Journal of Geometry and Physics 86 (2014) 211–221

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Deforming the orthosymplectic Lie superalgebra inside theLie superalgebra of superpseudodifferential operatorsOthmen Ncib, Salem Omri ∗Département de Mathématiques, Faculté des Sciences de Gafsa, Zarroug 2112 Gafsa, Tunisie

a r t i c l e i n f o

Article history:Received 21 April 2014Received in revised form 11 July 2014Accepted 4 August 2014Available online 11 August 2014

Keywords:Lie superalgebraOrthosymplectic Lie superalgebraDeformationSuperpseudodifferential operators

a b s t r a c t

We classify deformations of the standard embedding of the Lie algebra sl(2) into both theLie algebra Ψ DOL of pseudodifferential operators with polynomial coefficients and thePoisson Lie algebra P , we prove that any formal deformation is equivalent to its infinites-imal part. We study also the super analogue of this problem for the case of the standardembedding of the orthosymplectic Lie superalgebra osp(n|2) on the (1, n)-dimensional su-perspace R1|n into the Lie superalgebra SΨ DO(n) of superpseudodifferential operatorswith polynomial coefficients, where n = 1, 2 getting the necessary and sufficient condi-tions for its integrability. Finally, by using the contract procedurewe deduce similar resultsfor the standard embedding into the Poisson Lie superalgebra SP (n).

© 2014 Elsevier B.V. All rights reserved.

Contents

1. Introduction............................................................................................................................................................................................. 2122. Definitions and notation......................................................................................................................................................................... 212

2.1. Pseudodifferential operators on R ............................................................................................................................................. 2122.2. The Lie superalgebra of contact vector fields on R1|n ............................................................................................................... 2122.3. Superpseudodifferential operators on R1|n ............................................................................................................................... 213

3. Statement of the problem....................................................................................................................................................................... 2143.1. Deformation theory .................................................................................................................................................................... 2143.2. Infinitesimal deformations and the first cohomology.............................................................................................................. 214

4. The space H1(osp(n|2), SΨ DO(n)) ...................................................................................................................................................... 2154.1. A filtration on SΨ DO(n) ........................................................................................................................................................... 2154.2. H1(osp(n|2), SP (n)) .................................................................................................................................................................. 216

4.2.1. H1(sl(2), P ) ................................................................................................................................................................. 2164.2.2. H1(osp(n|2), SP (n)) ................................................................................................................................................... 216

4.3. H1(osp(n|2), SΨ DOh(n)) .......................................................................................................................................................... 2165. Integrability conditions .......................................................................................................................................................................... 217

5.1. Integrability conditions for the infinitesimal deformation of the standard embedding of sl(2) into both Ψ DO(1)L andP .................................................................................................................................................................................................. 217

5.2. Integrability conditions for the infinitesimal deformation of the standard embedding of osp(1|2) into SΨ DOh(1) ........ 2175.3. Integrability conditions for the infinitesimal deformation of the standard embedding of osp(2|2) into SΨ DOh(2) ........ 218Acknowledgements................................................................................................................................................................................. 221References................................................................................................................................................................................................ 221

∗ Corresponding author. Tel.: +216 97640814.E-mail addresses: othmenncib@yahoo.fr (O. Ncib), omri_salem@yahoo.fr (S. Omri).

http://dx.doi.org/10.1016/j.geomphys.2014.08.0010393-0440/© 2014 Elsevier B.V. All rights reserved.

212 O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221

1. Introduction

The study of multi-parameter deformations of the standard embedding π of the Lie algebra vect(1) of smooth vectorfields on S1 into the Lie algebra of pseudodifferential symbols Ψ D(S1) defined by π(f (x)∂x) = f (x)ξ was carried out byOvsienko and Roger in [1,2]. They have, essentially, studied polynomial deformations of the following form

π(c) = π +

k∈Z

πk(c)ξ k,

where πk(c) : vect(1) → C∞C (S1) satisfy πk(0) = 0 and πk ≡ 0 for k sufficiently large. The supercase for this problem was

studied by Ben Fraj and Omri in [3,4], they gave extra relations between parameters of deformation. Their works are basedon the spectral sequence theory.

In this paper, we are interested in the case of the symplectic Lie algebra sl(2) and the orthosymplectic Lie superalgebraosp(n|2) on the (1, n)-dimensional superspace R1|n for n = 1, 2. More precisely, we investigate the result due to Basdouriin [5] for the classical case to deduce the integrability condition for the infinitesimal deformation in the case of thesymplectic Lie algebra sl(2), we compute in the same way as in [5] the first cohomology space for the contract algebra ofsuperpseudodifferential operators and getting the integrability conditions for the infinitesimal deformation of the standardembedding of the orthosymplectic Lie superalgebra osp(n|2) on the (1, n)-dimensional superspace R1|n into SΨ DOh(n) forn = 1, 2 by using the Maurer–Cartan equation. A contraction procedure leads to the integrability condition for the PoissonLie superalgebra.

