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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.

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

Let Vect(Ã¯ÂÂ’Ã¯ÂÂ1) be the Lie algebra of smooth vector fields on Ã¯ÂÂ’Ã¯ÂÂ1. In this paper, we classify aff(1) -invariant linear differential operators from Vect(Ã¯ÂÂ’Ã¯ÂÂ1) to Ã¯ÂÂ„λ,μ;v vanishing on aff(1), where Ã¯ÂÂ„λ,μ;vÃ¢Â‰Â”Homdiff(Ã¯ÂÂ†λ⊗Ã¯ÂÂ†μ;Ã¯ÂÂ†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(Ã¯ÂÂ’Ã¯ÂÂ1) with coefficients in Ã¯ÂÂ„λ,μ;v

Differential operators; Transvectants; Lie algebra; Cohomology

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

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

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

In this paper, we will compute the first cohomology group

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

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

**The space of tensor densities on **

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

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

(1)

where the superscript ′ stands for *d*/*dx*.

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

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

Consider bilinear differential operators that act on tensor densities:

(2)

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

(3)

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

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

The coboundary operator is a g-map satisfying The kernel of δ_{n}, denoted , is the space of *h*-relative *n*- cocycles, among them, the elements in the range of *δ _{n}*

By definition, the *n ^{th}*

We will only need the formula of *δ _{n}* (which will be simply denoted

*V ^{h}*={

and for ϒ∈*C*^{1}(*g,h;V*),

*δ*(ϒ)(*x*,*y*)?*x*⋅ϒ(*y*)−*y*⋅ϒ(*x*)−ϒ([*x*,*y*]) for any *x*,*y*∈*g*.

**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() 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(). Thus, if ϒ(*X*)=0 for all *X*∈ *aff*(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 given by:

(5)

where and the coefficients *γ _{i,j}* are constants.

*Proof*. Any differential operator is of the form

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

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

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

(6)

where *i*+*j*+*l*=*k* and the coefficients *γ _{i}*

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

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

(7)

where *i*+*j*+*l*=*k* and the coefficients *γ _{i}*

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

(8)

where *γ _{i}*

(9)

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

(10)

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

(11)

**Cohomology of Vect() acting on**

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

**Theorem 5.1**

(i) For *v*−*μ*−*λ*≤11, the space has the following structure:

(1) If *v*−*μ*−*λ*=1, then

(12)

(2) If *v*−*μ*−*λ*=2, then

(13)

(3) If *v*−*μ*−*λ*=3, then

(14)

(4) If *v*−*μ*−*λ*=4, then

(15)

(5) If *v*−*μ*−*λ*=5, then

(16)

(6) If *v*−*μ*−*λ*=6, then

(17)

(7) If *v*−*μ*−*λ*=7, then

(18)

(8) If *v*−*μ*−*λ*=8, then

(19)

(9) If *v*−*μ*−*λ*=9, then

(20)

(10) If *v*−*μ*−*λ*=10, then

(21)

(11) If *v*−*μ*−*λ*=11, then

(22)

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

**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 , where *k*=*v*−*μ*−*λ*+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 *L _{X}B*,

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 .

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

Now, clearly the coboundary has the following form:

(23)

where

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

**Lemma 5.2**

where *α*≥2 and *β*≥0.

We need also the following Lemma.

**Lemma 5.3**

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

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

(24)

where *c _{i}*

*c*_{0,j,l}=*c*_{1,j,l}=0.

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

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

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

So, for(*λ*,*μ*)=(0,0), the coeffcient *c*_{2,0,0} 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 *c*_{2,0,0} can be eliminated because *β*_{2,0,0}≠0. Hence, the cohomology is zero-dimensional.

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

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

So, for , the cohomology space is one-dimensional, since only one of the coefficients *c*_{3,0,0}, *c*_{2,1,0} or *c*_{2,0,1} cannot be eliminated by adding a coboundary. While for , the coeffcient *c*_{3,0,0}, *c*_{2,1,0} and *c*_{2,0,1} can be eliminated because *β*_{3,0,0}, *β*_{2,1,0} and *β*_{2,0,1} are nonzero. Hence, the cohomology space is zero-dimensional.

**The case where v−μ−λ≥3:** this case, the 1-cocycle condition is equivalent to the system:

(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 *c _{i}*

**The case where v−μ−λ=3:**In this case, according to Proposition 4.4, the space of solutions is spanned by:

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

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

A direct computation proves that

where

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

**The case where v−μ−λ=4:** In this case, according to Proposition 4.4, the space of solutions is spanned by:

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

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

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

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

**The case where v−μ−λ=5:** In this case, according to Proposition 4.4, the space of solutions is spanned by:

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

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

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

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

**The case where v−μ−λ=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

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

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 v−μ−λ=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

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

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 v−μ−λ=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:

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

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 v−μ−λ=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

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

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 v−μ−λ=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

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

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 v−μ−λ=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

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

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 *λ* and *μ* are generic, one hase

In this paper, we classify aff(1) -invariant linear differential operators from vanishing on aff(1), where 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() with coefficients in.

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