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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On the First aff(1)-Relative Cohomology of the Lie Algebra of Vector Fields and Differential Operators

Meher A*

Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie

*Corresponding Author:
Meher A
Département de Mathématiques
Faculté des Sciences de Sfax
BP 802, 3038 Sfax, Tunisie
Tel: 71 872 600
Fax: 71 871 666
E-mail: [email protected]

Received Date: April 08, 2017; Accepted Date: June 21, 2017; Published Date: June 27, 2017

Citation: Meher A (2017) On the First aff(1)-Relative Cohomology of the Lie Algebra of Vector Fields and Differential Operators. J Generalized Lie Theory Appl 11: 269. doi: 10.4172/1736-4337.1000269

Copyright: © 2017 Meher A. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Let Vect(1) be the Lie algebra of smooth vector fields on 1. In this paper, we classify aff(1) -invariant linear differential operators from Vect(1) to λ,μ;v vanishing on aff(1), where λ,μ;v≔Homdiff(λ⊗μ;v) is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential aff(1)-relative cohomology of Vect(1) with coefficients in λ,μ;v

Keywords

Differential operators; Transvectants; Lie algebra; Cohomology

Introduction

Let Equation be a Lie algebra and let Equation and Equation be two Equation-modules. It is wellknown that nontrivial extensions of g-modules:

Equation

are classified by the first cohomology group Equation [1]. Any 1-cocycle Equation generates a new action on Equation as follows: for all Equation and for all (a,b)∈ Equation, we define Equation. For the space of tensor density of weight λ, Equation, viewed as a module over the Lie algebra of smooth vector fields Vect(Equation), the classification of nontrivial extensions

Equation

leads Feigin and Fuks [2] to compute the cohomology group Equation. Later, Ovsienko and Bouarroudj [3] have computed the corresponding relative cohomology group with respect to Equation, namely

Equation

In this paper, we will compute the first cohomology group

Equation

Vect (Equation)-Module Structures on the Space of Bilinear Differential Operators

Consider the standard (local) action of aff(1) on Equation by linearfractional transformations. Although the action is local, it generates global vector fields

Equation

that form a Lie subalgebra of Vect(Equation) isomorphic to the Lie algebra aff(1). This realization of aff(1) is understood throughout this paper.

The space of tensor densities on Equation

The space of tensor densities of weight λ (or λ-densities) on Equation, denoted by:

Equation

is the space of sections of the line bundle Equation. This space coincides with the space of functions and differential forms for λ=0 and for λ=1, respectively. The Lie algebra Vect(Equation) acts on Equation by the Lie derivative. For all Equation and for all Equation :

Equation (1)

where the superscript ′ stands for d/dx.

The space of bilinear differential operators as a Vect(Equation)-module

We are interested in defining a three-parameter family of Vect(Equation)1)-modules on the space of bilinear differential operators. The counterpart Vect(Equation)-modules of the space of linear differential operators is a classical object [4].

Consider bilinear differential operators that act on tensor densities:

Equation (2)

The Generalized Lie algebra Vect(Equation) acts on the space of bilinear differential operators as follows. For all Equation and for all Equation:

Equation (3)

where Equation is the action (1). We denote by Equation the space of bilinear differential operators (2) endowed with the defined Vect(Equation)-module structure (3).

Relative Cohomology

Let us first recall some fundamental concepts from cohomology theory [1]. Let g be a Lie algebra acting on a vector space V and let h be a sub- algebra of g. (If h is omitted it assumed to be {0}.) The space of h-relative n-cochains of g with values in V is the g-module

Equation

The coboundary operator Equation is a g-map satisfying Equation The kernel of δn, denoted Equation, is the space of h-relative n- cocycles, among them, the elements in the range of δn−1 are called h-relative n- coboundaries. We denote Equation the space of n-coboundaries.

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

Equation

We will only need the formula of δn (which will be simply denoted δ) in degrees 0,1 and 2: for Equation, where

Vh={vV|h.v=0 for all hh},

and for ϒ∈C1(g,h;V),

δ(ϒ)(x,y)?x⋅ϒ(y)−y⋅ϒ(x)−ϒ([x,y]) for any x,yg.

aff(1)-Invariant Differential Operators

The following steps to compute the relative cohomology has intensively been used in refs. [3,5-8]. First, we classify aff(1)-invariant differential operators, then we isolate among them those that are 1-cocycles. To do that, we need the following Lemma.

Lemma 4.1

Any 1−cocycle vanishing on the subalgebra aff(1) of Vect(Equation) is aff(1)-invariant.

The 1-cocycle condition of ϒ reads:

X⋅ϒ(Y)−Y⋅ϒ(X)−ϒ([X,Y])=0, (4)

where X,Y∈ Vect(Equation). Thus, if ϒ(X)=0 for all Xaff(1), eqn. (4) becomes

ϒ([X,Y])=X⋅ϒ(Y)

expressing the aff(1)-invariance property of ϒ.

