2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/96668The 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_{λ, ν;μ}, 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_{λ, ν;μ}.Latex2e, no figures, 16 pagesDifferential GeometryQuantum Algebratext