Central extensions of infinite-dimensional Lie groups
[Extensions centrales des groupes de Lie de dimension infinie]
Annales de l'Institut Fourier, Tome 52 (2002) no. 5, pp. 1365-1442.

Le principal résultat de cet article est une suite exacte pour le groupe abélien des extensions centrales d’un groupe de Lie connexe G de dimension infinie par un groupe abélien de Lie Z pour lequel la composante connexe est un quotient d’un espace vectoriel par un sous-groupe discret. Un point essentiel de ce résultat est qu’il n’est pas restreint aux groupes lissement paracompacts. Par conséquence, il s’applique à tous les groupes de Lie-Banach et de Lie-Fréchet. La suite exacte codifie en particulier les obstructions précises pour l’intégration d’un cocycle d’algèbre de Lie à un cocycle localement lisse des groupes de Lie.

The main result of the present paper is an exact sequence which describes the group of central extensions of a connected infinite-dimensional Lie group G by an abelian group Z whose identity component is a quotient of a vector space by a discrete subgroup. A major point of this result is that it is not restricted to smoothly paracompact groups and hence applies in particular to all Banach- and Fréchet-Lie groups. The exact sequence encodes in particular precise obstructions for a given Lie algebra cocycle to correspond to a locally group cocycle.

DOI : 10.5802/aif.1921
Classification : 22E65, 58B20, 58B05
Keywords: infinite-dimensional Lie group, invariant form, central extension, period map, Lie group cocycle, homotopy group, local cocycle, diffeomorphism group
Mot clés : groupe de Lie de dimension infinie, forme différentielle invariante, extension centrale, application de période, cocycle de groupe de Lie, groupe d'homotopie, cocycle local, groupes de difféomorphisme
Neeb, Karl-Hermann 1

1 Technische Universität Darmstadt, Fachbereich Mathematik AG5, Schlossgartenstrasse 7, 64289 Darmstadt (Allemagne)
@article{AIF_2002__52_5_1365_0,
     author = {Neeb, Karl-Hermann},
     title = {Central extensions of infinite-dimensional {Lie} groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1365--1442},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {5},
     year = {2002},
     doi = {10.5802/aif.1921},
     mrnumber = {1935553},
     zbl = {1019.22012},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1921/}
}
TY  - JOUR
AU  - Neeb, Karl-Hermann
TI  - Central extensions of infinite-dimensional Lie groups
JO  - Annales de l'Institut Fourier
PY  - 2002
SP  - 1365
EP  - 1442
VL  - 52
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1921/
DO  - 10.5802/aif.1921
LA  - en
ID  - AIF_2002__52_5_1365_0
ER  - 
%0 Journal Article
%A Neeb, Karl-Hermann
%T Central extensions of infinite-dimensional Lie groups
%J Annales de l'Institut Fourier
%D 2002
%P 1365-1442
%V 52
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1921/
%R 10.5802/aif.1921
%G en
%F AIF_2002__52_5_1365_0
Neeb, Karl-Hermann. Central extensions of infinite-dimensional Lie groups. Annales de l'Institut Fourier, Tome 52 (2002) no. 5, pp. 1365-1442. doi : 10.5802/aif.1921. http://www.numdam.org/articles/10.5802/aif.1921/

[Br93] G. E. Bredon Topology and Geometry, Graduate Texts in Mathematics, 139, Springer-Verlag, Berlin, 1993 | MR | Zbl

[Br97] G. E. Bredon Sheaf Theory, Graduate Texts in Mathematics, 170, Springer-Verlag, Berlin, 1997 | MR | Zbl

[Bry93] J.-L. Brylinski Loop Spaces, Characteristic Classes and Geometric Quantization, Progr. in Math., 107, Birkhäuser Verlag, 1993 | MR | Zbl

[Ca51] E. Calabi Sur les extensions des groupes topologiques, Brioschi Annali di Mat. Pura et Appl., Ser 4, Volume 32 (1951), pp. 295-370 | DOI | MR | Zbl

[Ca52a] E. Cartan Le troisième théorème fondamental de Lie (Oeuvres I), Volume 2 (1952), pp. 1143-1148

[Ca52b] E. Cartan La topologie des espaces représentifs de groupes de Lie (Oeuvres I), Volume 2 (1952), pp. 1307-1330

[Ca52c] E. Cartan Les représentations linéaires des groupes de Lie (Oeuvres I), Volume 2 (1952), pp. 1339-1350

[Ch46] C. Chevalley Theory of Lie Groups I, Princeton Univ. Press, 1946 | MR | Zbl

