Is there a name for this Lie algebra?

Question

Consider the three dimensional, complex Lie algebra with basis $\{a,a^\dagger, I\}$ and the following structure relations: \begin{align} [a,a^\dagger] = I, \qquad [a,I] = 0, \qquad [a^\dagger, I] = 0. \end{align} Is there a special name for this algebra in the mathematical literature?

This question is motivated by quantum mechanics where representations of this algebra are important for understanding the quantum harmonic oscillator and other systems.

Answer 1

It is the three dimensional Heisenberg Lie algebra. In the link provided $a = x$, $a^\dagger = y$, and $I = z$.