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Deforming K(1) superalgebra modules of symbols | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Deforming K(1) superalgebra modules of symbols

Faouzi AMMAR and Kaouthar KAMOUN

Faculte des sciences de Sfax, Universite de Sfax, Route Soukra, km 3.5, B.P. 1171, Sfax 3000, Tunisie
E-mails: [email protected], [email protected]

Received Date: April 16, 2008; Revised Date: December 05, 2008

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Abstract

We study nontrivial deformations of the natural action of the Lie superalgebra K(1) of contact vector elds on the (1; 1)-dimensional superspace R1j1 on the space of symbols e Sn  = Ln k=0 F􀀀k2 . We calculate obstructions for integrability of in nitesimal multiparameter deformations and determine the complete local commutative algebra corresponding to the miniversal deformation.

1 Introduction

We consider the superspace Image equipped with the contact 1-formImage, where θ is the odd variable, the Lie superalgebra K(1) of contact polynomial vector fields on Image (also called superconformal Lie algebra see [16]), and the K(1)-module of symbols , whereImageis the module of the weighted densities on Image. As Lie superalgebra K(1) is rigid like the Lie algebra of Virasoro [13], so one tries deformations of its modules. We will use the frame- work of Fialowski [1,3] and Fialowski-Fuchs [2] (see also [10]) and consider (multiparameter) deformations over complete local commutative algebras related to this deformation. The first step of any approach to the deformation theory consists in the determination of infinitesimal deformations. According to Nijenhuis-Richardson [4], infinitesimal deformations of the action of a Lie algebra on some module are classi ed by the first cohomology space of the Lie algebra with values in the module of endomorphisms of that module. In our case,

Image

where Image is the superspace of linear differential operatImageors from the superspace of weighted densities Image to Image, and hereafterImage.

While the obstructions for integrability of this infinitesimal deformations belong to the second cohomology space,

Image

The odd first space Image was calculated in [9]: our task, therefore, is to calculate the even first space Imageand the obstructions.

We shall give concrete explicit examples of the deformed action.

2 Definitions and notations

2.1 The Lie superalgebra of contact vector fields on Image

Let Image be the superspace with coordinates (x, θ), where θ is the odd variables (θ2 = 0). We consider the superspace Image[x, θ] of superpolynomial functions on Image:

Image

where Image[x] is the space of polynomial functions on Image. The superspace Image[x, θ] has a structure of superalgebra given by the contact bracket

Image (2.1)

where Image and p(F) is the parity of F. Note thatImageso Image is sometimes called a "square root" of Image.

Any contact structure on Image can be defined by the following 1-form:

Image

LetImagebe the superspace of superpolynomial vector fields on Image :

Image

where Imagestands for Image and Image stands for Image, and consider the superspace K(1) of contact polynomial vector fields on Image defined by

Image, for someImage

where Image is the Lie derivative of along the vector fieldsImage. Any contact superpolynomial vector field on Image can be given by the following explicit form:

Image

2.2 The space of polynomial weighted densities on Image

Recall the definition of the Image-module of polynomial weighted densities on Image, where Image is the Lie algebra of polynomial vector fields on Image. Consider the 1-parameter action of Image on the space of polynomial functions Image[x] given by

Image

where λ ∈Image. Denote byImagethe Image-module structure on Image[x] defined by this action. Geometrically, Image is the space of polynomial weighted densities of weight λ on Image , i.e.,

Image (2.2)

Now, in supersetting, we have an analogous definition of weighted densities (see [9]) with dx replaced by Image. Consider the 1-parameter action of K(1) on Image[x, θ] given by the rule

Image (2.3)

where F,G ∈Image[x, θ] and Image or, in components,

Image (2.4)

where Image . In particular, we have

Image

We denote this K(1)-module byImage, the space of all polynomial weighted densities on Image of weight λ:

Image (2.5)

Let Image be the K(1)-module of linear di erntial superoperators, the K(1)- action on this superspace is given by

Image (2.6)

Obviously,

(1) The adjoint K(1)-module is isomorphic to Image (2) As a Image-module, Image, where Image is the Image-module of polynomial weighted densities of weight λ and Π is the functor of the change of parity.

Proposition 2.1. As a Image-module, we have for the homogeneous relative parity compo- nents,

Image

2.3 The supertransvectants: Explicit formula

Definition 2.2 (see [12]). The supertransvectants are the bilinearImage-invariant maps

Image

where Image. These operators were calculated in [11] (see also [19]), let us give their explicit formula. One has

Image (2.7)

where the numeric coecients are

Image (2.8)

where [a] denotes the integer part of a ∈ Image, as above, the binomial coecients Image are well defined if b is integer. It can be deduced directly that those operators are, indeed, Image-invariant.

2.4 The first cohomology spaceImage

Let us first recall some fundamental concepts from cohomology theory (see [10]). Let Imagebe a Lie superalgebra acting on a super vector space Image. The spaceImage is Image-graded via

Image (2.9)

Let

Image

be the space of 1-cocycles for the Chevalley-Eilenberg differential. According to the Image-grading (2.9), any 1-cocycleImage is broken toImage. The first cohomology space Image inherits the Image -grading from (2.9) and it decomposes into odd and even parts as follows:

Image

The odd first space Image was calculated in [9]; we calculate, here, the even first space Image .

