

( j))

+∑

i+ j=k−3

(−1)p(F)�τ,λi, j,k−1η1η2η3(F (i))η1η2η3(G( j)),

(4.2)where �i

j = ( j + 2)( j + 1)(i + 1).

(ii) If τ, λ ∈ Ik+2 = {0, − 12 , −1, . . . , − [k]+2

2 }, then Jτ,λ,3k is the unique (up to a scalar factor)

osp(3|2)-invariant bilinear differential operator F3τ ⊗ F3

λ −→ F3τ+λ+k .

(iii) If one of the weights τ or λ belongs to Ik + 2 but the second one does not, then Jτ,λ,3k is the

unique (up to a scalar factor). If, say, τ = 1−m4 for some odd m, then the corresponding bilinear

osp(3|2)-invariant operator is given by

Jτ,λ,3k (F, G) = J

m−34 ,λ,3

k− m−22

(ηm−41 η2η3(F), G). (4.3)

If, say, λ = 1−�4 for some odd �, then the corresponding space of osp(3|2)-invariant bilinear differ-

ential operators is spanned by

Jτ,λ,3k (F, G) = J

τ, �−34 ,3

k− �−22

(F, η�−41 η2η3(G)). (4.4)

(iv) If τ = 1−m4 for some odd m and λ = 1−�

4 for some odd �, and if �+m2 > k + 2, then J

τ,λ,3k is

still unique (up to a scalar factor) and is of the form:

Jτ,λ,3k (F, G) = J

m−34 , 1−�

4 ,3

k− m−22

(ηm−41 η2η3(F), G) = J

1−m4 , �−3

4 ,3

k− �−22

(F, η�−41 η2η3(G)). (4.5)

(v) If τ = 1−m4 for some odd m and λ = 1−�

4 for some odd �, and if �+m2 ≤ k + 2, then the space

of osp(3|2)-invariant bilinear differential operators F3τ ⊗ F3

λ −→ F3τ+λ+k is two dimensional and

spanned by

Jτ,λ,3k (F, G) = J

m−34 , 1−�

4 ,3

k− m−22

(ηm−41 η2η3(F), G) = J

1−m4 , �−3

4 ,3

k− �−22

(F, η�−41 η2η3(G)), (4.6)

Jτ,λ,3k (F, G) = (−1)p(F)J

m−34 , �−3

4 ,3

k− m+�2 +2

(ηm−41 η2η3(F), η�−4

1 η2η3(G)). (4.7)

Proof: (i) Let T : F3τ ⊗ F3

λ −→ F3μ be an osp(3|2)-invariant bilinear differential operator. Ob-

serve that the osp(3|2)-invariance of T is equivalent to invariance with respect just to the subalgebrasosp(2|2) and osp(2|2)i , i = 1, 2. Obviously, the osp(2|2)-invariant elements of (D2

τ,λ;μ) can be de-duced from those given in (3.11) and (3.12) by using the following osp(2|2)-isomorphism:

D2τ,λ;μ → (D2

τ,λ;μ), A �→ (A ◦ (σ ⊗ σ )). (4.8)

Now, by isomorphism (2.11), we exhibit the osp(2|2)-invariant elements of D3τ,λ;μ. Of course, these

elements are identically zero if 2(μ − λ − τ ) /∈ N. So, the parameters τ , λ and μ must satisfy

μ = λ + τ + k, where k ∈ 1

2N.

The corresponding operators will be denoted Jτ,λ,3k . Obviously, if k is integer, then the operator

Jτ,λ,3k is even and if k is semi-integer, then the operator J

τ,λ,3k is odd. More precisely, for k ∈ N, any

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 12: Supertransvectants, cohomology, and deformations

023501-11 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

osp(3|2)-invariant element Jτ,λ,3k of D3

τ,λ;τ+λ+k can be expressed as follows:

Jτ,λ,3k = �τ,λ,τ+λ+k

(�0

τ,λ,kJτ,λ,2k + �1

τ,λ,kJτ+ 1

2 ,λ+ 12 ,2

k−1 + �2τ,λ,kJ

τ,λ+ 12 ,2

k + �3τ,λ,kJ

τ+ 12 ,λ,2

k

)+

�τ,λ,τ+λ+k

(�0

τ,λ,k Jτ,λ,2k + �1

τ,λ,k Jτ+ 1

2 ,λ+ 12 ,2

k−1 + �2τ,λ,k J

τ,λ+ 12 ,2

k + �3τ,λ,k J

τ+ 12 ,λ,2

k

),

(4.9)

where Jτ,λ,2k and J

τ,λ,2k are defined by (3.11) and (3.12). The coefficients ��

τ,λ,k and ��τ,λ,k, � = 0,

1, 2, 3, are, a priori, arbitrary constants, but the invariance of Jτ,λ,3k with respect osp(2|2)i , i = 1, 2,

imposes some supplementary conditions over these coefficients and determines thus completely theeven space of osp(3|2)-invariant elements of D3

