This article is cited in 9 papers

Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces

S. P. Baranovskii, I. V. Shirokov

Omsk State University

Abstract: 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 $\mathbb R^1$.

Keywords: Lie groups, homogeneous spaces, vector fields, Lie algebra cohomologies.

Received: 01.04.2002

DOI: 10.4213/tmf171

 English version: Theoretical and Mathematical Physics, 2003, 135:1, 510–519

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