On a spectral sequence for the cohomology of a nilpotent Lie algebra

TL;DR

This paper introduces a spectral sequence for computing the cohomology of nilpotent Lie algebras, providing a new tool for understanding their structure and cohomological properties.

Contribution

It constructs a spectral sequence from a filtration of the Chevalley-Eilenberg complex that converges to Lie algebra cohomology, with explicit descriptions for certain cases.

Findings

Spectral sequence converges to Lie algebra cohomology.

Provides a grading for cohomology groups, except in specific degrees.

Explicit computations for all real nilpotent Lie algebras up to dimension six.

Abstract

Given a nilpotent Lie algebra $\mathfrak n$ we construct a spectral sequence which is derived from a filtration of its Chevalley-Eilenberg differential complex and converges to the Lie algebra cohomology of $\mathfrak n$. The limit of this spectral sequence gives a grading for the Lie algebra cohomology, except for the cohomology groups of degree 0, 1, $\dim \mathfrak n-1$ and $\dim\mathfrak n$ as we shall prove. We describe the spectral sequence associated to a nilpotent Lie algebra which is a direct sum of two ideals, one of them of dimension one, in terms of the spectral sequence of the co-dimension one ideal. Also, we compute the spectral sequence corresponding to each real nilpotent Lie algebra of dimension less than or equal to six.