2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129417In the present paper we determine for each parallelizable smooth compact manifold $M$ the cohomology spaces $H^2(V_M,\barΩ^p_M)$ of the Lie algebra $V_M$ of smooth vector fields on $M$ with values in the module $\barΩ^p_M = Ω^p_M/dΩ^{p-1}_M$. The case of $p=1$ is of particular interest since the gauge algebra $C^\infty (M,k)$ has the universal central extension with center $\barΩ^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2(V_M, \barΩ^1_M)$ classifies twists of the semidirect product of $V_M$ with the universal central extension $C^\infty (M,k) \oplus \barΩ^1_M$.Representation TheoryRings and Algebras17B56; 17B65; 17B68text