On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators

Let Vect(ℝ) be the Lie algebra of smooth vector fields on ℝ. The space of symbols Pol(T*ℝ) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(ℝ)-module that becomes trivial once the action is restricted to sl(2) ⊂ Vect(ℝ). The deformations of Pol(T*ℝ), which become trivial once the action is restricted to sl(2) and such that the Vect(ℝ)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of H²_diff(Vect(ℝ),sl[(2);D _λ,μ), where Hⁱ_diff denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D_λμ = Hom _diff(F_λ, F_μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning H²(g; D_λ,μ) for g = Vect(ℝ) and sl(2).