On cohomology of split Lie algebra extensions

TL;DR

This paper introduces compatible actions to compute the cohomology of split Lie algebra extensions, providing a new resolution and an improved spectral sequence with sharper bounds.

Contribution

It presents a novel approach using compatible actions to construct resolutions and refine the Hochschild-Serre spectral sequence for split Lie algebra extensions.

Findings

Constructed a new resolution for cohomology computation.

Provided an alternative construction of the Hochschild-Serre spectral sequence.

Obtained sharper bounds for the spectral sequence length.

Abstract

We introduce the notion of compatible actions in the context of split extensions of finite dimensional Lie algebras over a field. Using compatible actions, we construct a new resolution to compute the cohomology of semi-direct products of Lie algebras. We also give an alternative way to construct the Hochschild-Serre spectral sequence associated to a split extension of finite dimensional Lie algebras and obtain a sharper bound for the length of this spectral sequence.