As our 1-cocycles vanish on aff(1), we will investigate aff(1)- invariant linear differential operators that vanish on aff(1).

Proposition 4.2: There exist aff(1)-invariant bilinear differential operators Equation given by:

Equation (5)

where Equation and the coefficients γi,j are constants.

Proof. Any differential operator Equation is of the form

Equation

The osp(1|2) -invariant property of the operators Equation with respect to the vector field Equation yields:

Equation

So, we see that the corresponding operator can be expressed as (5).

Proposition 4.3: There exist aff(1)-invariant trilinear differential operators Equation given by:

Equation (6)

where i+j+l=k and the coefficients γi,j,l are constants

If τ, λ and μ are generic, then the space of solutions is Equation-dimensional.

Proposition 4.4: There exist aff(1)-invariant trilinear differential operators Equation that vanishe on aff(1) given by:

Equation (7)

where i+j+l=k and the coefficients γi,j,l are constants but γ0,j,kj=γ1,j,kj−1=0. Moreover, the space of solutions is Equation- dimensional, for all λ and μ.

Proof of Proposition 4.3 and 4.4: We are going to prove Proposition 4.3 and 4.4 simultaneously. Any differential operator Equation is of the form

Equation (8)

where γi,j,l are functions. The aff(1) -invariant property of the operators Equation reads as follows.

Equation (9)

The invariant property with respect to the vector field X = d/dx implies that Equation. On the other hand, the invariant property with respect to the vector fields Equation implies that v=τ+λ+μ+k. If τ, λ and μ are generic, then the space of solutions is Equation -dimensional, spanned by

Equation (10)

Now, the proof of Proposition 4.4 follows as above by putting τ−1. In this case, the space of solutions is Equation -dimensional, spanned by

Equation (11)

Cohomology of Vect(Equation) acting onEquation

In this section, we will compute the first cohomology group of Vect (Equation) with values in Equation, vanishing on aff(1). Our main result is the following:

Theorem 5.1

(i) For vμλ≤11, the space Equation has the following structure:

(1) If vμλ=1, then

Equation (12)

(2) If vμλ=2, then

Equation (13)

(3) If vμλ=3, then

Equation (14)

(4) If vμλ=4, then

Equation (15)

(5) If vμλ=5, then

Equation (16)

(6) If vμλ=6, then

Equation (17)

(7) If vμλ=7, then

Equation (18)

(8) If vμλ=8, then

Equation (19)

(9) If vμλ=9, then

Equation (20)

(10) If vμλ=10, then

Equation (21)

(11) If vμλ=11, then

Equation (22)

(ii) If vμ−λ is semi-integer but λ and μ are generic then,

Equation

Proof of Theorem 5.1: To proof Theorem (5.1) we proceed bye following the three steps:

• We will investigate the dimension of the space of operators that satisfy the 1-cocycle condition. By Proposition (4.4), its dimension is at most Equation, where k=vμλ+1, since any 1-cocycle that vanishes on aff(1) is certainly aff(1)-invariant.

• We will study all trivial 1-cocycles, namely, operators of the form LXB,

where B is a bilinear operator. As our 1-cocycles vanish on the Lie algebra aff(1), it follows that the operator B coincides with the transvectant Equation.

• By taking into account Part 1 and Part 2 and depending on λ and μ the dimension of the cohomology group Equation will be equal to

Equation

Now, clearly the coboundary Equation has the following form:

Equation (23)

where

Equation

The following Lemma is proved directly which will be useful in the proof of Theorem 5.1.

Lemma 5.2

Equation

Equation

where α≥2 and β≥0.

We need also the following Lemma.

Lemma 5.3

Every 1-cocycle on Vect(Equation) with values in Equation) is differentiable Proof [7].

Now we are in position to prove Theorem (5.1). By Lemma (5.3), any 1-cocycle on Vect(Equation) should retains the following general form:

Equation (24)

where ci,j,l are constants. The fact that this 1-cocycle vanishes on aff(1) implies that

c0,j,l=c1,j,l=0.

The 1-cocycle condition reads as follows: for all Equation and for all Equation, one has

Equation

The case where vμλ=1: In this case, according to Proposition 4.4, the 1-cocycle (24) can be expressed as follows:

Equation

By a direct computation, we can see that the 1-cocycle condition is always satisfied. Let us study the triviality of this 1-cocycle. A direct computation proves that

Equation

So, for(λ,μ)=(0,0), the coeffcient c2,0,0 cannot be eliminated by adding a coboundary. Hence, the cohomology space is one-dimensional. While for (λ,μ)≠(0,0), we can see that the coeffcient c2,0,0 can be eliminated because β2,0,0≠0. Hence, the cohomology is zero-dimensional.