2. Definitions and notation

2.1. Pseudodifferential operators on R

To every pseudodifferential operator F on R one associates its symbol [6–8]. Let Ψ DO(1) be the space ofpseudodifferential symbols on R [1,2]. The order of Ψ DO(1) is defined to be

ord(F) = {sup k ∈ Z|fk(x) = 0} for any F(x, ξ , ξ−1) =

k∈Z

ξ kfk(x) ∈ Ψ DO(1),

where fk(x) ∈ R[x] with fk = 0 for k sufficiently large. By means of the two natural derivations on Ψ DO(1)

∂ξ :

k

fkξ k−→

k

kfkξ k−1 and ∂x :

k

fkξ k−→

k

f ′

kξk (2.1)

one defines a natural Poisson bracket:

{F ,G} =∂F∂ξ

∂G∂x

−∂F∂x

∂G∂ξ

for any F ,G ∈ Ψ DO(1). (2.2)

On the space of pseudodifferential symbols an associative algebra structure is defined by the rule ([9])

F ◦ G =

k≥0

1k!

:∂kF∂ξ k

∂kG∂xk

: (2.3)

where : · : stands for the ‘‘normal ordering’’ defined as

: f (x)ξ kg(x)ξ ℓ:= f (x)g(x)ξ k+ℓ. (2.4)

This is a natural generalization of theWick product. As usual,we can also define the Lie bracket associatedwith the ◦-product[F ,G] = F ◦G−G◦F ; we denote the Lie algebra with this structure byΨ DO(1)L to distinguish from the Poisson Lie algebrastructure.

2.2. The Lie superalgebra of contact vector fields on R1|n

Let R1|n be the superspace with local coordinates (x; θ1, . . . , θn), where θ = (θ1, . . . , θn) are the odd variables. Anycontact structure on R1|n can be given by the following 1-form:

αn = dx +

ni=1

θidθi. (2.5)

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

{F ,G} = FG′− F ′G −

12(−1)|F |

ni=1

ηi(F) · ηi(G), (2.6)

O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221 213

where the superscript ′ stands for ∂∂x , ηi =

∂∂θi

−θi∂∂x and |F | is the parity of F . Note that the derivations ηi are the generators

of n-extended supersymmetry and generate the kernel of the form (2.5) as a module over the ring of polynomial functions.Let VectPol(R1|n) be the superspace of polynomial vector fields on R1|n:

VectPol(R1|n) =

F0∂x +

ni=1

Fi∂i | Fi ∈ R[x, θ] for all i

,

where ∂i =∂

∂θiand ∂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 singled out by the 1-form αn:

K(n) =X ∈ VectPol(R1|n) | there exists F ∈ R[x, θ ] such that LX (αn) = Fαn

where LX is the Lie derivative along the vector field X .

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

XF = F∂x −12

ni=1

(−1)|F |ηi(F)ηi, where F ∈ R[x, θ].

The bracket in K(n) can be written as:

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

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

osp(n|2) = SpanX1, Xx, Xx2 , Xθi , Xxθi , Xθiθj

, 1 ≤ i, j ≤ n.

called the orthosymplectic Lie superalgebra on the (1, n)-dimensional superspace R1|n.The case n = 0 corresponds to the classical setting: K(0) = VectPol(R) = {F∂x|F ∈ R[x]} and the corresponding

orthosymplectic Lie algebra osp(0|2) is nothing but the classical Lie algebra sl(2) which is isomorphic to the Lie subalgebraof VectPol(R) generated by {∂x, x∂x, x2∂x}. Of course, osp(n − 1|2) can be viewed as a subalgebra of osp(n|2).

2.3. Superpseudodifferential operators on R1|n

Let T ∗R1|n be the cotangent bundle on R1|n with local coordinates (x, θ1, . . . , θn, ξ , θ1, . . . , θn), where |θi| = 1. Thesuperspace of the supercommutative algebra SP (n) of pseudodifferential symbols on R1|n with its natural multiplication isspanned by the series

SP (n) =

∞

k=−M

ϵ=(ϵ1,...,ϵn)

ak,ϵ(x, θ)ξ−kθϵ11 . . . θ ϵn

n | ak,ϵ ∈ R[x, θ]; ϵi = 0, 1;M ∈ N

.

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

{A, B} = ∂ξA∂xB − ∂xA∂ξB − (−1)|A|

2i=1

∂iA∂θi

B + ∂θiA∂iB

.

Of course SP (0) is the classical spaces of symbols, usually denoted

P =

F(x, ξ)| F(x, ξ) =

mk=−∞

fk(x)ξ k

.

The associative superalgebra of pseudodifferential operators SΨ D(n) on R1|n has the same underlying vector space asSP (n), but the multiplication is defined by the following rule:

A ◦ B =

α≥0,νi=0,1

(−1)νi(|A|+1)

α!(∂α

ξ ∂νiθiA)(∂α

x ∂νii B).

Denote by SΨ DO(n) the Lie superalgebra with the same superspace as SΨ D(n) and the supercommutator defined onhomogeneous elements by:

[A, B] = A ◦ B − (−1)|A||B|B ◦ A.

