On quadratic Lie algebras containing the Heisenberg Lie algebra

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 $\h_m$.