Central extensions of infinite-dimensional Lie groups

Neeb, Karl-Hermann

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.

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