On quadratic Lie algebras containing the Heisenberg Lie algebra

TL;DR

This paper characterizes quadratic Lie algebras containing the Heisenberg algebra as an ideal, providing construction methods and conditions for their structure and metric properties.

Contribution

It introduces a construction procedure for such Lie algebras and establishes necessary and sufficient conditions for their classification.

Findings

A universal construction method for quadratic Lie algebras with Heisenberg ideals.

Conditions to identify when an algebra is a Heisenberg extension.

Analysis of metric properties of quotients and nilradicals.

Abstract

In this work we study quadratic Lie algebras that contain the Heisenberg Lie algebra $\h_m$ as an ideal. We give a procedure for constructing these kind of quadratic Lie algebras and prove that any quadratic Lie algebra $\g$ that contains the Heisenberg Lie algebra as an ideal is constructed by using this procedure. We state necessary and sufficiency conditions to determine whether an indecomposable quadratic Lie algebra is the Heisenberg Lie algebra extended by a derivation. In addition, we state necessary and sufficiency conditions to determine whether the quotient $\g/\h_m$ admits an invariant metric and we also study the case when the nilradical of the Lie algebra $\g$ is equal to $\h_m$.