Completing Lie algebra actions to Lie group actions
Authors:
Franz W. Kamber and Peter W. Michor
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
Full-text PDF Free Access
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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.
- D. V. Alekseevsky and Peter W. Michor, Differential geometry of $\mathfrak g$-manifolds, Differential Geom. Appl. 5 (1995), no. 4, 371–403. MR 1362865, DOI https://doi.org/10.1016/0926-2245%2895%2900023-2
2 G. Hector, Private communication, 2002.
- 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. MR 1783093, DOI https://doi.org/10.1090/S1079-6762-00-00083-4
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
- Ivan Kolář, Peter W. Michor, and Jan Slovák, Natural operations in differential geometry, Springer-Verlag, Berlin, 1993. MR 1202431
- Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, Providence, RI, 1997. MR 1471480
- Richard S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (1957), iii+123. MR 121424
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.
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.
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 Kolář, Jan Slovák, and Peter W. Michor, Natural operators in differential geometry, Springer-Verlag, Heidelberg, Berlin, New York, 1993.
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/.
7 Richard S. Palais, A global formulation of the Lie theory of transformation groups, Mem. AMS 22 (1957).
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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
MR Author ID:
124340
Email:
michor@esi.ac.at
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