τ,λ;τ+λ+k . By a direct computation, we get

�0τ,λ,k = �1

τ,λ,k = �2τ,λ,k = −�3

τ,λ,k,

and all other coefficients are identically zero. Therefore, we easily check that Jτ,λ,3k is expressed as

in (4.2). Now, for k ∈ N + 12 , any osp(3|2)-invariant element J

τ,λ,3k of D3

τ,λ;μ with μ = τ + λ + k,can be expressed as follows:

Jτ,λ,3k = �τ,λ,μ

(

((�0

τ,λ,kJτ,λ,2k+ 1

2+ �1

τ,λ,kJτ,λ+ 1

2 ,2

k− 12

+ �2τ,λ,kJ

τ+ 12 ,λ,2

k− 12

+ �3τ,λ,kJ

τ+ 12 ,λ+ 1

2 ,2

k− 12

)◦ (σ ⊗ σ )

))+ �τ,λ,μ

(

((�0

τ,λ,k Jτ,λ,2k+ 1

2+ �1

τ,λ,k Jτ,λ+ 1

2 ,2

k− 12

+ �2τ,λ,k J

τ+ 12 ,λ,2

k− 12

+ �3τ,λ,k J

τ+ 12 ,λ+ 1

2 ,2

k− 12

)◦ (σ ⊗ σ )

)),

(4.10)where J

τ,λ,2k± 1

2and J

τ,λ,2k± 1

2are defined by (3.11) and (3.12). The coefficients ��

λ,μ,k, and ��λ,μ,k, � = 0,

1, 2, 3 are, a priori, arbitrary constants, but the invariance of Jτ,λ,3k with respect osp(2|2)i , i = 1, 2,

shows that

(k + 12 )�1

τ,λ,k = (2τ + k + 12 )�0

τ,λ,k,

(k + 12 )�2

τ,λ,k = −(2λ + k + 12 )�0

τ,λ,k,

(k + 12 )�3

λ,μ,k = −(2τ + 2λ + k + 12 )�0

τ,λ,k,

and all other coefficients are identically zero. Therefore, we easily check that Jτ,λ,3k is expressed as

in (4.1). This completes the proof of Theorem 4.1 part (i).(ii) The uniqueness of 1|3-supertansvectants follows from the uniqueness of 1|2-

supertansvectants.(iii) For k ∈ N and for fixed τ = 1−m

4 for some odd m, any osp(3|2)-invariant element Jτ,λ,3k

of D3τ,λ;τ+λ+k can be expressed as in (4.9) with J

τ,λ,2k and J

τ,λ,2k are as in (3.13)–(3.14). The same

arguments as before show that

(2λ + k − m−32 )�1

τ,λ,k = −k�0τ,λ,k,

(2λ + k − m−32 )�2

τ,λ,k = (2λ + k)�0τ,λ,k,

(2λ + k − m−32 )�3

τ,λ,k = −(k − m−12 )�0

τ,λ,k,

and all other coefficients are identically zero. Therefore, we easily check that Jτ,λ,3k is expressed as

in (4.3). Now, for k ∈ N + 12 and for fixed τ = 1−m

4 for some odd m, any osp(3|2)-invariant elementJ

τ,λ,3k of D3

τ,λ;τ+λ+k can be expressed as in (4.10) with Jτ,λ,2k and J

τ,λ,2k are as in (3.13) and (3.14).

Again the same arguments as before show that

�0τ,λ,k = �1

τ,λ,k = �2τ,λ,k = �3

τ,λ,k,

and all other coefficients are identically zero. Therefore, we easily check that Jτ,λ,3k is expressed as

in (4.3).

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 13: Supertransvectants, cohomology, and deformations

023501-12 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

Finally, the properties (iv) and (v) can be checked directly, by the same arguments asbefore. �

Remark 4.2: For N = 2 and 3, and for k ∈ 12 (N + 2N + 3), the space of osp(N |2)-invariant

linear differential operators from K(N ) to DNλ,λ+k−1 vanishing on osp(N |2) is one dimensional.

V. COHOMOLOGY

Let us first recall some fundamental concepts from cohomology theory (see, e.g., Ref. 10). Letg = g0 ⊕ g1 be a Lie superalgebra acting on a superspace V = V0 ⊕ V1 and let h be a subalgebra ofg. (If h is omitted it assumed to be {0}.) The space of h-relative n-cochains of g with values in V isthe g-module

Cn(g, h; V ) := Homh(�n(g/h); V ).

The coboundary operator δn : Cn(g, h; V ) −→ Cn+1(g, h; V ) is a g-map satisfying δn◦δn − 1 = 0.The kernel of δn, denoted Zn(g, h; V ), is the space of h-relative n-cocycles, among them, theelements in the range of δn − 1 are called h-relative n-coboundaries. We denote Bn(g, h; V ) the spaceof n-coboundaries.

By definition, the nth h-relative cohomolgy space is the quotient space

Hn(g, h; V ) = Zn(g, h; V )/Bn(g, h; V ).