Of course SΨ DO(0) = Ψ DO(1)L.Consider the family of associative laws on SΨ D(n) depending on a parameter h ∈]0, 1]:

A ◦h B =

α≥0,νi=0,1

(−1)νi(|A|+1)hνi+α−1

α!(∂α

ξ ∂νiθiA)(∂α

x ∂νii B).

214 O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221

Denote by SΨ DOh(n) the associative superalgebra of pseudodifferential operators onR1|n equippedwith themultiplication◦h. It is clear that all the associative superalgebras SΨ DOh(n) are isomorphic to each other and SΨ DO1(n) = SΨ DO(n).

For the supercommutator [A, B]h := A ◦h B − (−1)|A||B|B ◦h A, one has:

[A, B]h = {A, B} + O(h),

and therefore limh→0[A, B]h = {A, B}, wherewe identify SP (n)withSΨ DO(n) as vector spaces. Hence the Lie superalgebraSΨ DO(n) contracts to the Poisson Lie superalgebra SP (n) (cf. [10]).

3. Statement of the problem

The main purpose of this paper is to study deformations of the canonical embedding

ρ0 : osp(n|2) → SΨ DOh(n) (3.7)

defined byρ0(XF ) = Fξ −12

ni=1(−1)|F |ηi(F)ζi,where ζi = θi − θiξ, for n = 1, 2, and ρ0(XF ) = Fξ for n = 0.

3.1. Deformation theory

Deformation theory of Lie algebra homomorphisms was first considered with one parameter ([11,12]). Recently,multiparameter deformations of Lie (super)algebras and their modules were intensively studied ([13–15,3,4,1,2]). Recallthe basics of this theory.

Let ρ0 : osp(n|2) → SΨ DO(n) be an embedding of Lie superalgebras. A formal deformation of ρ0 is a formal series

ρ(t) = ρ0 +

mi=1

tiρi +

mi,j=1

titjρ(2)ij + · · · , (3.8)

where the ρi, ρ(2)ij , . . . are linear maps from osp(n|2) to SΨ DO(n) with |ρi| = |ti|, |ρ

(2)ij | = |titj|, and so on (indeed, all the

terms in the formal deformation (3.8) are even), such that the map

ρ(t) : osp(n|2) → C[[t1, . . . , tm]] ⊗ SΨ DO(n), (3.9)

satisfies the homomorphism condition in any order in t = (t1, . . . , tm).Two formal deformations ρ andρ are said to be equivalent ifρ = It ◦ ρ for an inner automorphism

It : C[[t]] ⊗ SΨ DO(n) → C[[t]] ⊗ SΨ DO(n)

of the form

It = exp

mi=1

tiadFi +i,j

titjadFi,j + · · ·

for some Fi, Fi,j, . . . , Fi1...ik ∈ SΨ DO(n). (3.10)

In many cases we have to consider polynomial deformations instead of formal ones: the formal series can sometimesbe cut off, and the formal parameter be replaced by complex coefficients. More precisely, according to [3,1], a polynomialdeformation of the standard embedding (3.7) is a Lie superalgebra homomorphism ρ(c) : osp(n|2) → SΨ DO(n) of thefollowing form:

ρ(c) = ρ0 +

k∈Z

ρk(c), (3.11)

where ρk(c) : osp(n|2) → SPk are even linear maps with SPk as in (4.17), polynomial in parameter c = (c1, . . . , cm) ∈ Cm

and such that ρk ≡ 0 for k sufficiently large.The theory of polynomial deformations seems to be richer than that of formal ones. The equivalence problem for

polynomial deformations has additional interesting aspects related to parameter transformations.

3.2. Infinitesimal deformations and the first cohomology

Deformations (3.8) and (3.11), modulo second order terms in t and c respectively, are called infinitesimal. Infinitesimaldeformations of a Lie superalgebra homomorphism ρ from g into b are classified by the first cohomology H1(g, b), whereb is a g-module through ρ. Namely, the first order terms ρi in (3.8) and ∂ρ(c)

∂ci|c=0 in (3.11) are 1-cocycles. Two infinitesimal

deformations are said to be equivalent if the corresponding cocycles are cohomologous. Conversely, given a Lie superalgebrahomomorphism ρ : g → b, an arbitrary 1-cocycle ρ1 ∈ Z1(g, b) defines an infinitesimal deformation of ρ.

The obstructions for existence of a deformation (3.8) lie in H2(osp(n|2), SΨ DO(n)). Indeed, setting

ϕt = ρ − ρ0, ρ(1)=

ti ρi, ρ(2)

=

titjρ

(2)ij , . . . ,

O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221 215

we can rewrite the relation (3.8) 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. (3.12)

The first three terms are (δϕt)(x, y), where δ stands for the coboundary. For arbitrary linear maps γ1, γ2 : g −→ b, considerthe standard cup-product: [[γ1, γ2]] : g ⊗ g −→ b defined by:

[[γ1, γ2]](x, y) = (−1)|γ2|(|γ1|+|x|)[γ1(x), γ2(y)] + (−1)|γ1||x|[γ2(x), γ1(y)]. (3.13)

The relation (3.12) becomes now equivalent to:

δϕt = −12[[ϕt , ϕt ]]. (3.14)

Expanding (3.14) in power series in t , we obtain the following equation for ρ(k):

δρ(k)= −

12

i+j=ki,j>0

[[ρ(i), ρ(j)]]. (3.15)

The right hand side of (3.15) is a 2-cocycle whose coefficients are homogeneous of degree k polynomials in t , so thecohomology class of this cocycle is an obstruction for existence of the solutions. The first non-trivial relation δρ(2)

=

−12 [[ρ

(1), ρ(1)]] gives the first obstruction to integration of an infinitesimal deformation. Thus, considering the coefficient of

titj, we get

δρ(2)i,j = −

12[[ρi, ρj]].

