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

References in corpus (1)

• Abelian extensions of infinite-dimensional Lie groups