Sur l'algèbre de Lie des sections d'un fibré en algèbres de Lie

Pierre Lecomte

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 4, page 35-50
  • ISSN: 0373-0956

Abstract

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One studies the natural Lie algebra structure of the space of sections of class C k of a locally trivial fibre bundle whose typical fibre is a Lie algebra L ; in particular, its derivations and its automorphisms are described. The algebras L for which this structure characterizes the differentiable structure of the base manifold of the bundle are determined.

How to cite

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Lecomte, Pierre. "Sur l'algèbre de Lie des sections d'un fibré en algèbres de Lie." Annales de l'institut Fourier 30.4 (1980): 35-50. <http://eudml.org/doc/74473>.

@article{Lecomte1980,
abstract = {On étudie la structure naturelle d’algèbre de Lie de l’espace des sections de classe $C_k$ d’un fibré localement trivial dont la fibre-type est une algèbre de Lie $L$; on décrit, en particulier, ses dérivations et ses automorphismes. On détermine les algèbres de Lie $L$ pour lesquelles cette structure caractérise la structure différentiable de la base du fibré.},
author = {Lecomte, Pierre},
journal = {Annales de l'institut Fourier},
keywords = {Lie algebra structure of the space of sections of a locally trivial fibre bundle whose typical fibre is a Lie algebra; differentiable structure},
language = {fre},
number = {4},
pages = {35-50},
publisher = {Association des Annales de l'Institut Fourier},
title = {Sur l'algèbre de Lie des sections d'un fibré en algèbres de Lie},
url = {http://eudml.org/doc/74473},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Lecomte, Pierre
TI - Sur l'algèbre de Lie des sections d'un fibré en algèbres de Lie
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 4
SP - 35
EP - 50
AB - On étudie la structure naturelle d’algèbre de Lie de l’espace des sections de classe $C_k$ d’un fibré localement trivial dont la fibre-type est une algèbre de Lie $L$; on décrit, en particulier, ses dérivations et ses automorphismes. On détermine les algèbres de Lie $L$ pour lesquelles cette structure caractérise la structure différentiable de la base du fibré.
LA - fre
KW - Lie algebra structure of the space of sections of a locally trivial fibre bundle whose typical fibre is a Lie algebra; differentiable structure
UR - http://eudml.org/doc/74473
ER -

References

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  1. [1] I. AMEMIYA, Lie algebra of vector fields and complex structure, J. of Math. Soc. Japan, vol. 27, n° 4 (oct. 1975), 545. Zbl0311.57012MR53 #4087
  2. [2] A. A. KIRILLOV, Local Lie algebras, Russian Math. Surveys, 31, 4 (1976), 55. Zbl0357.58003MR55 #11304a
  3. [3] S. KOBAYASKI, K. NOMIZU, Foundations of Differential Geometry, Interscience Publishers, 15, vol. 1, New York, 1963. Zbl0119.37502
  4. [4] A. KORIYAMA, On Lie algebras of vector fields with invariant submanifolds, Nagoya Math. J., vol. 55 (1974), 91. Zbl0273.22016MR51 #6874
  5. [5] P. LECOMTE, Algèbres de Lie d'ordre zéro sur une variété, Thèse de doctorat, Liège, 1979. 
  6. [6] P. LECOMTE, On a class of local Lie algebras over a manifold, A paraître dans Letters in Mathematical Physics. Zbl0414.58001
  7. [7] P. LECOMTE, Derivations of linear endomorphisms of the tangent bundle, Bull. Soc. Roy. Sc. Liège, 47e année, 11-12 (1978), 329. Zbl0411.58002MR80i:58008
  8. [8] P. LECOMTE, On some ideals of a Lie algebra of order zero, A paraître dans Bull. Soc. Roy. Sc. Liège. Zbl0442.17006
  9. [9] R. NARASIMHAN, Analysis on Real and Complex Manifolds, Masson et Cie, Paris, 1973. 
  10. [10] H. OMORI, Infinite dimensional Lie transformation group, Lecture Notes in Mathematics, 427, Springer-Verlag, 1976. Zbl0328.58005
  11. [11] L. E. PURSELL, M. E. SHANKS, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc., vol. 5 (1954), 468. Zbl0055.42105MR16,331a

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