Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces

We introduce the notion of the $\mathfrak{gl}(V)$-prolongation of Lie algebras of differential operators on homogeneous spaces. The $\mathfrak{gl}(V)$-prolongations are topological invariants that coincide with one-dimensional cohomologies of the corresponding Lie algebras in the case where $V$ is a homogeneous space. We apply the obtained results to the spaces $S^1$ (the Virasoro algebra) and $\mathbb R^1$.