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, pp. 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 : 10.5802/aif.2402
Classification : 17B56, 17B65, 17B68
Keywords: Lie algebra of vector fields, Lie algebra cohomology, Gelfand-Fuks cohomology, extended affine Lie algebra
Mot clés : algèbre de Lie des champs vectoriels, cohomologie de l’algèbre de Lie, cohomologie de Gelfand-Fuks, algèbre de Lie affine étendu
Billig, Yuly 1 ; Neeb, Karl-Hermann 2

1 Carleton University School of Mathematics and Statistics 1125 Colonel By Drive Ottawa, Ontario, K1S 5B6 (Canada)
2 Technische Universität Darmstadt Schlossgartenstrasse 7 64289 Darmstadt (Deutschland)
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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. http://www.numdam.org/articles/10.5802/aif.2402/

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