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Completing Lie algebra actions to Lie group actions
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## Completing Lie algebra actions to Lie group actions

### Franz W. Kamber and Peter W. Michor

**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.

*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 Frderung der wissenschaftlichen Forschung, Projekt P 14195 MAT'

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