We will only need the formula of δn (which will be simply denoted δ) in degrees 0 and 1: forv ∈ C0(g, h; V ) = V h, δv(g) := (−1)p(g)p(v)g · v, where

V h = {v ∈ V | h · v = 0 for all h ∈ h},and for ϒ ∈ C1(g, h; V ),

δ(ϒ)(g, h) := (−1)p(g)p(ϒ)g · ϒ(h) − (−1)p(h)(p(g)+p(ϒ))h · ϒ(g) − ϒ([g, h]) for any g, h ∈ g.

A. The space H1diff(K(N), osp(N|2); DN

λ,μ)

In this subsection, we will compute the first differential cohomology spacesH1

diff(K(N ), osp(N |2); DNλ,μ) for N = 2 and 3. Our main result is the following:

Theorem 5.1: (i) dimH1diff(K(2), osp(2|2); D2

λ,μ) = 1 if

μ − λ = 1 and λ = − 12 ,

μ − λ = 2 and λ = −1.

dimH1diff(K(2), osp(2|2); D2

λ,μ) = 2 if μ − λ = 2 and λ = − 1. Otherwise,

H1diff(K(2), osp(2|2); D2

λ,μ) = 0.

The corresponding spaces H1diff(K(2), osp(2|2); D2

λ,λ+k) are spanned by the cohomology

classes of ϒ2λ,λ+k = J

−1,λ,2k+1 , where k ∈ {1, 2}.

(ii) dimH1diff(K(3), osp(3|2); D3

λ,μ) = 1 if

μ − λ = 12 and λ = − 1

2 ,

μ − λ = 32 and λ = −1.

Otherwise, H1diff(K(3), osp(3|2); D3

λ,μ) = 0.The corresponding spaces H1

diff(K(3), osp(3|2); D3λ,λ+ k

2) are spanned by the cohomology

classes of ϒ3λ,λ+ k

2= J

−1,λ,3k2 +1

, where k ∈ {1, 3}.

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 14: Supertransvectants, cohomology, and deformations

023501-13 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

To prove Theorem 5.1, we need first the following Lemmas:

Lemma 5.2: As a K(N − 1)-module, we have

(DNλ,μ)0 � D

N−1λ,μ ⊕ D

N−1λ+ 1

2 ,μ+ 12

and (DNλ,μ)1 � (DN−1

λ+ 12 ,μ

⊕ DN−1λ,μ+ 1

2).

Proof: Observe that the osp(N − 1|2) isomorphism (2.10) is also a K(N − 1)-isomorphism.Thus, by isomorphism (2.10), we deduce a K(N − 1)-isomorphism,

�λ,μ : DN−1λ,μ ⊕ D

N−1λ+ 1

2 ,μ+ 12⊕

(D

N−1λ,μ+ 1

2⊕ D

N−1λ+ 1

2 ,μ

)→ DN

λ,μ

A �→ ϕ−1μ ◦ A ◦ ϕλ.

(5.1)

Here, we identify the K(N − 1)-modules via the following isomorphisms:

(D

N−1λ,μ+ 1

2

)→ Homdiff

(F

N−1λ , (FN−1

μ+ 12))

(A) �→ ◦ A,

(D

N−1λ+ 1

2 ,μ

)→ Homdiff

( (FN−1

λ+ 12),FN−1

μ

) (A) �→ A ◦ ,

DN−1λ+ 1

2 ,μ+ 12

→ Homdiff

( (FN−1

λ+ 12), (FN−1

μ+ 12))

A �→ ◦ A ◦ .

�Lemma 5.3: Any 1-cocycle ϒ ∈ Z1

diff(K(N ); DNλ,μ) vanishing on osp(N |2) is osp(N |2)-invariant.

Proof: The 1-cocycle relation of ϒ reads:

(−1)p(F)p(ϒ)Lλ,μ

X Fϒ(XG) − (−1)p(G)(p(F)+p(ϒ))L

λ,μ

XGϒ(X F ) − ϒ([X F , XG]) = 0, (5.2)

where X F , XG ∈ K(N ). Thus, if ϒ(XF) = 0 for all X F ∈ osp(N |2), Eq. (5.2) becomes

(−1)p(F)p(ϒ)Lλ,μ

X Fϒ(XG) − ϒ([X F , XG]) = 0 (5.3)

expressing the osp(N |2)-invariance of ϒ . �We also need Proposition 5.1

Proposition 5.1:6

(1) For λ = 0 or λ = μ, any element of Z1diff(K(2),D2

λ,μ) is a coboundary over K(2) if and onlyif at least one of its restrictions to the subalgebras K(1)1 or K(1)2 is a coboundary.

(2) For (λ,μ) = (− 12 , 0), any element of Z1

diff(K(3); D3λ,μ) is a coboundary over K(3) if and only

if at least one of its restrictions to the subalgebras K(2)i is a coboundary.

