Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces.

We introduce the notion of the <MATH>\goth{gl}(V)</MATH>-prolongation of Lie algebras of differential operators on homogeneous spaces. The <MATH>\goth{gl}(V)</MATH>-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 S1 (the Virasoro algebra) and <MATH>\mathbb{R}^1</MATH>. [ABSTRACT FROM AUTHOR]