Reach Us
+44-1522-440391

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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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 Fk2 . We calculate obstructions for integrability of innitesimal multiparameter deformations and determine the complete local commutative algebra corresponding to the miniversal deformation.

We consider the superspace equipped with the contact 1-form, where θ is the odd variable, the Lie superalgebra K(1) of contact polynomial vector fields on (also called superconformal Lie algebra see [16]), and the K(1)-module of symbols , whereis the module of the weighted densities on . 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 classied by the first cohomology space of the Lie algebra with values in the module of endomorphisms of that module. In our case,

where is the superspace of linear differential operators from the superspace of weighted densities to , and hereafter.

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

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

We shall give concrete explicit examples of the deformed action.

**2.1 The Lie superalgebra of contact vector fields on**

Let be the superspace with coordinates (x, θ), where θ is the odd variables (θ^{2} = 0). We consider the superspace [x, θ] of superpolynomial functions on :

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

(2.1)

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

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

Letbe the superspace of superpolynomial vector fields on :

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

, for some

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

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

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

(2.2)

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

(2.3)

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

(2.4)

where . In particular, we have

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

(2.5)

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

(2.6)

Obviously,

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

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

**2.3 The supertransvectants: Explicit formula**

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

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

(2.7)

where the numeric coecients are

(2.8)

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

**2.4 The first cohomology space**

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

(2.9)

Let

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

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

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

The following theorem recalls the result.

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

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

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

where, here and below, .

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

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

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

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

(2) The space is isomorphic to the following:

(2.10)

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

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

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

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

• For, the generator can be chosen as follows:

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

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

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

• For , we have. Then the component of γ is broken on, where

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

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

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

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**

Let be an action of a Lie superalgebra g on a vector superspace V . When
studying deformations of the g-action ρ_{0}, one usually starts with infinitesimal deformations:

(3.1)

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

(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

If dim, then one can choose 1-cocyclesas a basis ofand consider the following infinitesimal deformation:

(3.3)

where are independent formal parameters with t_{i} and γ_{i} are the same parity, i.e.,

.

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

(3.4)

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

(3.5)

where and .

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 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

(3.6)

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

(3.7)

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

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

(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 .

**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 with the same base A are called equivalent if there
exists a formal inner automorphism of the associative superalgebra such that

where is the unity of the superalgebra .

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

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

So the first cohomology space determines and classies infinitesimal deforma- tions up to equivalence.

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

(4.1)

where

By setting

we can rewrite the relation (3.2) as follows:

(4.2)

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

(4.3)

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

(4.4)

wheredenotes the parity.

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

(4.5)

The first nontrivial relation is

(4.6)

Therefore, it is easy to check that for any two 1-cocycles γ1 and , the bilinear map is 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 is a 2-coboundary. We, therefore, naturally deduce that the operation (4.4) defines a bilinear map:

(4.7)

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

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

(4.8)

where .

Let us consider the 2-cochains

(4.9)

then it is easy to see that

(4.10)

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

and

and

and

and

and

and

and

and

and

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

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

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

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

(4.11)

where and

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

For , we need the following lemma.

**Lemma 4.3.** Let

be a 1-cochain. If

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

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

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

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

where

Also one can check that

where

Also

where

Also one can check that

where

Finally

where

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):

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

It seems to be an interesting open problem to compute the full cohomology ring . 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 has the following structure:

We study deformations of K(1)-modules for any and for arbitrary generic.

**Example 7.1.** Let us consider the K(1)-modules and.

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

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

(7.1)

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

(7.2)

(7.3)

but, by a direct computation, we show that for all λ, then and as a consequence.

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

(7.4)

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

(7.5)

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

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

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

(7.6)

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

The condition of integrability is

(7.7)

where , i.e., .

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

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

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

We are grateful to Claude Roger for his constant support.