This relation is precisely the condition for the 2-cocycle [[ρi, ρj]] to be a coboundary.As before, the bracket [ , ]h defines a cup-product [[ , ]]h, which coincides with the standard cup-product [[ , ]] at h = 1.In the following we denote also by [[ , ]], the cup-product [[ , ]]h as h → 0.

4. The space H1(osp(n|2), SΨ DO(n))

In this section we give a description of the first cohomology space H1(osp(n|2), SΨ DOh(n)) for n = 0, 1, 2.

4.1. A filtration on SΨ DO(n)

The natural embeddings of osp(n|2) into SΨ DO(n) and into SP (n) given by formula (3.7) induce an osp(n|2)-modulestructure on both SΨ DO(n) and on SP (n).

Setting deg x = deg θi = 0, deg ξ = deg θi = 1 for all i, we endow the Poisson superalgebra SP (n) with a Z-grading:

SP (n) =

p∈ZSPp, (4.16)

wherep∈Z = (

p<0)

p≥0 and

SPp =

Fξ−p

+ G1ξ−p−1θ1 + G2ξ

−p−1θ2 + · · · + H1,2ξ−p−2θ1θ2 + · · · | F , Gi, Hi,j ∈ R[x, θ]

(4.17)

is the homogeneous subspace of degree −p. Each element of SΨ DO(n) can be expressed as

A =

k∈Z

(Fk + G1kξ

−1θ1 + · · · + H1,2k ξ−2θ1θ2 + · · ·)ξ−k,

where Fk, Gik, H i,j

k ∈ R[x, θ]. We define the order of A to be

ord(A) = sup{k | Fk = 0 or Gik = 0 or H i,j

k = 0}.This definition of order equips SΨ DO(n) with a decreasing filtration as follows: set

Fp = {A ∈ SΨ DO(n) | ord(A) ≤ −p},where p ∈ Z. So we have

· · · ⊂ Fp+1 ⊂ Fp ⊂ · · · . (4.18)This filtration is compatible with the multiplication and the super Poisson bracket, that is, for A ∈ Fp and B ∈ Fq, one hasA ◦ B ∈ Fp+q and {A, B} ∈ Fp+q−1. This filtration makes SΨ DO(n) an associative filtered superalgebra. Moreover, thisfiltration is compatible with the natural osp(n|2)-action on SΨ DO(n). Indeed,

XF (A) = [XF , A] ∈ Fp for any XF ∈ osp(n|2) and A ∈ Fp.The induced osp(n|2)-module structure on the quotient Fp/Fp+1 is isomorphic to that of the osp(n|2)-moduleSPn. Therefore,

SP (n) ≃

p∈ZFp/Fp+1.

216 O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221

4.2. H1(osp(n|2), SP (n))

Note that H1(osp(n|2), SP (n)) =

p∈Z H1(osp(n|2), SPp) for n = 0, 1, 2were calculated in [5]. The result is as follows.

4.2.1. H1(sl(2), P )

The nontrivial space H1(sl(2), P ) is spanned by the cohomology classes of the 1-cocycles χn defined by:

χ0(Xf ) = f ′ and χ1(Xf ) = f ′′ξ−1. (4.19)

4.2.2. H1(osp(n|2), SP (n))For n = 1, the nontrivial space H1(osp(1|2), SP (1)) is spanned by the cohomology classes of the 1-cocycles εn defined

by:

ε0(XF ) = F ′ and ε1(XF ) = η1(F ′)ξ−1ζ1. (4.20)

For n = 2, the nontrivial space H1(osp(2|2), SP (2)) is spanned by the cohomology classes of the 1−cocyclesΥn defined by:

Υ1(XF ) = (−1)|F |η1η2(F)ξ−1ζ1ζ2,

Υ2(XF ) = F ′ξ−1ζ1ζ2,

Υ3(XF ) = F ′,

Υ4(XF ) = (−1)|F |η1η2(F),

Υ5(XF ) = (−1)|F |+1η1(F ′)ζ1 + η2(F ′)ζ2

ξ−1,

Υ6(XF ) = F ′′ξ−2ζ1ζ2 + (−1)|F |

η2(F ′)ζ1 − η1(F ′)ζ2

ξ−1.

