×

Perspectives sur la théorie des pseudogroupes de transformations analytiques de type infini. (French) Zbl 0885.22023

The authors study the local structure of Lie pseudogroups of infinite type by using the Lie groups of infinite dimension. They consider the formal integration and some formal pseudogroups and obtain a fundamental theorem of the third Lie theorem type in the theory of Lie groups, for a class of transitive Lie pseudogroups of infinite type.
Reviewer: V.Oproiu (Iaşi)

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
17B65 Infinite-dimensional Lie (super)algebras
Full Text: DOI

References:

[1] Avez, A.; Lichnerowicz, A.; Diaz-Miranda, A., Sur l’algèbre de Lie des automorphismes infinitésimaux d’une variéte symplectique, J. Diff. Geom., 9, 1-40 (1974) · Zbl 0283.53033
[2] Arnold, V., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Grenoble, 16, 1, 319-361 (1966) · Zbl 0148.45301
[3] Banyaga, A.; Donato, P., Some remarks on the integration of the Poisson algebra (1993), C.P.T. Luniny Marseille, prépublication
[4] Cartan, E., (Euvres Complètes d’E. Cartan, Partie II, Vol. 2 (1953), Gauthier-Villars: Gauthier-Villars Paris), 571-624 · JFM 35.0176.04
[5] Cartan, E., (Euvres Complètes d’E. Cartan, Partie II, Vol. 2 (1953), Gauthier-Villars: Gauthier-Villars Paris), 625-714 · JFM 36.0223.03
[6] Cartan, E., (Euvres Complètes d’E. Cartan, Partie II, Vol. 2 (1953), Gauthier-Villars: Gauthier-Villars Paris), 719-856 · JFM 39.0206.04
[7] Cartan, E., (Euvres Complètes d’E. Cartan, Partie II, Vol. 2 (1953), Gauthier-Villars: Gauthier-Villars Paris), 857-925 · JFM 40.0193.02
[8] Cartan, E., (Euvres Complètes d’E. Cartan, Partie II, Vol. 2 (1953), Gauthier-Villars: Gauthier-Villars Paris), 1335-1384 · JFM 33.0161.04
[9] Chern, S. S., Pseudo-groupes continus infinis, (Colloque de géométrie différentielle. Colloque de géométrie différentielle, Strasbourg (1954), Editions du CNRS: Editions du CNRS Paris)
[10] Colombeau, J. F., Differential Calculus and Holomorphy, (Mathematical Studies, Vol. 64 (1982), North-Holland: North-Holland Amsterdam) · Zbl 0441.46037
[11] Coste, A.; Dazord, P.; Weinstein, A., Groupoïdes symplectiques, (Publications du Département de Mathématiques de l’Univ. de LYON, 1 (2/A-1987)) · Zbl 0668.58017
[12] Dazord, P., Lie groups and algebras in infinite dimension: A new approach, (XXXIII Taniguchi Symp. on Symplectic Geometry and Its applications (1993)) · Zbl 0834.22018
[13] Dazord, P., Sur l’intégration des algèbres de Lie locales et la préquantification, Prepub. Inst. Girard Desargues U.R.A. CNRS, 746 (5/1995)
[14] Douady, A.; Lazard, M., Espaces fibrés en algèbres de Lie et en groupes, Invent Math., 1, 133-151 (1966) · Zbl 0144.01804
[15] Ebin, D. G.; Marsden, J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92, 1 (1970) · Zbl 0211.57401
[16] Ecalle, J., Théorie itérative: Introduction à la théorie des invariants holomorphes, J. Math. Pure Appl., 54, 183-258 (1975) · Zbl 0285.26010
[17] Freifeld, C., One-parameter subgroups do not fill a neighborhood of the identity in an inifinite-dimensional Lie (pseudo-) Group, Battelle Rencontres,, (Lectures in Mathematics and Physics (1967), Benjamin: Benjamin New York), 538-543 · Zbl 0174.05701
[18] Frölicher, A.; Kriegl, A., Linear Spaces and Differentiation Theory, (Pure and Applied Mathematics (1988), Wiley: Wiley Chichester) · Zbl 0658.46002
[19] Goldschmidt, H., Sur la structure des équations de Lie: I. Le troisième théorème fondamental, J. Diff. Geom., 6, 67-95 (1972) · Zbl 0273.58015
[20] Grabowski, J., Free subgroups of diffeomorphism groups, Fund. Math., 131, 103-121 (1988) · Zbl 0666.58011
[21] Grabowski, J., Derivative of the exponential mapping for infinite dimensional Lie groups (1993), preprint · Zbl 0836.22028
[22] Guillemin, V.; Sternberg, S., An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc., 70, 1, 16-47 (1964) · Zbl 0121.38801
[23] Hamilton, R., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7, 65-222 (1982) · Zbl 0499.58003
[24] Kamran, N.; Robart, T., Abstract structure for Lie pseudogroups of infinite type, C. R. Acad. Sci. Paris, Série I, 324, 1395-1399 (1997), Ce résultat a été annoncé dans · Zbl 0894.58075
[25] Kuranishi, M., On the local theory of continuous infinite pseudo groups I & II, Nagoya Math. J., 19, 55-91 (1961) · Zbl 0212.56501
