On the cohomology of vector fields on parallelizable manifolds

Billig, Yuly; Neeb, Karl-Hermann

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 $\overline{\Omega }{}_M^p={\Omega }_M^p/d\Omega {}_M^{p-1}$. 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 ${\overline{\Omega }}_M^1$, généralisant les algèbres de Kac-Moody affines. L’espace $H^2({𝒱}_M,{\overline{\Omega }}_M^1)$ 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 $\overline{\Omega }{}_M^p={\Omega }_M^p/d{\Omega }_M^{p-1}$. 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 ${\overline{\Omega }}_M^1$, generalizing affine Kac-Moody algebras. The second cohomology $H^2({𝒱}_M,{\overline{\Omega }}_M^1)$ classifies twists of the semidirect product of ${𝒱}_M$ with the universal central extension of a gauge Lie algebra.

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

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