(4.21)

4.3. H1(osp(n|2), SΨ DOh(n))

Recall that SΨ DOh(n) has the same underlying vector space as SP (n). The result of [5] is a specialization at h = 1 ofthe following theorem obtained as in [5]:

Theorem 4.1. The spaces H1(osp(n|2), SΨ DOh(n)), where n = 0, 1, 2 are purely even. An explicit description of these spacesis the following:

1.

H1(sl(2), Ψ DOh(1)L) ≃ R2. (4.22)

It is spanned by the classes of the following nontrivial 1-cocycles

Ξ0(Xf ) = f ′,

Ξ1(Xf ) = f ′′ξ−1.

2.

H1(osp(1|2), SΨ DOh(1)) ≃ R2. (4.23)

It is spanned by the classes of the following nontrivial 1-cocycles

∧0(XF ) = F ′,

∧1(XF ) = η1(F ′)ξ−1ζ1 − hF ′′ξ−1.

3.

H1(osp(2|2), SΨ DOh(2)) ≃ R6. (4.24)

It is spanned by the classes of the following nontrivial 1-cocycles

Θ1(XF ) = (−1)|F |η1η2(F)ξ−1ζ1ζ2,

Θ2(XF ) = F ′ξ−1ζ1ζ2,

Θ3(XF ) = F ′,

Θ4(XF ) = (−1)|F |η1η2(F),

Θ5(XF ) = (−1)|F |+1η1(F ′)ζ1 + η2(F ′)ζ2ξ−1

+ hF ′′ξ−1,

Θ6(XF ) = F ′′ξ−2ζ1ζ2 + (−1)|F |

η2(F ′)ζ1 − η1(F ′)ζ2

ξ−1.

O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221 217

5. Integrability conditions

In this section we obtain the integrability conditions for the infinitesimal deformation of the standard embedding (3.7)

5.1. Integrability conditions for the infinitesimal deformation of the standard embedding of sl(2) into both Ψ DO(1)L and P

The main result of this subsection is the following.

Theorem 5.1. - Any formal deformation of the standard embedding of sl(2) intoΨ DO(1)L is equivalent to its infinitesimal part.- Any formal deformation of the standard embedding of sl(2) into P is equivalent to its infinitesimal part.

Proof. Consider a formal deformation of the embedding (3.7) for n = 0:ρ = ρ0 + t0Ξ0 + t1Ξ1 +

m≥2

i+j=m

t i0tj1ρ

(m)i,j ,

where the highest-order terms ρ(m)i,j are linear maps from sl(2) to Ψ DO(1)L.

By a direct computation, we check that [[Ξ0, Ξ0]] = [[Ξ0, Ξ1]] = [[Ξ1, Ξ1]] = 0. The solution ρ(2)i,j of (3.15) is defined

up to a 1-cocycle and it has been shown in [13,11] that different choices of solutions of (3.15) correspond to equivalentdeformations. Thus, one can always kill ρ(2)

i,j . Then, by recurrence, the highest-order terms satisfy the equation δρ(m)i,j = 0

and can also be killed.As before we prove the result concerning the deformation of the standard embedding of sl(2) into P from the fact that

we have by a direct computation [[χ0, χ0]] = [[χ0, χ1]] = [[χ1, χ1]] = 0. �

5.2. Integrability conditions for the infinitesimal deformation of the standard embedding of osp(1|2) into SΨ DOh(1)

We consider the infinitesimal deformation of the embedding (3.7) for n = 1:

ρ(t) = ρ0 + t0 ∧0 +t1 ∧1 . (5.25)

Assume that it can be integrated to a formal deformation

ρ(t) = ρ0 + t0 ∧0 +t1 ∧1 +

m≥2

i+j=m

t i0tj1ρ

(m)i,j (5.26)

where the highest-order terms ρ(m)i,j are even linear maps from osp(1|2) to SΨ DOh(1).

The main result in this subsection is the following.

Theorem 5.2. Any formal deformation (5.35) of the standard embedding (3.7) for n = 1 is equivalent to the polynomialdeformation

ρ(XF ) = ρ0(XF ) + t0 ∧0(XF ) + t1 ∧1(XF ) + (2t0t1 − 2t21 )F′′ξ−1, (5.27)

where t0, t1 are the deformation parameters.

Proof. By a direct computation we have:(i) In the first step

[[∧0, ∧0 ]]h(XF , XG) = 0,

[[∧0, ∧1 ]]h(XF , XG) = −12[[∧1, ∧1 ]]h(XF , XG)

=

F ′′η1(G′) − η1(F ′)G′′

ζ1ξ

−2+ 2(−1)|F |+|G|η1(F ′)η1(G′)ξ−1

= δ(b)(XF , XG),

where b(XF ) = −2F ′′ξ−1. So we can choose ρ(2)(XF ) = (2t0t1 − 2t21 )F′′ξ−1.

(ii) In the second step

[[b, ∧0 ]]h = [[b, ∧1 ]]h = 0,

so we can reduce ρ(3), to zero by equivalence. Then, by recurrence, the terms ρ(k), for k ≥ 3, satisfy the equation δρ(k)= 0,

and can also be reduced to the identically zero map. �

Remark 5.3. The classification found in Theorem5.2 is obtained under any dependence of h, so passing to the limit as h → 0in the deformation (5.27), we obtain a similar classification for the polynomial deformation of the infinitesimal deformationof the standard embedding of osp(1|2) into SP (1) generated by the 1-cocycles ε0, ε1.

