On infinite Lie groups

Alexandre A. Martins Rodrigues

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 3, page 245-274
  • ISSN: 0373-0956

Abstract

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Under some regularity conditions one proves that quotients and kernels of infinitesimal analytic Lie pseudo-groups by invariant fiberings are again infinitesimal Lie pseudo-groups. The regularity conditions are shown to be necessary and sufficient if one wishes both quotient and kernel to be infinitesimal Lie pseudo-groups. One defines and proves the existence of the quotient of an infinitesimal Lie pseudo-group by a normal sub-pseudo group. An equivalence relation for germs of infinitesimal Lie pseudo-groups is introduced and the notions of morphism kernel and quotient are defined for the equivalence classes. In the special case of transitive pseudo-groups or of pseudo-groups of finite type the regularity conditions are always satisfied.

How to cite

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Rodrigues, Alexandre A. Martins. "On infinite Lie groups." Annales de l'institut Fourier 31.3 (1981): 245-274. <http://eudml.org/doc/74506>.

@article{Rodrigues1981,
abstract = {Under some regularity conditions one proves that quotients and kernels of infinitesimal analytic Lie pseudo-groups by invariant fiberings are again infinitesimal Lie pseudo-groups. The regularity conditions are shown to be necessary and sufficient if one wishes both quotient and kernel to be infinitesimal Lie pseudo-groups. One defines and proves the existence of the quotient of an infinitesimal Lie pseudo-group by a normal sub-pseudo group. An equivalence relation for germs of infinitesimal Lie pseudo-groups is introduced and the notions of morphism kernel and quotient are defined for the equivalence classes. In the special case of transitive pseudo-groups or of pseudo-groups of finite type the regularity conditions are always satisfied.},
author = {Rodrigues, Alexandre A. Martins},
journal = {Annales de l'institut Fourier},
keywords = {analytic Lie pseudo-groups},
language = {eng},
number = {3},
pages = {245-274},
publisher = {Association des Annales de l'Institut Fourier},
title = {On infinite Lie groups},
url = {http://eudml.org/doc/74506},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Rodrigues, Alexandre A. Martins
TI - On infinite Lie groups
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 3
SP - 245
EP - 274
AB - Under some regularity conditions one proves that quotients and kernels of infinitesimal analytic Lie pseudo-groups by invariant fiberings are again infinitesimal Lie pseudo-groups. The regularity conditions are shown to be necessary and sufficient if one wishes both quotient and kernel to be infinitesimal Lie pseudo-groups. One defines and proves the existence of the quotient of an infinitesimal Lie pseudo-group by a normal sub-pseudo group. An equivalence relation for germs of infinitesimal Lie pseudo-groups is introduced and the notions of morphism kernel and quotient are defined for the equivalence classes. In the special case of transitive pseudo-groups or of pseudo-groups of finite type the regularity conditions are always satisfied.
LA - eng
KW - analytic Lie pseudo-groups
UR - http://eudml.org/doc/74506
ER -

References

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  1. [1] E. CARTAN, La structure des groupes infinis, Ţuvres complètes : II, vol. 2, Gauthier-Villars, Paris, 1953, pp. 1335-1384. 
  2. [2] H. GOLDSCHMIDT, Existence theorems for analytic linear partial differential equations, Ann. of Math., 86, (1967), 246-270. Zbl0154.35103MR36 #2933
  3. [3] H. GOLDSCHMIDT, Prolongations of linear partial differential equations : I-A conjecture of E. Cartan, Ann. Sci. Ec. Norm. Sup., (4) 1 (1968), 417-444. Zbl0167.09402MR38 #3888
  4. [4] M. KURANISHI, A.A.M. RODRIGUES, Quotients of pseudo-groups by invariant fiberings, Nagoya Math., J., 24 (1964), 109-128. Zbl0163.45301MR29 #5962
  5. [5] M. KURANISHI, On the local theory of continuous infinite pseudo-groups. I, II, Nagoya Math. J., 15 (1959), 225-260 ; 19 (1961), 55-91. Zbl0212.56501
  6. [6] M. KURANISHI, Lectures on involutive systems of partial differential equations, Publicações da Sociedade de Matemática de São Paulo, São Paulo, (1967). Zbl0163.12001
  7. [7] A. PETITJEAN, A.A.M. RODRIGUES, Correspondance entre algèbres de Lie abstraites et pseudo-groupes de Lie transitifs, Annals of Math., vol. 101 (1975), 268-279. Zbl0333.22010MR52 #6800
  8. [8] Ngô Van QUÊ, Du prolongement des espaces fibrés et des structures infinitésimales, Ann. Inst. Fourier, (Grenoble), 17 (1967), 157-223. Zbl0157.28506MR36 #4468
  9. [9] QUILLEN, Formal properties of over-determined system of linear partial differential equations (Ph. D. thesis, Harvard, 1964). 
  10. [10] A.A.M. RODRIGUES, G-structures et pseudo groupes de Lie, Notes d'un cours donné à l'Université de Grenoble (1968). 
  11. [11] A.A.M. RODRIGUES, Sur le noyau d'un pseudo-groupe de Lie infinitésimal involutif transitif par rapport à une fibration invariante, C.R. Acad. Sci., Paris, Sér. A, 269 (1969), 1154-1155. Zbl0194.52704MR41 #2722
  12. [12] A.A.M. RODRIGUES, Sur le quotient d'un pseudo-groupe de Lie infinitésimal transitif par une fibration invariante, C.R. Acad. Sci., Paris, Sér. A, 269 (1969), 1211-1213. Zbl0194.52801MR41 #2723
  13. [13] I.M. SINGER, S. STERNBERG, The infinite groups of Lie and Cartan, I, J. D'Analyse Math., 15 (1965). Zbl0277.58008MR36 #911

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