The center of the universal enveloping algebra of a nilpotent or solvable Lie algebra

Question

My questions are the following.

• Prove that the center of the universal enveloping algebra of a nilpotent Lie algebra is generated by the center of the Lie algebra.
• Give a solvable Lie algebra such that the center of its universal enveloping algebra is not generated by the center of the Lie algebra.

I heard these two are classical results, however I finally could not find the proof. Thank you.

P.S.

I know that there is a counterexample in the semi-simple Lie algebra case and I already calculated the center of universal enveloping algebras of several nilpotent Lie algebras (Heisenberg algebras, ladder algebras and so on). I want to know a general proof in the nilpotent case and I could not find such a question in the sugested.

Answer 1

Excuse me but I got an answer by myself. In fact, I got a mistake but the ladder Lie algebra is a counterexample of my first question.

The ladder Lie algebra is a Lie algebra $\mathfrak{g} := \langle X_0, X_1, X_2, X_3 \rangle$ whose Lie bracket is defined by the following:

\begin{align} [X_0,X_1] = X_2, [X_0,X_2] = X_3, [X_0,X_3] = [X_1,X_2] = [X_1,X_3] = [X_2,X_3] = 0. \end{align}

In this case, the center of $\mathfrak{g}$ is $\langle X_3 \rangle$, however $X_2^2 - 2 X_1 X_3$ is contained in the center of $U(\mathfrak{g})$.