More precisely, we obtain :

218 O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221

Theorem 5.4. Any nontrivial polynomial deformation of the infinitesimal deformation of the standard embedding of osp(1|2)into SP (1) generated by the 1-cocycles ε0, ε1 is equivalent to the deformationρ(XF ) = ρ0(XF ) + t0ε0(XF ) + t1ε1(XF ) + (2t0t1 − 2t21 )F

′′ξ−1.

where t0 and t1 are the deformation parameters.

5.3. Integrability conditions for the infinitesimal deformation of the standard embedding of osp(2|2) into SΨ DOh(2)

We consider the infinitesimal deformation of the embedding (3.7) for n = 2:

ρ(t) = ρ0 + t1Θ1 + t2Θ2 + t3Θ3 + t4Θ4 + t5Θ5 + t6Θ6. (5.28)

Assume that it can be integrated to a formal deformation

ρ(t) = ρ0 +

6i=1

tiΘi +

6i,j=1

titjρ(2)i,j + · · · , (5.29)

where thehighest-order termsρ(2)i,j , ρ

(3)i,j,k, . . . are even linearmaps from osp(2|2) toSΨ DO(2)h, with |ρ

(2)i,j | = |titj|, |ρ

(3)i,j,k| =

|titjtk|, . . ..The homomorphism condition gives for the second-order terms the following (Maurer–Cartan) equation δρ(2)

= −12

[[ρ(1), ρ(1)]], where ρ(1)

=6

i=1 tiΘi, ρ(2)=6

i,j=1 titjρ(2)i,j and ρ

(2)i,j is of the form

ρ(2)i,j =

Mk=∞

0≤i,j≤4

ai,jk,εξkηi

1ηj2 +

Mk=∞

0≤i,j≤4

bi,jk,ε θ1ξkηi

1ηj2 +

Mk=∞

0≤i,j≤4

c i,jk,ε θ2ξkηi

1ηj2 +

Mk=∞

0≤i,j≤4

di,jk,ε θ1θ2ξkηi

1ηj2, (5.30)

where ai,jk,ε, bi,jk,ε, c i,jk,ε, di,jk,ε ∈ R[x, θ1, θ2], which represent the first obstruction for the integrability condition of the inf-initesimal deformation (5.28). So we have to compute in the first step all the cup-products [[Θi, Θj ]]h .

Remark 5.5. It is easy to see that if [[Θi, Θj ]]h = 0, so, there are no integrability conditions involving ti and tj.

In the following, we study, successively, the second and the third-order integrability conditions for the deformation (5.29)

Lemma 5.6. The system {[[Θ1, Θ3 ]]h, [[Θ1, Θ4 ]]h, [[Θ1, Θ6 ]]h, [[Θ2, Θ3 ]]h, [[Θ2, Θ4 ]]h, [[Θ2, Θ5 ]]h, [[Θ2, Θ6 ]]h, δρ(2)ij } is lin-

early independent.

Proof. By a straightforward computation, we show that

[[Θ1, Θ3 ]]h(XF , XG) = −12[[Θ1, Θ5 ]]h(XF , XG)

= (−1)|F |+|G|

F ′′η1η2(G) − η1η2(F)G′′

ζ1ζ2ξ

−2

+

(−1)|F |η1η2(F)η2(G′) − η2(F ′)η1η2(G)

ζ1ξ

−1

+

η1(F ′)η1η2(G) − (−1)|F |η1η2(F)η1(G′)

ζ2ξ

−1,

[[Θ1, Θ4 ]]h(XF , XG) =

(−1)|F |η1η2(F)η1(G′) − η1(F ′)η1η2(G)

ζ1ξ

−1

+

(−1)|F |η1η2(F)η2(G′) − η2(F ′)η1η2(G)

ζ2ξ

−1,

[[Θ1, Θ6 ]]h(XF , XG) = 2(−1)|G|+1η1(F ′)η1(G′) + η2(F ′)η2(G′)

ζ1ζ2ξ

−2

+ 2η1(F ′)η1η2(G) + (−1)|F |+1η1η2(F)η1(G′)

ζ1ξ

−1

+ 2η2(F ′)η1η2(G) + (−1)|F |+1η1η2(F)η2(G′)

