On the cohomology of vector fields on parallelizable manifolds

Abstract

In the present paper we determine for each parallelizable smooth compact manifold $M$ the cohomology spaces $H^2(V_M,\bar\Omega^p_M)$ of the Lie algebra $V_M$ of smooth vector fields on $M$ with values in the module $\bar\Omega^p_M = \Omega^p_M/d\Omega^{p-1}_M$. The case of $p=1$ is of particular interest since the gauge algebra $C^\infty (M,k)$ has the universal central extension with center $\bar\Omega^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2(V_M, \bar\Omega^1_M)$ classifies twists of the semidirect product of $V_M$ with the universal central extension $C^\infty (M,k) \oplus \bar\Omega^1_M$.