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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 $\mathfrak{g}$ 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.


References [Enhancements On Off] (What's this?)

  • 1. D. V. Alekseevsky and Peter W. Michor, Differential geometry of $\mathfrak g$-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

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