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.
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.