Invariant bilinear operators and the second 𝔰𝔩(2)-relative cohomology of the Lie algebra of vector fields on ℝ

Abstract

Let [Formula: see text] be the Lie algebra of smooth vector fields on [Formula: see text] and [Formula: see text] be the space of bilinear differential operators acting on weighted densities. In this paper, we classify [Formula: see text]-invariant skew-symmetric binary differential operators from [Formula: see text] to [Formula: see text] vanishing on [Formula: see text]. This result allows us to compute the second differential [Formula: see text]-relative cohomology of [Formula: see text] with coefficients in [Formula: see text].