On the cohomology of vector fields on parallelizable manifolds

Billig, Yuly; Neeb, Karl-Hermann

doi:10.5802/aif.2402

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)

Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 1937-1982.

Abstract
Résumé

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 $\bar{Ω}_{M}^{p} = Ω_{M}^{p} / d Ω_{M}^{p - 1}$ . 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 ${\bar{Ω}}_{M}^{1}$ , generalizing affine Kac-Moody algebras. The second cohomology $H^{2} (𝒱_{M}, {\bar{Ω}}_{M}^{1})$ classifies twists of the semidirect product of $𝒱_{M}$ with the universal central extension of a gauge Lie algebra.

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 $\bar{Ω}_{M}^{p} = Ω_{M}^{p} / d Ω_{M}^{p - 1}$ . 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 ${\bar{Ω}}_{M}^{1}$ , généralisant les algèbres de Kac-Moody affines. L’espace $H^{2} (𝒱_{M}, {\bar{Ω}}_{M}^{1})$ classifie des torsions du produit semi-direct de $𝒱_{M}$ avec l’extension centrale universelle d’une algèbre de Lie de jauge.

MR Zbl

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

Author's affiliations:

Billig, Yuly ¹; Neeb, Karl-Hermann ²

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     title = {On the cohomology of vector fields on parallelizable manifolds},
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     year = {2008},
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Billig, Yuly; Neeb, Karl-Hermann. On the cohomology of vector fields on parallelizable manifolds. Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 1937-1982. doi : 10.5802/aif.2402. https://aif.centre-mersenne.org/articles/10.5802/aif.2402/

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