Extensions of Lie algebras

TL;DR

This paper reviews non-abelian extensions of Lie algebras, highlighting a cohomological obstruction and drawing analogies to differential geometry, serving as a comprehensive overview rather than new research.

Contribution

It provides a summarized review of known results on Lie algebra extensions, emphasizing the cohomological aspects and geometric analogies.

Findings

Identifies a 3-dimensional cohomological obstruction to Lie algebra extensions

Draws an analogy between Lie algebra extensions and differential geometry concepts

Most results are known; the paper serves as a review

Abstract

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. In the new version references added: Most of the results are known. So this paper will not be submitted to a journal, it can be regarded as a review paper.