On Solvable Quadratic Lie algebras having an Abelian descending central ideal

TL;DR

This paper classifies solvable quadratic Lie algebras with an Abelian descending central ideal, providing a detailed construction method, conditions for invariant metrics, and a specific classification example involving Heisenberg algebras.

Contribution

It introduces a canonical construction and classification framework for solvable quadratic Lie algebras with Abelian descending central ideals, using extension theory and cocycles.

Findings

All such Lie algebras can be constructed from canonical ideals.

Necessary and sufficient conditions for invariant metrics are established.

A classification example involving Heisenberg Lie algebras is provided.

Abstract

Solvable Lie algebras having at least one Abelian descending central ideal are studied. It is shown that all such Lie algebras can be built up from canonically defined ideals. The nature of such ideals is elucidated and their construction is provided in detail. An approach to study and to classify these Lie algebras is given through the theory of extensions via appropriate cocycles and representations on which a group action is naturally defined. Also, necessary and sufficient conditions for the existence of invariant metrics on the studied extensions are given. It is shown that any solvable quadratic Lie algebra $\g$ having an Abelian descending central ideal is of the form $\g=\h\oplus\a\oplus\h^*$, where $\h^*\simeq \ide(\g)$ and $\a \oplus \h^{*}\simeq \j(\g)$ are in fact two canonically defined Abelian ideals of $\g$ satisfying $\ide(\g)^\perp=\j(\g)$. As an example, a classification of this type of quadratic Lie algebras is given assuming that $\j(\g)/\ide(\g)$ is an $r$-dimensional vector space and $\g/\j(\g)$ is the 3-dimensional Heisenberg Lie algebra.