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 $\phi :\mathcal{H}⟶\mathcal{H}$, the Lie algebra $\mathcal{H}$ is generated by the generators of $\mathfrak{g}$, together with their images under $\phi$, and that $\mathcal{H}$ is the sum of $\mathfrak{g}$, $\phi (\mathfrak{g})$ and $\left[\mathfrak{g},\phi (\mathfrak{g})\right]$.

Cited by in corpus (3)

• Revisiting the Jaynes-Cummings model with time-dependent coupling
• Reproducing the effects of quantum deformation in the undeformed Jaynes-Cummings model
• Lie polynomials in a $q$-deformed universal enveloping algebra of a low-dimensional Lie algebra