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ISSN 1079-6762



Completing Lie algebra actions to Lie group actions

Authors: Franz W. Kamber and Peter W. Michor
Translated by:
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 1-10
MSC (2000): Primary 22F05, 37C10, 54H15, 57R30, 57S05
Published electronically: February 18, 2004
MathSciNet review: 2048426
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Abstract: 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.

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Additional Information

Franz W. Kamber
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801

Peter W. Michor
Affiliation: Institut für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria, and Erwin Schrödinger Institut für Mathematische Physik, Boltzmanngasse 9, A-1090 Wien, Austria

Keywords: $\mathfrak{g}$-manifold, $G$-manifold, foliation
Received by editor(s): October 27, 2003
Published electronically: February 18, 2004
Additional Notes: FWK and PWM were supported by ‘Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MAT’
Communicated by: Alexandre Kirillov
Article copyright: © Copyright 2004 American Mathematical Society