Cohomology of the vector fields Lie algebras on RP^1 acting on bilinear differential operators

Abstract

The main topic of this paper is two folds. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(RP^1), with coefficients in the space of bilinear differential operators that act on tensor densities, D_{\lambda, \nu;\mu}, vanishing on the Lie algebra sl(2,R). Second, we compute the first cohomology group of the Lie algebra sl(2,R) with coefficients in D_{\lambda, \nu;\mu}.