On a filtration of the second cohomology of nilpotent Lie algebras

TL;DR

This paper investigates a specific filtration of the second cohomology group of finite-dimensional nilpotent Lie algebras, providing new characterizations, bounds, and criteria related to their cohomological properties and extensions.

Contribution

It offers a novel characterization of the filtration of second cohomology and derives bounds and criteria for central extensions of nilpotent Lie algebras.

Findings

New expression for the second Betti number

Bounds for the second Betti number

Cohomological criterion for central extensions

Abstract

We study a known filtration of the second cohomology of a finite dimensional nilpotent Lie algebra $\mathfrak{g}$ with coefficients in a finite dimensional nilpotent $\mathfrak{g}$-module $M$, that is based upon a refinement of the correspondence between $\mathrm{H}^2(\mathfrak{g},M)$ and equivalence classes of abelian extensions of $\mathfrak{g}$ by $M$. We give a different characterization of this filtration and as a corollary, we obtain an expression for the second Betti number of $\mathfrak{g}$. Using this expression, we find bounds for the second Betti number and derive a cohomological criterium for the existence of certain central extensions of $\mathfrak{g}$.