A note on the Chevalley-Eilenberg Cohomology for the Galilei and Poincare Algebras

TL;DR

This paper systematically computes the Chevalley-Eilenberg cohomology for Galilei and Poincare groups, identifying extensions and non-trivial forms relevant to space-time symmetries.

Contribution

It provides a comprehensive method to determine cohomology and extensions for space-time symmetry groups, applicable to various groups beyond those studied.

Findings

Complete cohomology at degrees two, three, and four for Galilei and Poincare groups.

Identification of all possible central and non-central extensions for these groups.

Method applicable to any space-time symmetry group.

Abstract

We construct in a systematic way the complete Chevalley-Eilenberg cohomology at form degree two, three and four for the Galilei and Poincare groups. The corresponding non-trivial forms belong to certain representations of the spatial rotation (Lorentz) group. In the case of two forms they give all possible central and non-central extensions of the Galilei group (and all non-central extensions of the Poincare group). The procedure developed in this paper can be applied to any space-time symmetry group.