[DL66] A. Douady; M. Lazard Espaces fibrés en algèbres de Lie et en groupes, Invent. Math, Volume 1 (1966), pp. 133-151 | DOI | MR | Zbl

[dlH72] P. de la Harpe Classical Banach Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space, Lecture Notes in Math., 285, Springer-Verlag, Berlin, 1972 | MR | Zbl

[EK64] W. T. van Est; Th. J. Korthagen Non enlargible Lie algebras, Proc. Kon. Ned. Acad. v. Wet. A, Volume 67 (1964), pp. 15-31 | MR | Zbl

[EL88] W. T. van Est; M. A. M. van der Lee Enlargeability of local groups according to Malcev and Cartan-Smith, Hermann, Paris, 1988 | MR | Zbl

[EML43] S. Eilenberg; S. MacLane Relations between homology and homotopy theory, Proc. Nat. Acad. Sci. U.S.A, Volume 29 (1943), pp. 155-158 | DOI | MR | Zbl

[EML47] S. Eilenberg; S. MacLane Cohomology theory in abstract groups. II, Annals of Math, Volume 48 (1947) no. 2, pp. 326-341 | DOI | MR | Zbl

[Est54] W. T. van Est A group theoretic interpretation of area in the elementary geometries, Simon Stevin, Wis. en Natuurkundig Tijdschrift, Volume 32 (1954), pp. 29-38 | MR | Zbl

[Est62] W. T. van Est Local and global groups, Indag. Math. (Proc. Kon. Ned. Akad. v. Wet. Series A), Volume 24 (1962), pp. 391-425 | Zbl

[Est88] W. T. van Est; P. Dazord et al. eds Une démonstration de E. Cartan du troisième théorème de Lie, Séminaire Sud-Rhodanien de Géométrie VIII: Actions Hamiltoniennes de Groupes; Troisième Théorème de Lie (1988) | Zbl

[Fu70] L. Fuchs Infinite Abelian Groups, I, Acad. Press, New York, 1970 | MR | Zbl

[Gl01a] H. Glöckner; A. Strasburger et al eds. Infinite-dimensional Lie groups without completeness restriction, Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups, Volume 55 (2002), pp. 43-59 | Zbl

[Gl01b] H. Glöckner Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups (J. Funct. Anal., to appear) | MR | Zbl

[Gl01c] H. Glöckner Algebras whose groups of units are Lie groups (2001) (Preprint) | MR | Zbl

[Go86] V. V. Gorbatsevich The construction of a simply connected Lie group with a given Lie algebra, Russian Math. Surveys, Volume 41 (1986), pp. 207-208 | DOI | MR | Zbl

[God71] C. Godbillon Eléments de Topologie Algébrique, Hermann, Paris, 1971 | MR | Zbl

[Ha82] R. Hamilton The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc, Volume 7 (1982), pp. 65-222 | DOI | MR | Zbl

[He73] A. Heller Principal bundles and groups extensions with applications to Hopf algebras, J. Pure and Appl. Algebra, Volume 3 (1973), pp. 219-250 | DOI | MR | Zbl

[Hi76] M. W. Hirsch Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, 1976 | MR | Zbl

[Ho51] G. Hochschild Group extensions of Lie groups I, II, Annals of Math, Volume 54 ; 54 (1951 ; 1951) no. 1 ; 3, p. 96-109 ; 537-551 | DOI | MR | Zbl

[HoMo98] K. H. Hofmann; S. A. Morris The Structure of Compact Groups, Studies in Math., de Gruyter, Berlin, 1998 | MR | Zbl

[Hub61] P. J. Huber Homotopical Cohomology and Cech Cohomology, Math. Annalen, Volume 144 (1961), pp. 73-76 | MR | Zbl

[KM97] A. Kriegl; P. Michor The Convenient Setting of Global Analysis, Math. Surveys and Monographs, 53, Amer. Math. Soc., 1997 | MR | Zbl

[La99] S. Lang Fundamentals of Differential Geometry, Graduate Texts in Math, 191, Springer-Verlag, 1999 | MR | Zbl

[Ma01] P. Maier; A. Strasburger et al eds. Central extensions of topological current algebras, Geometry and Analysis on Finite- and Infinite-Dimensional Lie groups, Volume 55 (2002), pp. 61-76 | Zbl

[Ma57] G. W. Mackey Les ensembles boréliens et les extensions des groupes, J. Math, Volume 36 (1957), pp. 171-178 | MR | Zbl

[MacL63] S. MacLane Homological Algebra, Springer-Verlag, 1963

[Mi59] E. Michael Convex structures and continuous selections, Can. J. Math, Volume 11 (1959), pp. 556-575 | DOI | MR | Zbl

