ON THE COHOMOLOGY OF VECTOR FIELDS ON PARALLELIZABLE MANIFOLDS

In the present paper we determine for each parallelizable smooth compact manifold M the second cohomology spaces of the Lie algebra nu(M) of smooth vector fields on M with values in the module (Omega) over bar (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 (Omega) over bar (1)(M) generalizing affine Kac-Moody algebras. The second cohomology H(2)(nu(M), (Omega) over bar (1)(M)) classifies twists of the semidirect product of nu(M) with the universal central extension of a gauge Lie algebra.