Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces

We introduce the notion of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathfrak{g}}{\mathfrak{l}(V)}$$ \end{document}-prolongation of Lie algebras of differential operators on homogeneous spaces. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathfrak{g}}{\mathfrak{l}(V)}$$ \end{document}-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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{R}}^1 $$ \end{document}.