Proof. (Theorem 5.1): (i) Note that, by Lemma 5.3, the osp(2|2)-relative cocycles are super-transvectants and by Proposition 5.1, they are related to the osp(1|2)-relative cocycles. Ben Frajet al.3 showed that

H1diff(K(1), osp(1|2); D1

λ,λ+k) �

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩R if

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

k = 32 and λ = − 1

2 ,

k = 2 for allλ,

k = 52 and λ = −1,

k = 3 and λ = 0,− 52 ,

k = 4 and λ = − 7±√33

4 ,

0 otherwise.

(5.4)

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 15: Supertransvectants, cohomology, and deformations

023501-14 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

These spaces are generated by the cohomology classes of the following non-trivial osp(1|2)-relative1-cocycles:

ϒ1λ,λ+ 3

2(XG)(Fαλ) = η(G ′′)Fαλ+ 3

2 for λ = − 12 ,

ϒ1λ,λ+ 5

2(XG)(Fαλ) = (

2λη(G(3))F − 3η(G ′′)F ′ − (−1)p(G)G(3)η(F))αλ+ 5

2 for λ = −1,

ϒ1λ,λ+2(XG)(Fαλ) = (

23λG(3) F − (−1)p(G)η(G ′′)η(F)

)αλ+2 for all λ,

ϒ1λ,λ+3(XG)(Fαλ) =

((−1)p(G)η(G ′′)η(F ′) − 2λ+1

3

((−1)p(G)η(G(3))η(F) + G(3) F ′)

+ λ(2λ+1)6 G(4) F

)αλ+3 for λ = 0, − 5

2 ,

ϒ1λ,λ+4(XG)(Fαλ) =

((−1)p(G)η(G ′′)η(F ′′) − 2(λ+1)

3

(2(−1)p(G)η(G(3))η(F ′) + G(3) F ′′)

+ (λ+1)(2λ+1)6

((−1)p(G)η(G(4))η(F) + 2G(4) F ′)

− λ(λ+1)(2λ+1)15 G(5) F

)αλ+4 for λ = −7±√

334 .

Now, let us study the relationship between any 1-cocycle of K(2) and its restriction to thesubalgebra K(1). More precisely, we study the relationship between H1

diff(K(2), osp(2|2); D2λ,μ) and

H1diff(K(1), osp(1|2); D2

λ,μ). By Lemma 5.2, and (5.4), we see that H1diff(K(1), osp(1|2); D2

λ,μ) canbe deduced from the spaces H1

diff(K(1), osp(1|2); D1λ,μ):

H1diff

(K(1), osp(1|2); D2

λ,μ

) � H1diff

(K(1), osp(1|2); D1

λ,μ

) ⊕ H1diff

(K(1), osp(1|2); D1

λ+ 12 ,μ+ 1

2

)⊕

H1diff

(K(1), osp(1|2); (D1

λ,μ+ 12))

⊕ H1diff

(K(1), osp(1|2); (D1

λ+ 12 ,μ

)).

(5.5)So, we see first that for λ = 0 or λ = μ, if 2(μ − λ) ∈{2, . . . , 9}, the corresponding cohomologyH1

diff(K(2), osp(2|2); D2λ,μ) vanish. Indeed, let ϒ be any element of Z1

diff(K(2), osp(2|2); D2λ,μ).

Then by (5.4) and (5.5), up to a coboundary, the restriction of ϒ to osp(1|2) vanishes, so ϒ = 0 byProposition 5.1.

For 2(μ − λ) ∈ {2, . . . , 9} or μ = λ = 0, we study the supertranvectant J−1,λ,2μ−λ+1. If it is

a non-trivial 1-cocycle, then the corresponding cohomology space is one-dimensional, otherwiseit is zero. To study any supertranvectant J

−1,λ,2μ−λ+1 satisfying δ(J−1,λ,2

μ−λ+1) = 0, we consider the twocomponents of its restriction to osp(1|2), which we compare with ϒ1

λ,μ and ϒ1λ+ 1

2 ,μ+ 12

or ϒ1λ+ 1

2 ,μ

and ϒ1λ,μ+ 1

2depending on whether μ − λ is integer or semi-integer. More precisely, we get the

following non-trivial 1-cocycles:

ϒ2λ,λ+1(XG) = η1η2(G ′) if λ = − 1

2 ,

ϒ2λ,λ+2(XG) = (2λ + 1)

(2λ3 G ′′′ − HG ′′

) − 2λη2η1(G ′)η2η1 for all λ,

ϒ2λ,λ+2(XG) = MG ′′ + 2λη2η1(G ′′) − 2η2η1(G ′)∂x if λ = −1,

where, for G ∈ R[x, θ ],HG = (−1)p(G) ∑2

i=1 ηi (G)ηi and MG = ∑2i=1(−1)p(G)+iη3−i (G)ηi .

