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

TL;DR

This work addresses the problem of classifying and constructing the Chevalley-Eilenberg cohomology for the Galilei and Poincaré algebras up to degree $4$. It develops an algorithmic use of Maurer-Cartan forms to identify non-trivial closed forms at degrees $2$, $3$, and $4$, organizing them into rotation representations and building extended algebras with higher-level generators. The main contributions are the complete lists of $2$-, $3$-, and $4$-form CE cocycles for both groups, explicit parametrizations and potentials, and the identification of physical roles such as Bargmann central extensions, magnetic backgrounds, Wess-Zumino terms, and non-relativistic string/branes; the method is general and applicable to any space-time symmetry group. Overall, the paper provides a systematic framework for understanding symmetry extensions and their physical realizations in both non-relativistic and relativistic contexts.

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.