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For a finite-dimensional Lie algebra $\mathfrak{g}$ of vector fields on a manifold $M$ we show that $M$ can be completed to a $G$-space in a universal way, which however is neither Hausdorff nor $T_1$ in general. Here $G$ is a connected Lie group with Lie-algebra $\mathfrak{g}$. For a transitive $\mathfrak{g}$-action the completion is of the form $G/H$ for a Lie subgroup $H$ which need not be closed. In general the completion can be constructed by completing each $\mathfrak{g}$-orbit.

Copyright 2004 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 10 (2004), pp. 1-10
• Publisher Identifier: S 1079-6762(04)00124-6
• 2000 Mathematics Subject Classification. Primary 22F05, 37C10, 54H15, 57R30, 57S05
• Key words and phrases. $\mathfrak{g}$-manifold, $G$-manifold, foliation
• Received by editors October 27, 2003
• Posted on February 18, 2004
• Communicated by Alexandre Kirillov
• Comments (When Available)

Franz W. Kamber

Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801

E-mail address: kamber@math.uiuc.edu

Peter W. Michor

Institut für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria, and Erwin Schrödinger Institut für Mathematische Physik, Boltzmanngasse 9, A-1090 Wien, Austria

E-mail address: michor@esi.ac.at

FWK and PWM were supported by `Fonds zur Fšrderung der wissenschaftlichen Forschung, Projekt P 14195 MAT'