Lemma 2.3 (see [9]). The 1-cocycle Imageis a coboundary over K(1) if and only if Imageis a coboundary overImage .

The following theorem recalls the result.

Theorem 2.4. (1) The space Image is isomorphic to the following:

Image

The space Image is generated by the cohomology classes of the 1-cocycles:

• For λ = μ, the generator can be chosen as follows:

Image

where, here and below, Image .

• For μ − λ = 2 and λ≠ − 1the generator can be chosen as follows:

Image

• For μ − λ = 3 and λ = 0, the generator can be chosen as follows:

Image

• For μ − λ = 3 and Image , the generator can be chosen as follows:

Image

• For μ − λ = 4 and Image, the generator can be chosen as follows:

Image

(2) The spaceImage is isomorphic to the following:

Image (2.10)

The space Imageis generated by the cohomology classes of the 1-cocycles:

• For λ = 0 and Image, the generators can be chosen as follows:

Image

• ForImage and Image, the generator can be chosen as follows:

Image

• For Image and Image, the generator can be chosen as follows:

Image

• ForImage, the generator can be chosen as follows:

Image

Proof. The odd cohomology Image was calculated in [9].

Now, we are interested in the even cohomology. The adjoint K(1)-module is Image-isomorphic toImage, so the even 1-cocycleImage decomposes into two components: Image, where

Image

• For λ = μ, a straightest computation shows thatImage is prolongation ofImage calculated by Feigen and Fuchs in [2].

• For Image, we haveImage. Then the componentImage of γ is broken onImage, where

Image

If the component Image is a differential operator with degree ≥ 2, then it vanish on Image, thus Image is a supertransvectant by the following lemma.

Lemma 2.5 (see [8, Theorem 3.1]). Up to coboundary, any even 1-cocycle Image vanishing onImage is Image-invariant. That is, ifImage, then the restriction of γ to Image is trivial.

As the adjoint K(1)-module is isomorphic to Image, the 1-cocycleImage can be looked as a differential operatorImage. We consider the supertransvectantsImage as k = μ − λ. If μ − λ ≥ 2, we look for those which are nontrivial 1-cocycles. In this way we can deduce Image , and Image, whereImage.

3 Deformation theory and cohomology

Deformation theory of Lie algebra homomorphisms was first considered for one-parameter de- formations [2,4,14]. Recently, deformations of Lie (super)algebras with multiparameters were intensively studied (e.g., [1,2,3,5,6,17,18]). Here we give an outline of this theory.

3.1 Infinitesimal deformations

LetImage be an action of a Lie superalgebra g on a vector superspace V . When studying deformations of the g-action ρ0, one usually starts with infinitesimal deformations:

Image (3.1)

whereImageis a linear map and t is a formal parameter. The homomorphism condition

Image(3.2)

where x, y ∈ g, is satisfied in order 1 in t if and only if γ is a 1-cocycle. That is, the map γ satisfies

Image

If dimImage, then one can choose 1-cocyclesImageas a basis ofImageand consider the following infinitesimal deformation:

Image (3.3)

whereImage are independent formal parameters with ti and γi are the same parity, i.e.,

Image.

To study deformations of K(1)-action on Image,we must consider the spaceImage, End(Image)). Any infinitesimal deformation of the K(1)-module Image is then of the form

Image (3.4)

where Image is the Lie derivative of Image along the vector fieldImagedefined by (2.3), and

Image (3.5)

whereImage and Image.

Let us emphasize that we restrict our study to the deformation (3.4) for generic values of λ.

3.2 Integrability conditions

Consider the supercommutative associative superalgebra Image with unity and consider the problem of integrability of infinitesimal deformations. Starting with the infinitesimal defor- mation (3.3), we look for a formal series

Image (3.6)

where the highest-order termsImageare linear maps from g to End(V) with ImageImage such that the map

Image (3.7)

satisfies the homomorphism condition (3.2) at any order in Image.

However, quite often the above problem has no solution. Following [1] and [5], we must impose extra algebraic relations on the parameters Image in order to get the full deformation. Let R be an ideal in Image generated by some set of relations, the quotient

Image (3.8)

is a supercommutative associative superalgebra with unity, and one can speak about deforma- tions with base A (see [2] for details). The map (3.7) sends g to Image.

3.3 Equivalence and the first cohomology

The notion of equivalence of deformations over commutative associative algebras has been con- sidered in [1].

Definition 3.1. Two deformations ρ and Image with the same base A are called equivalent if there exists a formal inner automorphism Image of the associative superalgebra Image such that

Image

where Image is the unity of the superalgebra Image.

As a consequence, two infinitesimal deformations Image and Image are equivalent if and only if Image is a coboundary:

Image

where Image and Image stands for the cohomological Chevalley-Eilenberg coboundary for cochains on g with values in End(V ) [10,4].

So the first cohomology space Imagedetermines and classi es infinitesimal deforma- tions up to equivalence.

