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

TL;DR

This paper establishes a connection between the de Rham complex on a bisimplicial manifold associated with a semi-direct product Lie group and its continuous cohomology, providing a new computational approach.

Contribution

It demonstrates that the total complex of a specific double complex is isomorphic to the continuous cohomology of the semi-direct product Lie group.

Findings

The de Rham complex's total complex computes the cohomology of the classifying space.

The double complex's total complex is isomorphic to the continuous cohomology of the group.

Provides a new method for calculating continuous cohomology of semi-direct product Lie groups.

Abstract

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