how to calculate the Casimir function of the Heisenberg Lie algebra?

Question

Given a Heisenberg Lie algebra of dimension $2n+1$ with generators $X_i$, how can I calculate the Casimir function of the Heiseneberg Lie algebra ?

Answer 1

A Casimir operator can also be defined for non-semisimple Lie algebras:

Definition: Let $\mathfrak{g}$ be a complex Lie algebra und $U(\mathfrak{g})$ be its universal enveloping algebra. An element $C\in U(\mathfrak{g})$ is called a Casimir operator, if $[C,X]=0$ for all $X\in U(\mathfrak{g})$. In other words, $C$ is in the center of the universal enveloping algebra.

For the $3$-dimensional Heisenberg Lie algebra, with basis $(e_1,e_2,e_3)$ and $[e_1,e_2]=e_3$, we have that $C=e_3$ is a Casimir operator. The $2n+1$-dimensional Heisenberg can be done similarly, see here:

Reference: Casimir operators for non-semisimple Lie algebras.