(ii) Using arguments similar to those of the proof of Theorem 5.1 part (i), we deduce thatdimH1

diff(K(3), osp(3|2); D3λ,μ) ≤ 1. It is 1 only if μ − λ = 1

2 and λ = − 12 or μ − λ = 3

2 andλ = − 1. The cohomology classes of the following 1-cocycles generate the corresponding spaces:

ϒ3λ,λ+ 1

2(XG) = η3η2η1(G) if λ = − 1

2 ,

ϒ3λ,λ+ 3

2(XG) = �G ′ + 2λη3η2η1(G ′) + η3η2η1(G)η2

1 if λ = −1,

where, for G ∈ R[x, θ ],�G = (−1)p(G) ∑1≤i< j≤3(−1)i+ jη jηi (G)η6−i− j .

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 16: Supertransvectants, cohomology, and deformations

023501-15 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

VI. DEFORMATION THEORY AND COHOMOLOGY

Deformation theory of Lie algebra homomorphisms was first considered with only one-parameter of deformation.9, 18, 21 Recently, deformations of Lie (super)algebras with multi-parameters were intensively studied ( see, e.g., Refs. 1, 2, 4, 5, 19, and 20). Here, we give an outlineof this theory.

A. Infinitesimal deformations and the first cohomology

Let ρ0 : g −→ End(V ) be an action of a Lie superalgebra g on a vector superspace V and let h

be a subagebra of g (If h is omitted it assumed to be {0}.) When studying h-trivial deformations ofthe g-action ρ0, one usually starts with infinitesimal deformations:

ρ = ρ0 + t ϒ, (6.1)

where ϒ : g → End(V ) is a linear map vanishing on h and t is a formal parameter with p(t) = p(ϒ).The homomorphism condition

[ρ(x), ρ(y)] = ρ([x, y]), (6.2)

where x, y ∈ g, is satisfied in order 1 in t if and only if ϒ is a h-relative 1-cocycle. That is, the mapϒ satisfies

(−1)p(x)p(ϒ)[ρ0(x), ϒ(y)] − (−1)p(y)(p(x)+p(ϒ))[ρ0(y), ϒ(x)] − ϒ([x, y]) = 0.

Moreover, two h-trivial infinitesimal deformations ρ = ρ0 + t ϒ1, and ρ = ρ0 + t ϒ2, areequivalents if and only if ϒ1 − ϒ2 is h-relative coboundary:

(ϒ1 − ϒ2)(x) = (−1)p(x)p(A)[ρ0(x), A] := δA(x),

where A ∈ End(V )h and δ stands for differential of cochains on g with values in End(V ) (see, e.g.,Refs. 10 and 18). So, the space H1(g, h; End(V )) determines and classifies infinitesimal deformationsup to equivalence. If dim H1(g, h; End(V )) = m, then choose 1-cocycles ϒ1, . . . , ϒm representing abasis of H1(g, h; End(V)) and consider the infinitesimal deformation

ρ = ρ0 +m∑

i=1

ti ϒi , (6.3)

where t1, . . . , tm are independent parameters with p(ti) = p(ϒ i).In our study, we are interested in the osp(3|2)- trivial deformations of the K(3)-action on

S3,nβ :=

2n⊕k=0

F3β− k

2, where n ∈ 1

2N.

Thus, we consider the space H1diff(K(3), osp(3|2); End(S3,n

β )) spanned by the classes ϒ3λ,λ+ k

2, where

k = 1, 3, and 2(β − λ) ∈ {k, k + 1, . . . , 2n} for generic β. Any infinitesimal osp(3|2)-trivialdeformation of the K(3)-module S

3,nβ is then of the form

LX F = LX F + L(1)X F

, (6.4)

where LX F is the Lie derivative of S3,nβ along the vector field XF defined by (2.3), and

L(1)X F

=∑

λ

∑k=1,3

tλ,λ+ k2ϒ3

λ,λ+ k2(X F ), (6.5)

where the tλ,λ+ k2

are independent parameters with p(tλ,λ+ k2) = p(ϒλ,λ+ k

2) and 2(β − λ) ∈ {k, k

+ 1, . . . , 2n}.

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 17: Supertransvectants, cohomology, and deformations

023501-16 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

B. Integrability conditions and deformations over supercommutative algebras

Consider the supercommutative associative superalgebra with unityC[[t1, . . . , tm]] and considerthe problem of integrability of infinitesimal deformations. Starting with the infinitesimal deformation(6.3), we look for a formal series

ρ = ρ0 +m∑

i=1

ti ϒi +∑i, j

ti t j ρ(2)i j + · · · , (6.6)

where the higher order terms ρ(2)i j , ρ

(3)i jk, . . . are linear maps from g to End(V) with p(ρ(2)

i j )

= p(ti t j ), p(ρ(3)i jk) = p(ti t j tk), . . . such that the map

ρ : g → C[[t1, . . . , tm]] ⊗ End(V), (6.7)

satisfies the homomorphism condition (6.2).Quite often the above problem has no solution. Following Refs. 1 and 9, we will impose extra

algebraic relations on the parameters t1, . . . , tm. Let R be an ideal in C[[t1, . . . , tm]] generated bysome set of relations, and we can speak about deformations with base A = C[[t1, . . . , tm]]/R, (fordetails, see9). The map (6.7) sends g to A ⊗ End(V ).

