Let $\mathfrak{g}$ be a Lie algebra over a field $k$, and let $U(\mathfrak{g})$ be its universal enveloping algebra. $\mathfrak{g}$ is canonically embedded in $U(\mathfrak{g})$; identify it with its image there. Fixing (totally standard) notation, let
$$Z(\mathfrak{g}) = \{x\in \mathfrak{g}\mid [x,y]=0,\forall y\in\mathfrak{g}\}$$
be the center of $\mathfrak{g}$ (as a lie algebra) and let
$$Z(U(\mathfrak{g})) = \{f\in U(\mathfrak{g}) \mid fg = gf,\forall g\in U(\mathfrak{g})\}$$
be the center of $U(\mathfrak{g})$ (as an associative algebra). The canonical embedding of $\mathfrak{g}$ in $U(\mathfrak{g})$ also embeds $Z(\mathfrak{g})$ in $Z(U(\mathfrak{g}))$. Here is my question:
Is it the case that $Z(U(\mathfrak{g}))$ is generated as an associative $k$-algebra by $Z(\mathfrak{g})$? What's the proof?
Intuitively I believe that it should, and I have verified this by hand with an ad-hoc calculation in the special cases that $\mathfrak{g}$ is (1) the nonabelian 2-dimensional lie algebra over $\mathbb{C}$, and (2) the Heisenberg algebra. While my calculation "feels" a little general, I haven't been able to eke the general argument out of it.
Thanks in advance for your thoughts.
Best Answer
If the center of $\mathfrak{g}$ is 0, this only means that $U(\mathfrak{g})$ has no central elements of degree 1, but there could be central elements of higher degree. There is a natural $\mathfrak{g}$-module map $$\mathfrak{g} \otimes \mathfrak{g} \to U(\mathfrak{g}).$$ Now if, say, $\mathfrak{g}$ is semisimple so that it has a nondegenerate Killing form (a bilinear symmetric invariant form), then there are $\mathfrak{g}$-module isomorphisms $$\mathfrak{g} \otimes \mathfrak{g} \cong \mathfrak{g} \otimes \mathfrak{g}^* \cong \text{End}(\mathfrak{g})$$ Clearly $\text{End}(\mathfrak{g})$ has nonzero central elements, namely scalars, and if you take the corresponding element $\sum e_i \otimes f_i \in \mathfrak{g} \otimes \mathfrak{g}$ and its image $\sum e_i f_i \in U(\mathfrak{g})$, you get a central element of $U(\mathfrak{g})$ called the Casimir element.
(We are not quite done; we still have to show that the element we've constructed is $\ne 0$. This follows from the Poincare-Birkhoff-Witt theorem.)