Billig, Yuly; Neeb, Karl-Hermann
On the cohomology of vector fields on parallelizable manifolds  [ Sur la cohomologie des champs vectoriels sur les variétés parallélisables ]
Annales de l'institut Fourier, Tome 58 (2008) no. 6 , p. 1937-1982
MR 2473625 | Zbl 1157.17007
doi : 10.5802/aif.2402
URL stable : http://www.numdam.org/item?id=AIF_2008__58_6_1937_0

Classification:  17B56,  17B65,  17B68
Mots clés: algèbre de Lie des champs vectoriels, cohomologie de l’algèbre de Lie, cohomologie de Gelfand-Fuks, algèbre de Lie affine étendu
Dans le présent article, nous déterminons, pour chaque variété parallélisable compacte lisse M, les espaces de seconde cohomologie de l’algèbre de Lie 𝒱 M des champs vectoriels lisses sur M à valeurs dans le module Ω ¯ M p =Ω M p /dΩ M p-1 . Le cas p=1 est d’un intérêt particulier puisque l’algèbre de jauge des fonctions sur M à valeurs dans une algèbre de Lie simple de dimension finie possède l’extension centrale universelle avec le centre Ω ¯ M 1 , généralisant les algèbres de Kac-Moody affines. L’espace H 2 (𝒱 M ,Ω ¯ M 1 ) classifie des torsions du produit semi-direct de 𝒱 M avec l’extension centrale universelle d’une algèbre de Lie de jauge.
In the present paper we determine for each parallelizable smooth compact manifold M the second cohomology spaces of the Lie algebra 𝒱 M of smooth vector fields on M with values in the module Ω ¯ M p =Ω M p /dΩ M p-1 . The case of p=1 is of particular interest since the gauge algebra of functions on M with values in a finite-dimensional simple Lie algebra has the universal central extension with center Ω ¯ M 1 , generalizing affine Kac-Moody algebras. The second cohomology H 2 (𝒱 M ,Ω ¯ M 1 ) classifies twists of the semidirect product of 𝒱 M with the universal central extension of a gauge Lie algebra.

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