Setting

ϕt = ρ − ρ0, ρ(1) =∑

ti ϒi , ρ(2) =∑

ti t j ρ(2)i j , . . . ,

we can rewrite the relation (6.2) in the following way:

[ϕt (x), ρ0(y)] + [ρ0(x), ϕt (y)] − ϕt ([x, y]) +∑i, j>0

[ρ(i)(x), ρ( j)(y)] = 0. (6.8)

The first three terms are (δϕt)(x, y). For arbitrary linear maps γ1, γ2 : g −→ End(V ), consider thestandard cup-product: [[γ1, γ2]] : g ⊗ g −→ End(V ) defined by:

[[γ1, γ2]](x, y) = (−1)p(γ2)(p(γ1)+p(x))[γ1(x), γ2(y)] + (−1)p(γ1)p(x)[γ2(x), γ1(y)]. (6.9)

The relation (6.8) becomes now equivalent to

δϕt + 1

2[[ϕt , ϕt ]] = 0, (6.10)

Expanding (6.10) in power series in t1, . . . , tm, we obtain the following equation for ρ(k):

δρ(k) + 1

2

∑i+ j=k

[[ρ(i), ρ( j)]] = 0. (6.11)

The first non-trivial relation δρ(2) + 12 [[ρ(1), ρ(1)]] = 0 gives the first obstruction to integration

of an infinitesimal deformation. Thus, considering the coefficient of ti tj, we get

δρ(2)i j + 1

2[[ϒi , ϒ j ]] = 0. (6.12)

It is easy to check that for any two 1-cocycles γ 1 and γ2 ∈ Z1(g, h; End(V )), the bilinear map [[γ 1,γ 2]] is a h-relative 2-cocycle. The relation (6.12) is precisely the condition for this cocycle to be acoboundary. Moreover, if one of the cocycles γ 1 or γ 2 is a h-relative coboundary, then [[γ 1, γ 2]] isa h-relative 2-coboundary. Therefore, we naturally deduce that the operation (6.9) defines a bilinearmap

H1(g, h; End(V )) ⊗ H1(g, h; End(V )) −→ H2(g, h; End(V )). (6.13)

All the obstructions lie in H2(g, h; End(V )) and they are in the image of H1(g, h; End(V )) underthe cup-product.

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 18: Supertransvectants, cohomology, and deformations

023501-17 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

C. Equivalence

Two deformations, ρ and ρ ′ of a g-module V over A are said to be equivalent (see, e.g., Ref. 9)if there exists an inner automorphism � of the associative superalgebra A ⊗ End(V ) such that

� ◦ ρ = ρ ′ and �(I) = I,

where I is the unity of the superalgebra A ⊗ End(V ).The following notion of miniversal deformation is fundamental. It assigns to a g-module V a

canonical commutative associative algebra A and a canonical deformation over A. A deformation(6.6) over A is said to be miniversal if

(i) for any other deformation ρ ′ with base (local) A′, there exists a homomorphism ψ : A′ → Asatisfying ψ(1) = 1, such that

ρ = (ψ ⊗ Id) ◦ ρ ′.

(ii) under notation of (i), if ρ is infinitesimal, then ψ is unique.

If ρ satisfies only the condition (i), then it is called versal. This definition does not depend on thechoice 1-cocycles ϒ1, . . . , ϒm representing a basis of H1(g, h; End(V)).

The miniversal deformation corresponds to the smallest ideal R. We refer to Ref. 9 for aconstruction of miniversal deformations of Lie algebras and to Ref. 1 for miniversal deformationsof g-modules. Superization of these results is immediate: by the Sign Rule.

VII. INTEGRABILITY CONDITIONS

In this section, we obtain the integrability conditions for the infinitesimal deformation(6.4).

Theorem 7.1: (i) The following conditions are necessary and sufficient for integrability of theinfinitesimal deformation (6.4):

tλ,λ+ 12

tλ+ 12 ,λ+2 = 0 for 2(β − λ) ∈ {4, . . . , 2n} ,

tλ,λ+ 32

tλ+ 32 ,λ+2 = 0 for 2(β − λ) ∈ {4, . . . , 2n} ,

tλ,λ+ 32

tλ+ 32 ,λ+3 = 0 for 2(β − λ) ∈ {6, . . . , 2n} .

(7.1)

(ii) Any formal osp(3|2)-trivial deformation of the K(3)-module S3,nβ is equivalent to a defor-

mation of order 1, that is, to a deformation given by (6.4).

To prove Theorem 7.1, we need the following:

Lemma 7.1: Consider a linear differential operator b : K(3) −→ D3λ,μ. If b satisfies

δ(b)(X, Y ) = b(X ) = 0 for all X ∈ osp(3|2),

then b is a supertransvectant.