[Mi83] J. Milnor; B. DeWitt ed. Remarks on infinite-dimensional Lie groups, Proc. Summer School on Quantum Gravity, Les Houches (1983) | Zbl

[MN01] P. Maier; K.-H. Neeb Central extensions of current groups (2001) (Preprint) | MR | Zbl

[MT99] P. Michor; J. Teichmann Description of infinite dimensional abelian regular Lie groups, J. Lie Theory (1999), pp. 487-489 | MR | Zbl

[Ne01a] K.-H. Neeb; S. Huchleberry et al eds. Representations of infinite dimensional groups, Infinite Dimensional Kähler Manifolds (To appear in DMV Seminar), Volume 31 (2001)

[Ne01b] K.-H. Neeb Universal central extensions of Lie groups (Acta Appl. Math. to appear) | MR | Zbl

[Ne96] K.-H. Neeb A note on central extensions, J. Lie Theory (1996), pp. 207-213 | MR | Zbl

[Ne98] K.-H. Neeb Holomorphic highest weight representations of infinite dimensional complex classical groups, J. reine angew. Math, Volume 497 (1998), pp. 171-222 | DOI | MR | Zbl

[Omo97] H. Omori Infinite-Dimensional Lie Groups, Translations of Math. Monographs, 158, Amer. Math. Soc., 1997 | MR | Zbl

[Pa65] R. S. Palais On the homotopy type of certain groups of operators, Topology, Volume 3 (1965), pp. 271-279 | DOI | MR | Zbl

[Pa66] R. S. Palais Homotopy theory of infinite dimensional manifolds, Topology, Volume 5 (1965), pp. 1-16 | DOI | MR | Zbl

[PS86] A. Pressley; G. Segal Loop Groups, Oxford University Press, Oxford, 1986 | MR | Zbl

[Ro95] C. Roger Extensions centrales d'algèbres et de groupes de Lie de dimension infinie, algèbres de Virasoro et généralisations, Reports on Math. Phys, Volume 35 (1995), pp. 225-266 | DOI | MR | Zbl

[Se70] G. Segal Cohomology of topological groups, Symposia Math, Volume 4 (1970), pp. 377-387 | MR | Zbl

[Se81] G. Segal Unitary representations of some infinite-dimensional groups, Comm. Math. Phys, Volume 80 (1981), pp. 301-342 | DOI | MR | Zbl

[Sh49] A. Shapiro Group extensions of compact Lie groups, Annals of Math (1949), pp. 581-586 | DOI | MR | Zbl

[Si77] S. J. Sidney Weakly dense subgroups of Banach spaces, Indiana Univ. Math. Journal (1977), pp. 981-986 | DOI | MR | Zbl

[Sp66] E. H. Spanier Algebraic Topology, McGraw-Hill Book Company, New York, 1966 | MR | Zbl

[St78] J. D. Stasheff Continuous cohomology of groups and classifying spaces, Bull. of the Amer. Math. Soc (1978), pp. 513-530 | DOI | MR | Zbl

[Ste51] N. Steenrod The topology of fibre bundles, Princeton University Press, Princeton, New Jersey, 1951 | MR | Zbl

[tD91] T. Dieck Topologie, de Gruyter, Berlin -- New York, 1991 | MR | Zbl

[Te99] J. Teichmann Infinite-dimensional Lie Theory from the Point of View of Functional Analysis (1999) (Ph. D. Thesis, Vienna)

[Ti83] J. Tits Liesche Gruppen und Algebren, Springer, New York-Heidelberg, 1983 | MR | Zbl

[TL99] V. Toledano Laredo Integrating unitary representations of infinite-dimensional Lie groups, Journal of Funct. Anal., Volume 161 (1999), pp. 478-508 | DOI | MR | Zbl

[Tu95] G. M. Tuynman An elementary proof of Lie's Third Theorem (1995) (Unpublished note)

[TW87] G. M. Tuynman; W. A. J. J. Wiegerinck Central extensions and physics, J. Geom. Physics, Volume 4 (1987) no. 2, pp. 207-258 | DOI | MR | Zbl

[Va85] V. S. Varadarajan Geometry of Quantum Theory, Springer-Verlag, 1985 | MR | Zbl

[Wa83] F. W. Warner Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1983 | MR | Zbl

[We80] R. O. Wells Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, Springer-Verlag, 1980 | MR | Zbl

[We95] C. A. Weibel An introduction to homological algebra, Cambridge studies in advanced math, 38, Cambridge Univ. Press, 1995 | MR | Zbl

[We95] D. Werner Funktionalanalysis, Springer-Verlag, Berlin-Heidelberg, 1995 | MR | Zbl

Cité par Sources :