An explicit seven-term exact sequence for the cohomology of a Lie algebra extension

TL;DR

This paper constructs an explicit seven-term exact sequence for the low-degree cohomology of Lie algebra extensions, providing more accessible maps and cocycle descriptions to facilitate understanding and computation.

Contribution

It introduces alternative, elementary maps for the sequence, making the cohomology relations more explicit and easier to work with compared to spectral sequence methods.

Findings

Provides explicit cocycle descriptions of the maps

Constructs elementary methods for the sequence

Enhances understanding of Lie algebra cohomology

Abstract

We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra $\Lg$, an ideal $\Lh$ of $\Lg$ and the quotient $\Lg / \Lh$ with coefficients in a $\Lg$-module. The existence of such a sequence follows from the Hochschild-Serre spectral sequence associated to the Lie algebra extension. However, some of the maps occurring in this induced sequence are not always explicitly known or easy to describe. In this article, we give alternative maps that yield an exact sequence of the same form, making use of the interpretations of the low-dimensional cohomology spaces in terms of derivations, extensions etc. The maps are constructed using elementary methods. Although we don't know whether the new maps coincide with the ones induced by the spectral sequence, the alternative sequence can certainly be useful, especially since we include straight-forward cocycle descriptions of the constructed maps.