Proof: For all X, Y ∈ K(3), we have

δ(b)(X, Y ) := (−1)p(X )p(b)Lλ,μ

X (b(Y )) − (−1)p(Y )(p(X )+p(b))Lλ,μ

Y (b(X )) − b([X, Y ]).

Since δ(b)(X, Y ) = b(X ) = 0 for all X ∈ osp(3|2), we deduce that

(−1)p(X )p(b)Lλ,μ

X (b(Y )) − b([X, Y ]) = 0.

Thus, the map b is osp(3|2)-invariant. �

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 19: Supertransvectants, cohomology, and deformations

023501-18 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

We also need Proposition 7.2

Proposition 7.2: The cup products

Bλ,λ+2 = [[ϒ3λ+ 1

2 ,λ+2, ϒ3

λ,λ+ 12]] : K(3) ⊗ K(3) → D3

λ,λ+2,

Bλ,λ+2 = [[ϒ3λ+ 3

2 ,λ+2, ϒ3

λ,λ+ 32]] : K(3) ⊗ K(3) → D3

λ,λ+2,

Bλ,λ+3 = [[ϒ3λ+ 3

2 ,λ+3, ϒ3

λ,λ+ 32]] : K(3) ⊗ K(3) → D3

λ,λ+3

are generically nontrivial osp(3|2)-relative 2-cocycles. Moreover, the corresponding cohomologyclasses are linearly independent.

Proof: Each map Bλ,λ + k, where k = 2, 3 is the cup-product of two osp(3|2)-relative 1-cocycles,so, Bλ,λ + k is a osp(3|2)-relative 2-cocycle: Bλ,λ+k ∈ Z2(K(3), osp(3|2); D3

λ,λ+k). This 2-cocycle istrivial if and only if it is the coboundary of a linear differential operator

bλ,λ+k : K(3) −→ D3λ,λ+k

vanishing on osp(3|2). Consider bλ,λ + k as a bilinear map F3−1 ⊗ F3

λ −→ F3λ+k . So, according

to Lemma 7.1 and Theorem 4.1, the operator bλ,λ + k coincides (up to a scalar factor) withthe supertransvectant J

−1,λ,3k+1 . But, by a direct computation, we can check that Bλ,λ + k and

δ(J−1,λ,3k+1 ) are linearly independent. Moreover, by a direct computation, we can see that the sys-

tem(

Bλ,λ+2, Bλ,λ+2, δ(J

−1,λ,33

))is linearly independent. Thus, the cohomology classes of Bλ,λ + 2

and Bλ,λ+2 are linearly independent. This completes the proof. �Proof: (Theorem 7.1): (i) Assume that the infinitesimal deformation (6.4) can be integrated to

a formal deformation

LX F = LX F + L(1)X F

+ L(2)X F

+ · · · ·Considering the homomorphism condition, we compute the second order term L(2), which is asolution of the Maurer-Cartan equation

δ(L(2)) = −1

2[[L(1), L(1)]]. (7.2)

The right hand side of (7.2) yield the following maps:

�λ,λ+1 = tλ,λ+ 12

tλ+ 12 ,λ+1[[ϒ3

λ+ 12 ,λ+1

, ϒ3λ,λ+ 1

2]],

�λ,λ+2 = tλ,λ+ 12

tλ+ 12 ,λ+2 Bλ,λ+2 + tλ,λ+ 3

2tλ+ 3

2 ,λ+2 Bλ,λ+2,

�λ,λ+3 = tλ,λ+ 32

tλ+ 32 ,λ+3 Bλ,λ+3.

(7.3)

Clearly, [[ϒ3λ+ 1

2 ,λ+1, ϒ3

λ,λ+ 12]] = 0 and the same arguments, as in the proof of Proposition 7.2, show

that we must have

�λ,λ+2 = ω1(t)δ(J

−1,λ,33

)and �λ,λ+3 = ω2(t)δ

(J

−1,λ,34

),

where ω1, ω2 are some functions. So, by Proposition 7.2, we get the necessary integrability conditionsfor L(2). We thus proved that the conditions (7.1) are, indeed, necessary.

To prove that the conditions (7.1) are sufficient, we will find explicitly a deformation of LX F ,whenever the conditions (7.1) are satisfied. The solution L(2) of (7.2) can be chosen identically zero.Choosing the hightest-order terms L(m) with m ≥ 3, also identically zero, one obviously obtains adeformation, which is of order 1 in t. Theorem 7.1, part (i) is proved.

(ii) The solution L(2) of (7.2) is defined up to a 1-cocycle and it has been shown in Refs. 1and 9 that different choices of solutions of the Maurer-Cartan equation correspond to equivalentdeformations. Thus, we can always reduce L(2) to zero by equivalence. Then, by recurrence, the

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Page 20: Supertransvectants, cohomology, and deformations

023501-19 Ben Fraj, Laraiedh, and Omri J. Math. Phys. 54, 023501 (2013)

hightest-order terms L(m) satisfy the equation δ(L(m)) = 0 and can also be reduced to the identicallyzero map. This completes the proof of part (ii). �

Example 7.2: For n ∈ 12N and for arbitrary generic λ ∈ R, the following example is a 1-

parameter deformation of the K(3)-module S3,nλ+n:

LX F = LX F + t

(2n−1∑k=0

ϒ3λ+ k

2 ,λ+ k+12

),

that is, we put tλ+ k2 ,λ+ k+1

2= t and tλ+ k

2 ,λ+ k+32

= 0.

