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.
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.
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
@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{\textquoteright}institut Fourier}, volume = {58}, number = {6}, year = {2008}, doi = {10.5802/aif.2402}, zbl = {1157.17007}, mrnumber = {2473625}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2402/} }
TY - JOUR AU - Billig, Yuly AU - Neeb, Karl-Hermann TI - On the cohomology of vector fields on parallelizable manifolds JO - Annales de l'Institut Fourier PY - 2008 SP - 1937 EP - 1982 VL - 58 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2402/ DO - 10.5802/aif.2402 LA - en ID - AIF_2008__58_6_1937_0 ER -
%0 Journal Article %A Billig, Yuly %A Neeb, Karl-Hermann %T On the cohomology of vector fields on parallelizable manifolds %J Annales de l'Institut Fourier %D 2008 %P 1937-1982 %V 58 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2402/ %R 10.5802/aif.2402 %G en %F 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, Volume 58 (2008) no. 6, pp. 1937-1982. doi : 10.5802/aif.2402. http://archive.numdam.org/articles/10.5802/aif.2402/
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