An Explicit Seven-Term Exact Sequence for the Cohomology of a Lie Algebra Extension

Abstract

We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra 𝔤, an ideal 𝔥 of 𝔤, and the quotient 𝔤/𝔥 with coefficients in a 𝔤-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. This alternative approach to the seven term exact sequence can certainly be useful, especially since we include straightforward cocycle descriptions of the constructed maps.

Key Words::

• Crossed extension
• Derivation
• Lie algebra cohomology
• Lie algebra extension
• Low dimensional cohomology

2010 Mathematics Subject Classification:

• 17B56

ACKNOWLEDGMENT

The authors would like to thank the referee for his helpful remarks and suggestions.