ζ2ξ

−1,

[[Θ2, Θ3 ]]h(XF , XG) =

F ′′G′

− F ′G′′

ζ1ζ2ξ

−2+ (−1)|G|

F ′η1(G′) + (−1)|F |+1η1(F ′)G′

ζ2ξ

−1

+(−1)|G|

(−1)|F |η2(F ′)G′

− F ′η2(G′)ζ1ξ

−1+ 2

F ′G′′

− F ′′G′

θ2ζ1ξ

−1

+ 2F ′′G′

− F ′G′′

θ1ζ2ξ

−1,

O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221 219

[[Θ2, Θ4 ]]h(XF , XG) =

F ′η1(G′) + (−1)|G|+1η1(F ′)G′

ζ1ξ

−1

+

F ′η2(G′) + (−1)|G|+1η2(F ′)G′

ζ2ξ

−1,

[[Θ2, Θ5 ]]h(XF , XG) = 2F ′G′′

− F ′′G′

ζ1ζ2ξ

−2+ 2θ1

F ′′η1(G′) − η1(F ′)G′′

ζ1ζ2ξ

−2

+ 2θ2F ′′η2(G′) − η2(F ′)G′′

ζ1ζ2ξ

−2+(−1)|F |+|G|

+ 1

η1(F ′)η1(G′)

+ η2(F ′)η2(G′)ζ1ζ2ξ

−2+ 2

F ′η1(G′) + (−1)|G|+1η1(F ′)G′

ζ2ξ

−1

+ 2(−1)|G|η2(F ′)G′

− F ′η2(G′)ζ1ξ

−1+ h

F ′′η2(G′) − η2(F ′)G′′

ζ1ξ

−2

+ hη1(F ′)G′′

− F ′′η1(G′)ζ2ξ

−2,

[[Θ2, Θ6 ]]h(XF , XG) = −2F ′η1(G′) + (−1)|G|+1η1(F ′)G′

ζ1ξ

−1+ h

F ′′η1(G′) − η1(F ′)G′′

ζ1ξ

−2

− 2F ′η2(G′) + (−1)|G|+1η2(F ′)G′

ζ2ξ

−1+ h

F ′′η2(G′) − η2(F ′)G′′

ζ2ξ

−2,

where F = f0 + f1θ1 + f2θ2 + f12θ1θ2, G = g0 + g1θ1 + g2θ2 + g12θ1θ2 and f0, f1, f2, f12, g0, g1, g2, g12 ∈ R[x].Let us consider the equation

α[[Θ1, Θ3 ]]h(XF , XG) + β[[Θ1, Θ4 ]]h(XF , XG) + γ [[Θ1, Θ6 ]]h(XF , XG) + λ[[Θ2, Θ3 ]]h(XF , XG)

+ µ[[Θ2, Θ4 ]]h(XF , XG) + ν[[Θ2, Θ5 ]]h(XF , XG) + ϵ[[Θ2, Θ6 ]]h(XF , XG) = δρ(2)(XF , XG).(5.31)

Collecting the terms in f12g ′′

0 θ1θ2ξ−2, f12g ′

1θ1ξ−1, f12g ′

2θ2ξ−1, f ′

0g′′

0 θ1θ2ξ−2, f ′

0g′′

0 θ1θ1ξ−1, f ′

1g′

1θ1θ2ξ−2, f ′′

0 g′

1θ1θ2θ1ξ−1,

f ′′

0 g′

2θ1ξ−2, f ′′

0 g′

1θ1ξ−2, f ′

1g′′

0 θ2ξ−2, f ′′

0 g′

2θ2ξ−2, f ′

2g′′

0 θ1ξ−1, f ′

1g′′

0 θ1ξ−1, f ′′

0 g′

1θ2ξ−1, f ′

2g′′

0 θ2ξ−1, f ′

1g′

1ξ−1, and in f ′

1g′

1θ1θ1ξ−1,

we obtain the conditions

α = 0,−β + 2γ = a1,20,2 + a2,10,1,

−β + 2γ = −a1,20,2 − a2,10,1,

λ − 2ν = 0,µ − 2ϵ = 0,2(γ + ν) = c1,2

−1,0 − d0,4−2,0,

−2h(γ + ν) = hb2,1

−1,0 + b0,4−1,2

,

2hν = h−3b2,1

−1,0 − 2d0,4−2,0

,

2hϵ = h−3b1,2

−1,0 − a0,4−1,0

,

2hν = h3c1,2

−1,0 − 2d0,4−2,0

,

−2hν = h3c2,1

−1,0 + a0,4−1,0

,

2ν = c1,2−1,0 + c0,4

−1,1,

2ϵ = a0,4−1,0 + b1,2

−1,0 + b0,4−1,1,

−2ν = b2,1−1,0 + b0,4

−1,2,

2ϵ = a0,4−1,0 + c2,1

−1,0 + c0,4−1,2,

0 = b1,2−1,0 − a0,4

−1,0,

0 = b0,4−1,1.

(5.32)

Substituting expressions in the system (5.32), one gets α = β = γ = λ = µ = ν = ϵ = 0. �

Remark 5.7. We have

[[Θ2, Θ6 ]]h(XF , XG) = −2[[Θ2, Θ4 ]]h(XF , XG) +hF ′′η1(G′) − η1(F ′)G′′

ζ1ξ

−2

+hF ′′η2(G′) − η2(F ′)G′′

ζ2ξ

−2.(5.33)

Proposition 5.8. The second-order integrability conditions of the infinitesimal deformation (5.28) are the following:

t1 = t2 = 0, or t3 = t4 = t5 = t6 = 0, ort2 = t4 = t6 = 0,t3 = 2t5.

