Central extensions of infinite-dimensional Lie groups
Annales de l'Institut Fourier, Volume 52 (2002) no. 5, p. 1365-1442

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.

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.

DOI : https://doi.org/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
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {5},
     year = {2002},
     pages = {1365-1442},
     doi = {10.5802/aif.1921},
     zbl = {1019.22012},
     mrnumber = {1935553},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_5_1365_0}
}
Neeb, Karl-Hermann. Central extensions of infinite-dimensional Lie groups. Annales de l'Institut Fourier, Volume 52 (2002) no. 5, pp. 1365-1442. doi : 10.5802/aif.1921. http://www.numdam.org/item/AIF_2002__52_5_1365_0/

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