4 Computing the second-order Maurer-Cartan equation

Any infinitesimal deformation of the K(1)-module Image can be integrated to a formal deformation, such deformation is then of the form

Image(4.1)

where

Image

By setting

Image

we can rewrite the relation (3.2) as follows:

Image(4.2)

The first three terms give Image. The relation (4.2) becomes now equivalent to

Image(4.3)

Definition 4.1. LetImage be two arbitrary linear maps, we denote by Image the cup-product defined by

Image(4.4)

whereImagedenotes the parity.

Expanding (4.3) in power series in Image we obtain the following equation forImage :

Image (4.5)

The first nontrivial relation is

Image (4.6)

Therefore, it is easy to check that for any two 1-cocycles γ1 and Image, the bilinear map Imageis a 2-cocycle. The first nontrivial relation (4.6) is precisely the condition for this 2-cocycle to be a coboundary. Moreover, if one of the 1-cocycles γ1 or γ2 is a coboundary, then Image is a 2-coboundary. We, therefore, naturally deduce that the operation (4.4) defines a bilinear map:

Image (4.7)

All the potential obstructions are in the image of Image under the cup-product inImage.

The bilinear map (4.7) can be decomposed in homogeneous components as follows:

Image (4.8)

where Image.

4.1 Cup-products of the nontrivial 1-cocycles

Let us consider the 2-cochains

Image(4.9)

then it is easy to see that

Image (4.10)

and we we compute successively the 2-cocycles Image for Image andImage, two contact vectors, andImage. For generic values of λ, we have the following:

Image

and

Image

and

Image

and

Image

and

ImageImage

and

Image

and

Image

and

Image

and

Image

Image

and

Image

Proposition 4.2. (a) Each of the 2-cocycles

Image

defines a nontrivial cohomology class. Moreover, these classes are linearly independant.

(b) Each of the 2-cocycles Image is a coboundary.

 

Proof. A 2-cocycle Image is a coboundary if and only if it satisfies

Image(4.11)

where Image and

Image

For Image, a direct computation shows that thoseImage are nontrivial 2-cocycles.

For Image, we need the following lemma.

Lemma 4.3. Let

Image

be a 1-cochain. If

Image

then, for λ ≠ μ or Image or Imagewhere k is an integer, b is a supertransvectant.

Proof. The condition Image implies that b is a 1-cocycle on Image. From the result of [8, Theorem 3.1] the spaceImage if λ ≠ μ or Image or Image, where k is an integer. For such values of λ and μ, the condition of 1-cocycle

Image

is equivalent to the condition of Image-invariance. Then b is a supertransvectant.

Remark that the cup-products for Imageare Image-invariant, then, by Lemma 4.3, they are supertransvectant boundaries. A simple computation shows that

Image

where

Image

Also one can check that

Image

where

Image

Also

Image

where

Image

Also one can check that

Image

where

Image

Finally

Image

where

Image

5 Integrability conditions

In this section, we obtain the necessary second-order integrability conditions for the infinitesimal deformation (3.4). Theorem 5.1. The following conditions are necessary for integrability for the deformation (3.4):

Image

Proof. If we take account of the Proposition 4.2, we deduce the integrability conditions (1), (2) and (3) and we have

Image

6 An open problem

It seems to be an interesting open problem to compute the full cohomology ring ImageImage. The only complete result here concerns the first cohomology space. Proposition 4.2 provides a lower bound for the dimension of the second cohomology space. We formulate the following.

Conjecture 6.1. The space of second cohomology of K(1) with coecients in the superspace Image has the following structure:

Image

7 Examples

We study deformations of K(1)-modules Image for anyImage and for arbitrary genericImage.

Example 7.1. Let us consider the K(1)-modules Image andImage.

Proposition 7.2. Every deformation of K(1)-modules Image and Image is equivalent to infinitesimal one.

Proof. Let us consider the K(1)-module Image. Any infinitesimal deformation is given by

Image (7.1)

where Image is the Lie derivative of Image along the vector field Image defined by (2.3), and

Image (7.2)

Image (7.3)

but, by a direct computation, we show that Image for all λ, thenImage and as a consequenceImage.

Now consider the K(1)-module Image. Any infinitesimal deformation is given by

Image (7.4)

where Image is the Lie derivative of Image along the vector field Image defined by (2.3), and

Image (7.5)

By the same arguments, we can that show in this case Image, then the deformation is infinitesimal.

Example 7.3. Consider the K(1)-module Image. In this case,

Image

For Image, the deformation of this K(1)-module is of degree 1 given by

Image(7.6)

where Imageis the Lie derivative of Image along the vector field Image defined by (2.3),Image is defined as

Image

The condition of integrability is

Image (7.7)

where Image, i.e., Image.

In this case, we have Image, then this condition is necessary and sucient for integrability of the deformation (7.6).

Let, in this case (i.e., Image), A be the supercommutative associative superalgebra defined by the quotient ofImage by the ideal R generated by equation (7.7). Then we speak about a deformation with base A.

For Image, one hasImage, then the deformation of this K(1)-module is equivalent to infinitesimal one.

Acknowledgement

We are grateful to Claude Roger for his constant support.

References

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