Dans le présent article, nous déterminons, pour chaque variété parallélisable compacte lisse , les espaces de seconde cohomologie de l’algèbre de Lie des champs vectoriels lisses sur à valeurs dans le module . Le cas est d’un intérêt particulier puisque l’algèbre de jauge des fonctions sur à valeurs dans une algèbre de Lie simple de dimension finie possède l’extension centrale universelle avec le centre , généralisant les algèbres de Kac-Moody affines. L’espace classifie des torsions du produit semi-direct de avec l’extension centrale universelle d’une algèbre de Lie de jauge.
In the present paper we determine for each parallelizable smooth compact manifold the second cohomology spaces of the Lie algebra of smooth vector fields on with values in the module . The case of is of particular interest since the gauge algebra of functions on with values in a finite-dimensional simple Lie algebra has the universal central extension with center , generalizing affine Kac-Moody algebras. The second cohomology classifies twists of the semidirect product of with the universal central extension of a gauge Lie algebra.
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}, pages = {1937--1982}, publisher = {Association des Annales de l'institut Fourier}, volume = {58}, number = {6}, year = {2008}, 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. http://www.numdam.org/item/AIF_2008__58_6_1937_0/
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