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

DOI:
https://doi.org/10.1090/S1079-6762-04-00124-6

Published electronically:
February 18, 2004

MathSciNet review:
2048426

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Abstract | References | Similar Articles | Additional Information

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

**1.**D. V. Alekseevsky and Peter W. Michor,*Differential geometry of -manifolds.*, Differ. Geom. Appl.**5**(1995), 371-403, arXiv:math.DG/9309214. MR**96k:53035****2.**G. Hector,*Private communication*, 2002.**3.**Franz W. Kamber and Peter W. Michor,*The flow completion of a manifold with vector field*, Electron. Res. Announc. Amer. Math. Soc.**6**(2000), 95-97, math.DG/0007173. MR**2001k:37031****4.**Boris Khesin and Peter W. Michor,*The flow completion of Burgers' equation*, pp. 1-8, Walter de Gruyter, Berlin, 2004, IRMA Lectures in Mathematics and Theoretical Physics. http://www.mat.univie.ac.at/~michor/burgers.ps**5.**Ivan Kolár, Jan Slovák, and Peter W. Michor,*Natural operators in differential geometry*, Springer-Verlag, Heidelberg, Berlin, New York, 1993. MR**94a:58004****6.**Andreas Kriegl and Peter W. Michor,*The Convenient Setting for Global Analysis*, Math. Surveys and Monographs, vol. 53, AMS, Providence, 1997, www.ams.org/online_bks/surv53/. MR**98i:58015****7.**Richard S. Palais,*A global formulation of the Lie theory of transformation groups*, Mem. AMS**22**(1957). MR**22:12162**

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Additional Information

**Franz W. Kamber**

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

Email:
kamber@math.uiuc.edu

**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

Email:
michor@esi.ac.at

DOI:
https://doi.org/10.1090/S1079-6762-04-00124-6

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