(5.34)

220 O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221

Proof. The necessary integrability conditions for the second-order term ρ(2) are that some linear combinations ofexpressions titj[[Θi, Θj ]]h, must be a coboundary of a linear differential operator ρ

(2)i,j .

A straightforward computation leads to [[Θ1, Θ1 ]]h = [[Θ1, Θ2 ]]h = [[Θ2, Θ2 ]]h = [[Θ3, Θ3 ]]h = [[Θ3, Θ4 ]]h =

[[Θ3, Θ6 ]]h = [[Θ4, Θ4 ]]h = [[Θ4, Θ5 ]]h = [[Θ5, Θ6 ]]h = 0, and[[Θ3, Θ5 ]]h = [[Θ4, Θ6 ]]h = −

12 [[Θ5, Θ5 ]]h = −

12 [[Θ6, Θ6 ]]h = δ(b), where b(XF ) = −2F ′′ξ−1. So each coefficients titj

corresponding to a cup-product cited in Lemma 5.6 must vanish, except t1t3 and t1t5 where we imposed t1t3 = 2t1t5. Thatis, we get the first set of necessary integrability conditions, and we have to compute the third-order term ρ(3) in (3.15). Thiscompletes the proof. �

The main result in this subsection is the following.

Theorem 5.9. The conditions given in Proposition 5.8, are necessary and sufficient for the integrability of the infinitesimaldeformation (5.28).

Proof. Of course these conditions are necessary. Now, we show that these conditions are sufficient. The solution ρ(3), of theMaurer–Cartan equation is deduced from the cup-products [[b, Θi ]]h. By a direct computation we get that they are null. Sowe can reduce ρ(3), to zero by equivalence. Then, by recurrence, the terms ρ(k), for k ≥ 3, satisfy the equation δρ(k)

= 0,and can also be reduced to the identically zero map. �

Applying the contraction procedure as h → 0, one gets the corresponding necessary and sufficient conditions for theintegrability of the infinitesimal deformation

ρ(t) = ρ0 + t1Υ1 + t2Υ2 + t3Υ3 + t4Υ4 + t5Υ5 + t6Υ6 (5.35)

of the standard embedding ρ0 of osp(2|2) into SP (2). More precisely, we get:

Proposition 5.10. The second-order integrability conditions of the infinitesimal deformation (5.35) are the following:

t1 = t2 = 0 ort3 = 2t5,t4 = 2t6.

(5.36)

Proof. As h → 0:- Eq. (5.33) in Remark 5.7 implies [[Θ2, Θ6]] = −2[[Θ2, Θ4]], where [[, ]] denotes the cup product in SP (2).- The system (5.32) implies

α = 0,−β + 2γ = 0,λ − 2ν = 0,µ − 2ϵ = 0.

(5.37)

So the expression (5.31) becomes

γ2[[Θ1, Θ4]] + [[Θ1, Θ6]]

+ ν

2[[Θ2, Θ3]] + [[Θ2, Θ5]]

= δρ(2). (5.38)

Finally, by a straightforward computation we get

2[[Θ1, Θ4]] + [[Θ1, Θ6]] = 2[[Θ2, Θ3]] + [[Θ2, Θ5]]

= δk, (5.39)

where k(XF ) = −2F ′′ζ1ζ2ξ−2. That is, we deduce the first set of necessary integrability conditions, and we have to compute

the third-order term ρ(3) corresponding to the formal deformation of the standard embedding ρ0 of osp(2|2) into SP (2).This completes the proof. �

Theorem 5.11. The conditions given in Proposition 5.10, are necessary and sufficient for the integrability of the infinitesimaldeformation (5.35).

Proof. Of course the conditions given in deformation (5.36) are necessary.A direct computation shows that

[[Υ1, k]](XF , XG) = 0,[[Υ2, k]](XF , XG) = 0,

[[Υ3, k]](XF , XG) = 2η2(F ′)G′′

− F ′′η2(G′)ζ1ξ

−2

+ 2F ′′η1(G′) − η1(F ′)G′′

ζ2ξ

−2,

O. Ncib, S. Omri / Journal of Geometry and Physics 86 (2014) 211–221 221

[[Υ4, k]](XF , XG) = −2F ′′η1(G′) − η1(F ′)G′′

ζ1ξ

−2

− 2F ′′η2(G′) − η2(F ′)G′′

ζ2ξ

−2,

[[Υ5, k]](XF , XG) = −2[[Υ3, k]](XF , XG),

[[Υ6, k]](XF , XG) = −2[[Υ4, k]](XF , XG).

So collecting the terms in f ′

1g′′

0 θ1ξ−1, f ′

1g′′

0 θ1ξ−2, f ′

2g′′

0 θ2ξ−2, and f ′

2g′′

0 θ1ξ−2 we deduce that [[Υ3, k]], [[Υ4, k]] and δρ(2) are

linearly independent. This completes the proof. �

Acknowledgements

We would like to thank Mabrouk Ben Ammar and Imed Basdouri for many stimulating discussions.

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