[26] Leslie, J., On a differential structure for the group of diffeomorphisms, Topology, 6, 264-271 (1967) · Zbl 0147.23601
[27] Leslie, J., On the group of real analytic diffeomorphisms of a compact real analytic manifold, Trans. Amer. Math. Soc., 274, 2 (1982) · Zbl 0513.58017
[28] Leslie, J., Some integrable subalgebras of the Lie algebras of infinite-dimensional Lie groups, Trans. Amer. Math. Soc., 333, 423-443 (1992) · Zbl 0781.22015
[29] Leslie, J., On the integrability of some infinite dimensional Lie Algebras (1993), Howard University: Howard University Washington, DC, preprint
[30] Libermann, P., Pseudogroupes infinitésimaux attachés aux pseudogroupes de Lie, Bull. Soc. Math. France, 87, 409-425 (1959) · Zbl 0198.26801
[31] Lichnerowicz, A., Algèbre de Lie des automorphismes infinitesimaux d’une structure de contact, J. Math. Pure Appl., 52, 473-508 (1973) · Zbl 0277.53021
[32] Lichnerowicz, A., Algèbre de Lie des automorphismes infinitésimaux d’une structure unimodulaire, Ann. Inst. Fourier (Grenoble), 24, 3, 219-266 (1974) · Zbl 0289.58002
[33] Lichnerowicz, A., Les variétes de Poisson et leurs algèbres de Lie associees, J. Diff. Geom., 12, 253-300 (1977) · Zbl 0405.53024
[34] Lie, S., (Gesammelte Abhandlungen, Vol. 6 (1927), Teubner: Teubner Leipzig), 300-364 · JFM 23.0376.01
[35] Lie, S., (Gesammelte Abhandlungen, Vol. 6 (1927), Teubner: Teubner Leipzig), 396-493, also · JFM 26.0401.01
[36] Milnor, J., Remarks on infinite dimensional Lie groups, (Proc. Summer School on Quantum Gravity. Proc. Summer School on Quantum Gravity, les Houches (1983), North-Holland: North-Holland Amsterdam), Session XL · Zbl 0594.22009
[37] Natarajan, L.; Rodriguez-Carrington, E.; Wolf, J. A., (Proc. Symp. Pure Math., 56 (1994)), 377, 2
[38] Omori, H., Infinite Dimensional Lie Transformation Groups, (Lecture Notes in Mathematics, Vol. 427 (1974), Springer: Springer Berlin) · Zbl 0328.58005
[39] Omori, H.; de la Harpe, P., About interaction between Banach Lie groups and finite-dimensional manifolds, J. Math. Kyoto University, 12-13 (1972)
[40] Omori, H.; Maeda, Y.; Yoshioka, A.; Kobayashi, O., On regular Fréchet-Lie groups, Tokyo J. Math., 5, 2, 365-397 (1981)
[41] Palis, J., Vector fields generate few diffeomorphisms, Bull. Amer. Math. Soc., 80, 3, 503-505 (1974) · Zbl 0296.57008
[42] Robart, T., Groupes de Lie de dimension infinie. Second et troisième théorèmes de Lie. I - Groupes de première espèce, C.R. Acad. Sci. Paris, Série I, 322, 1071-1074 (1996) · Zbl 0870.57050
[43] Robart, T.; Kamran, N., Sur la théorie locale des pseudogroupes de transformations continus infinis, Math. Ann. (à paraître) (1997) · Zbl 0874.22019
[44] Shnider, S., The classification of real primitive infinite Lie algebras, J. Diff. Geom., 4, 1, 81-89 (1970) · Zbl 0244.17014
[45] Singer, I. M.; Sternberg, S., The infinite groups of Lie and Cartan. I. The transitive groups, J. d’Anal. Math., 15, 1-114 (1965) · Zbl 0277.58008
[46] Souriau, J-M., Groupes Différentiels, (Lecture Notes in Mathematics, Vol. 836 (1980), Springer: Springer Berlin), 91-128 · Zbl 0541.58002
[47] Souriau, J-M., Un algorithme générateur de structures quantiques, Soc. Math. France, Astérisque, hors série, 341-399 (1985) · Zbl 0608.58028
[48] Spencer, D. C., Deformation of structures on manifolds defined by transitive continuous pseudogroups, Ann. Math., 76, 2, 306-445 (1962) · Zbl 0124.38601
[49] Sternberg, S., On the structure of local homeomorphisms of Euclidean \(n\)-space, II, Amer. J. Math., 80, 623-631 (1958) · Zbl 0083.31406
[50] Tresse, A., Sur les invariants différentiels des groupes continus de transformations, Acta Math., 18, 1-88 (1894) · JFM 25.0641.01
[51] van Est, W. T., Rapport sur les S-Atlas, Astérique, 116, 235-292 (1984) · Zbl 0543.58003
[52] van Est, W. T.; Korthagen, T. J., Nonenlargeable Lie Algebras, (Proc. Konink. Nederl. Akad. Wetensch., Ser A-67 = Indag. Math., 26 (1964)), 15-31 · Zbl 0121.27503
[53] Weil, A., Œuvres Scientifiques, I (1926-1951) (1979), Springer-Verlag · Zbl 0424.01027
[54] Weinstein, A., Symplectic manifolds and their Lagrangian submanifolds, Adv. in Math., 6, 329-346 (1971) · Zbl 0213.48203
[55] Weinstein, A., Groupoids: Unifying internal and external symmetry, Notices of the Amer. Math. Soc., 43, 7, 744-752 (1996) · Zbl 1044.20507
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.