On the cohomology of vector fields on parallelizable manifolds

[Sur la cohomologie des champs vectoriels sur les variétés parallélisables]

Billig, Yuly ; Neeb, Karl-Hermann

Annales de l'Institut Fourier, Tome 58 (2008), p. 1937-1982 / Harvested from Numdam

Résumé

Dans le présent article, nous déterminons, pour chaque variété parallélisable compacte lisse $M$, les espaces de seconde cohomologie de l’algèbre de Lie ${\mathcal{V}}_M$ des champs vectoriels lisses sur $M$ à valeurs dans le module $\overline{\Omega }\,^p_M = \Omega ^p_M/d\Omega \, ^ {p-1} _M$. 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 $\overline{\Omega }^1_M$, généralisant les algèbres de Kac-Moody affines. L’espace $H^2 ({\mathcal{V}} _M, \overline{\Omega }^1_M)$ classifie des torsions du produit semi-direct de ${\mathcal{V}} _M$ avec l’extension centrale universelle d’une algèbre de Lie de jauge.

Abstract

In the present paper we determine for each parallelizable smooth compact manifold $M$ the second cohomology spaces of the Lie algebra ${\mathcal{V}}_M$ of smooth vector fields on $M$ with values in the module $\overline{\Omega }\,^p_M = \Omega ^p_M/d\Omega ^{p-1}_M$. 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 $\overline{\Omega }^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2({\mathcal{V}}_M, \overline{\Omega }^1_M)$ classifies twists of the semidirect product of ${\mathcal{V}}_M$ with the universal central extension of a gauge Lie algebra.

Détail

Publié le : 2008-01-01

MR 2473625 | Zbl 1157.17007

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

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