Lie structure of the Heisenberg-Weyl algebra

Abstract

As an associative algebra, the Heisenberg-Weyl algebra $\mathcal{H}$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\mathcal{H}$. We identify a non-nilpotent but solvable Lie subalgebra $\mathfrak{g}$ of $\mathcal{H}$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $\varphi:\mathcal{H}\longrightarrow\mathcal{H}$, the Lie algebra $\mathcal{H}$ is generated by the generators of $\mathfrak{g}$, together with their images under $\varphi$, and that $\mathcal{H}$ is the sum of $\mathfrak{g}$, $\varphi(\mathfrak{g})$ and $\left[ \mathfrak{g},\varphi(\mathfrak{g})\right]$.