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Yuly Billig; Karl-Hermann Neeb
On the cohomology of vector fields on parallelizable manifolds
(Sur la cohomologie des champs vectoriels sur les variétés parallélisables)
Annales de l'institut Fourier, 58 no. 6 (2008), p. 1937-1982, doi: 10.5802/aif.2402
Article PDF | Reviews MR 2473625 | Zbl 1157.17007
Class. Math.: 17B56, 17B65, 17B68
Keywords: Lie algebra of vector fields, Lie algebra cohomology, Gelfand-Fuks cohomology, extended affine Lie algebra

Résumé - 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.

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