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 by an abelian group 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 de dimension infinie par un groupe abélien de Lie 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.
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
@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://archive.numdam.org/articles/10.5802/aif.1921/} }
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%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://archive.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, Volume 52 (2002) no. 5, pp. 1365-1442. doi : 10.5802/aif.1921. http://archive.numdam.org/articles/10.5802/aif.1921/
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