Clearly, for n < 2, each of formal osp(3|2)-trivial deformations of K(3)-modules S3,nλ+n is

equivalent to its infinitesimal one, without any integrability condition. Of course, it is easy to givemany other examples of true deformations with several parameters.

ACKNOWLEDGMENTS

We would like to thank Valentin Ovsienko, Claude Roger, Dimitry Leites, Mabrouk Ben Ammar,and Christian Duval for their interest in this work.

1 Agrebaoui, B., Ammar, F., Lecomte, P., and Ovsienko, V., “Multi-parameter deformations of the module of symbols ofdifferential operators,” Int. Math. Res. Notices 2002, 847–869.

2 Agrebaoui, B., Ben Fraj, N., Ben Ammar, M., and Ovsienko, V., “Deformations of modules of differential forms,” J.Nonlinear Math. Phys. 10, 148–156 (2003).

3 Basdouri, I., Ben Ammar, M., Ben Fraj, N., Boujelbene, M., and Kammoun, K., “Cohomology of the Lie superalgebra ofcontact vector fields on K1|1 and deformations of the superspace of symbols,” J. Nonlinear Math. Phys. 16, 373 (2009).

4 Basdouri, I., Ben Ammar, M., Dali, B., and Omri, S., “Deformation of VectP(R)-modules of symbols,” J. Geom. Phys.60(3), 531–542 (2010); e-print arXiv:math.RT/0702664.

5 Ben Ammar, M. and Boujelbene, M., “sl(2)-Trivial deformations of VectP(R)-modules of symbols,” Symmetry, Integr.Geom.: Methods Appl. 4, 065 (2008).

6 Ben Ammar, M., Ben Fraj, N., and Omri, S., “The binary invariant differential operators on weighted densities on thesuperspace R1|n and cohomology,” J. Math. Phys. 51, 043504 (2010).

7 Bouarroudj, S., “On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators,” J. NonlinearMath. Phys. 14(1), 112–127 (2007).

8 Bouarroudj, S. and Ovsienko, V., “Three cocycles on Diff(S1) generalizing the Schwarzian derivative,” Int. Math. Res.Notices 1998, 25–39.

9 Fialowski, A. and Fuchs, D. B., “Construction of miniversal deformations of Lie algebras,” J. Funct. Anal. 161, 76–110(1999).

10 Fuchs, D. B., Cohomology of Infinite-Dimensional Lie Algebras (Plenum, New York, 1986).11 Gargoubi, H., Mellouli, N., and Ovsienko, V., “Differential operators on supercircle: Conformally equivariant quantization

and symbol calculus,” Lett. Math. Phys. 79, 51–65 (2007).12 Gargoubi, H. and Ovsienko, V., “Supertransvectants and symplectic geometry,” Int. Math. Res. Notes 2008 (2008); e-print

arXiv:0705.1411v1 [math-ph].13 Gieres, F. and Theisen, S., “Superconformally covariant operators and super W-algebras,” J. Math. Phys. 34, 5964–5985

(1993).14 Gordan, P., Invariantentheorie (Teubner, Leipzig, 1887).15 Grozman, P., Leites, D., and Shchepochkina, I., “Invariant operators on supermanifolds and standard models,” in Mul-

tiple Facets of Quantization and Supersymmetry, edited by M. Olshanetski and A. Vainstein (World Scientific, 2002),pp. 508–555.

16 Grozman, P., Leites, D., and Shchepochkina, I., “Lie superalgebras of string theories,” Acta Math. Vietnam. 26(1), 27–63(2001); e-print arXiv:hep-th/9702120.

17 Leites, D., “Introduction to the theory of supermanifolds,” Usp. Mat. Nauk 35(1), 3–57 (1980); Leites, D. [Russian Math.Surveys 35(1), 1–64 (1980) (in Russian)].

18 Nijenhuis, A. and Richardson, R. W., Jr., “Deformations of homomorphisms of Lie groups and Lie algebras,” Bull. Am.Math. Soc. 73, 175–179 (1967).

19 Ovsienko, V. and Roger, C., “Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of pseudodifferentialoperators on S1(AMS Transl. Ser. 2),” Adv. Math. Sci. Appl. 194, 211–227 (1999).

20 Ovsienko, V. and Roger, C., “Deforming the Lie algebra of vector fields on S1 inside the Poisson algebra on T ∗S1,”Commun. Math. Phys. 198, 97–110 (1998).

21 Richardson, R. W., “Deformations of subalgebras of Lie algebras,” J. Diff. Geom. 3, 289–308 (1969).

Downloaded 03 Oct 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions