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

Abstract

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