- Fialowski A (1988) An example of formal deformations of Lie algebras. In Proceedings of NATO Conference on Deformation Theory of Algebras and Applications ( Ciocco, Italy, 1986). Kluwer, Dordrecht pp: 375-401.
- Fialowski A, Fuchs DB (1999) Construction of miniversal deformations of Lie algebras. J. Func.
- Fialowski A (1986) Deformations of Lie algebras. Mat. Sbornyik USSR, 127: 476-482; English
- Nijenhuis A, Richardson RW (1967) Deformations of homomorphisms of Lie groups and Lie algebras. Bull. Amer. Math. Soc 73: 175-179.
- Agrebaoui B, Ammar F, Lecomte P, Ovsienko V (2002) Multi-parameter deformations of the module of symbols of differential operators. Int. Math. Res. Notices 16: 847-869.
- Agrebaoui B, Fraj NB, Ammar MB, Ovsienko V (2003) Deformations of modules of differential
- Ammar M Ben, Boujelbene M (2008) sl(2)-trivial deformations of VectPol(R)-modules of symbols.
- Basdouri I, Ammar M Ben (2007) Cohomology of osp(1|2) acting on Linear Differential Operators on the supercercle S 1|1 . Lett Math Phys. 81: 239-251.
- Basdouri I, Ammar M Ben, Fraj N Ben, Boujelben M, Kamoun K, Cohomology of the Lie superalgebra of contact vector fields on R 1|1 and deformations of the superspace of symbols. Preprint arXiv:math/0702645v1.
- Fuchs DB (1986) Cohomology of infinite-dimensional Lie algebras. Plenum Publ, New York.
- Gieres F, Theisen S (1993) Superconformally covariant operators and super W-algebras. J. Math.
- Gargoubi H, Ovsienko V (2008) Supertransvectants and symplectic geometry. Int. Math. Res. Not. IMRN.
- Guieu L, Roger C (2007) L’alg`ebre et le groupe de Virasoro. Publication of CRM, Montr´eal, ISBN 2-921120-44-5.
- Richardson RW (1969) Deformations of subalgebras of Lie algebras. J. Diff. Geom. 3: 289–308.
- Bouarroud S, Ovsienko V (1998) Three cocycles on Diff(S1) generalizing the Schwarzsian derivative. Int. Math. Res. Notices 1: 25–39.
- Kac VG, Van De Leur W (1989) On classification of superconformal algebras. In Strings 88.
- Ovsienko V, Roger C (1999) Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of pseudodifferential operators on S1 . AMS Transl. Ser. 2 (Adv. Math. Sci.) 194: 211-227.
- Ovsienko V, Roger C (1998) Deforming the Lie algebra of vector fields on S1 inside the Poisson algebra on T? *S1. Comm. Math. Phys. 198: 97-110.
- Huang WJ (1994) Superconformal covariantization of superdifferential operator on (1|1) superspace and classical N = 2W superalgebras. J. Math. Phys 35: 2570-2582.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebra
- Algebraic Geometry
- Analytical Geometry
- Applied Mathematics
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Combinatorics
- Complex Analysis
- Computational Model
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Deformations Theory
- Differential Equations
- Differential Transform Method
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Geometry
- Hamilton Mechanics
- Harmonic Analysis
- Homological Algebra
- Homotopical Algebra
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Latin Squares
- Lie Algebra
- Lie Superalgebra
- Lie Theory
- Lie Triple Systems
- Loop Algebra
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Operad Theory
- Physical Mathematics
- Quantum Group
- Quantum Mechanics
- Quantum electrodynamics
- Quasi-Group
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Representation theory
- Riemannian Geometry
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Super Algebras
- Symmetric Spaces
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topologies
- Topology
- mirror symmetry
- vector bundle

- Total views:
**11902** - [From(publication date):

May-2009 - Aug 24, 2019] - Breakdown by view type
- HTML page views :
**8110** - PDF downloads :
**3792**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2019-20