The case where vμλ=2: In this case, according to Proposition 4.4, the 1-cocycle (24) can be expressed as follows:

Equation

By a direct computation, we can see that the 1-cocycle condition is always satisfied. Let us study the triviality of this 1-cocycle. A direct computation proves that

Equation

So, for Equation, the cohomology space is one-dimensional, since only one of the coefficients c3,0,0, c2,1,0 or c2,0,1 cannot be eliminated by adding a coboundary. While for Equation, the coeffcient c3,0,0, c2,1,0 and c2,0,1 can be eliminated because β3,0,0, β2,1,0 and β2,0,1 are nonzero. Hence, the cohomology space is zero-dimensional.

The case where vμλ≥3: this case, the 1-cocycle condition is equivalent to the system:

Equation (25)

where α+β+γ+a=k+1, α>β≥2, α>γ and α>a, obtained from the coefficient of X(α)Y(β)(γ)(a).

This system can be deduced by a simple computation. Of course, such a system has at least one solution in which the solutions ci,j,l are just the coefficients βi,j,l of the coboundaries (23).

The case where vμλ=3:In this case, according to Proposition 4.4, the space of solutions is spanned by:

Equation

Moreover, by formula (25), we readily obtain:

Equation

Thus, we have just proved that the coefficients of every 1-cocycle is expressed in terms of

Equation

A direct computation proves that

Equation

where

Equation

So, for Equation, the cohomology space is one-dimensional, since only one of the coefficients Equation cannot be eliminated by adding a coboundary. While for Equation, the coeffcient Equation can be eliminated because EquationEquation are nonzero. Hence, the cohomology space is zero dimensional.

The case where vμλ=4: In this case, according to Proposition 4.4, the space of solutions is spanned by:

Equation

Moreover, by formula (25), we readily obtain:

Equation

Thus, we have just proved that the coefficients of every 1-cocycle is expressed in terms of

Equation

direct computation confirms that, the coefficients of Equation are expressed in terms of:

Equation

So, for Equation the cohomology space is one-dimensional, since only one of the coefficients Equation cannot be eliminated by adding a coboundary. While for Equation the coeffcientEquation can be eliminated because Equation are nonzero. Hence, the cohomology space is zero-dimensional.

The case where vμλ=5: In this case, according to Proposition 4.4, the space of solutions is spanned by:

Equation

Moreover, by formula (25), we readily obtain:

Equation

Thus, we have just proved that the coefficients of every 1-cocycle is expressed in terms of

Equation

A direct computation confirms that, the coefficients of Equation are expressed in terms of:

Equation

So, forEquation the cohomology space is one-dimensional, since only one of the coefficientsEquation cannot be eliminated by adding a coboundary. While forEquation the coeffcientEquation can be eliminated because Equation are nonzero. Hence, the cohomology space is zero-dimensional.

The case where vμλ=6: In this case, according to Proposition 4.4 together with formulas (25), we check that the coefficients of every 1-cocycle are expressed in terms of

Equation

A direct computation confirms that, the coefficients of Equation are expressed in terms of:

Equation

So, in the same way as before, by Lemma 5.2, we can see, with the help of the maple, that the cohomology space is given as in (17).

The case where vμλ=7: In this case, according to Proposition 4.4 together with formulas (25), we check that the coefficients of every 1-cocycle are expressed in terms of

Equation

A direct computation confirms that, the coefficients of Equation are expressed in terms of:

Equation

So, in the same way as before, by Lemma 5.2, we can see, with the help of the maple, that the cohomology space is given as in (18).

The case where vμλ=8: In this case, according to Proposition 4.4 together with formulas (25), we check that the coefficients of every 1-cocycle are expressed in terms of:

Equation

A direct computation confirms that, the coefficients of Equation are expressed in terms of:

Equation

So, in the same way as before, by Lemma 5.2, we can see, with the help of the maple, that the cohomology space is given as in (19).

The case where vμλ=9: In this case, according to Proposition 4.4 together with formulas (25), we check that the coefficients of every 1-cocycle are expressed in terms of

Equation

A direct computation confirms that, the coefficients of Equation are expressed in terms of:

Equation

So, in the same way as before, by Lemma 5.2, we can see, with the help of the maple, that the cohomology space is given as in (20).

The case where vμλ=9: In this case, according to Proposition 4.4 together with formulas (25), we check that the coefficients of every 1-cocycle are expressed in terms of

Equation

A direct computation confirms that, the coefficients o f Equation are expressed in terms of:

Equation

So, in the same way as before, by Lemma 5.2, we can see, with the help of the maple, that the cohomology space is given as in (21).

The case where vμλ=11: In this case, according to Proposition 4.4 together with formulas (25), we check that the coefficients of every 1-cocycle are expressed in terms of

Equation

A direct computation confirms that, the coefficients of Equation are expressed in terms of:

Equation

So, in the same way as before, by Lemma 5.2, we can see, with the help of the maple, that the cohomology space is given as in (22). This completes the proof.

Conjecture 5.1

For Equation λ and μ are generic, one hase

Equation

Conclusion

In this paper, we classify aff(1) -invariant linear differential operators fromEquation vanishing on aff(1), where Equation is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential aff(1)-relative cohomology of Vect(Equation) with coefficients inEquation.

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