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
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.
DOI : https://doi.org/10.5802/aif.2402
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
@article{AIF_2008__58_6_1937_0,
     author = {Billig, Yuly and Neeb, Karl-Hermann},
     title = {On the cohomology of vector fields on parallelizable manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {6},
     year = {2008},
     pages = {1937-1982},
     doi = {10.5802/aif.2402},
     mrnumber = {2473625},
     zbl = {1157.17007},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_6_1937_0}
}
Billig, Yuly; Neeb, Karl-Hermann. On the cohomology of vector fields on parallelizable manifolds. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 1937-1982. doi : 10.5802/aif.2402. https://www.numdam.org/item/AIF_2008__58_6_1937_0/

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