On the First aff(1)-Relative Cohomology of the Lie Algebra of Vector Fields and Differential Operators

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.


Vect()-Module Structures on the Space of Bilinear Differential Operators
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  1
The space of tensor densities of weight λ (or λ-densities) on  1 , 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( 1 ) acts on  λ by the Lie derivative. For all X∈ Vect( 1 ) and for all φ∈  λ : where the superscript ′ stands for d/dx.

The space of bilinear differential operators as a Vect( 1 )module
We are interested in defining a three-parameter family of Vect( 1 )-modules on the space of bilinear differential operators. The counterpart Vect( 1 )-modules of the space of linear differential operators is a classical object [4].
Consider bilinear differential operators that act on tensor densities: The Generalized Lie algebra Vect( 1 ) acts on the space of bilinear differential operators as follows. For all φ∈ λ and for all ψ∈  µ : where X L λ is the action (1). We denote by  λ,µ;v the space of bilinear differential operators (2) endowed with the defined Vect( 1 )-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 The coboundary operator δ n :C n (g,h;V)→C n+1 (g,h;V) is a g-map satisfying n δ n1 =0. The kernel of δ n , denoted Z n (g,h;V), is the space of h-relative n-cocycles, among them, the elements in the range of δ n−1 are called h-relative n-coboundaries. We denote B n (g,h;V) the space of n-coboundaries.
By definition, the n th h-relative cohomolgy space is the quotient space We will only need the formula of δ n (which will be simply denoted δ) in degrees 0,1 and 2: for v∈C 0 for any x,y∈g.
Proof. Any differential operator , : where i+j+l=k and the coefficients γ i,j,l are constants.

Proof of Proposition 4.3 and 4.4:
We are going to prove Proposition 4.3 and 4.4 simultaneously. Any differential operator , , : where γ i,j,l are functions. The aff(1) -invariant property of the operators , , k K τ λ µ reads as follows.
, , , The invariant property with respect to the vector field . On the other hand, the invariant property with respect to the vector fields = d X x dx implies that v=τ+λ+µ+k. If τ, λ and µ are generic, then the space of solutions is Now, the proof of Proposition 4.4 follows as above by putting τ−1. In this case, the space of solutions is

Cohomology of Vect( 1 ) acting on  λ,µ;v
In this section, we will compute the first cohomology group of Vect( 1 ) with values in  λ,µ;v , vanishing on aff(1). Our main result is the following:  aff  has the following structure: (2) If v−µ−λ=2, then . .
(11) If v−µ−λ=11, then (ii) If v−µ−λ is semi-integer but λ and µ are generic then, • We will study all trivial 1-cocycles, namely, operators of the form 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 , k J λ µ .
The following Lemma is proved directly which will be useful in the proof of Theorem 5.1.
We need also the following Lemma.
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 The case where v−µ−λ≥3: In this case, the 1-cocycle condition is equivalent to the system: 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,j,l are just the coefficients β i,j,l of the coboundaries (23).
A direct computation proves that