On the Continuous Cohomology of a semi-direct product Lie group

Abstract

Let $G$ be a Lie group and $H$ be a subgroup of it. We can construct a bisimplicial manifold $NG(\ast )⋊NH(\ast )$ and the de Rham complex ${\Omega }^{\ast }(NG(\ast )⋊NH(\ast ))$ on it. This complex is a triple complex and the cohomology of its total complex is isomorphic to $H^{\ast }(B(G⋊H))$. In this paper, we show that the total complex of the double complex ${\Omega }^q(NG(\ast )⋊NH(\ast ))$ is isomorphic to the continuous cohomology $H_c^{\ast }(G⋊H;S^q{\mathcal{G}}^{\ast }\otimes S^q{\mathcal{H}}^{\ast })$ for any fixed $q$.

9 pages