3
$\begingroup$

Let $A$ be a commutative ring with identity and $L$ be a Lie algebra which is also a free module over $A$. I have seen the following statements:

  1. The universal enveloping algebra $U(L)$ is isomorphic (as $A$ modules) to the symmetric algebra $S(L)$. (This is stated in the wiki article of PBW Theorem).
  2. The associated graded of the center of $U(L)$ is the Poisson center of $S(L)$.

They are usually stated as the consequence or corollary of the (generalized of) PBW theorem. I would like to have a reference for these results. Mostly the proofs are given in the standard textbook with either $A$ being field of characteristic zero or $L$ being finite-dimensional Lie algebra.

Also, any kind of reference related to the above results will be extremely helpful.

Thanks in advance.

Improve this question
$\endgroup$
3
  • 3
    $\begingroup$ 1. This is Theorem 5.9 (a) in my diploma thesis; a few paragraphs down below I give a list of references to proofs. The list is about 10 years out of date now (e.g., Semmes's notes have the proof too), but there should be enough for anyone there. $\endgroup$
    – darij grinberg
    Commented Jul 16, 2021 at 9:53
  • $\begingroup$ 2. I have never seen this considered part of the PBW theorem (and I wouldn't be surprised if it is not true in this generality). What is the Poisson structure on $S\left(L\right)$? Is it the Lie bracket of $L$ extended to $S\left(L\right)$ as a biderivation (i.e., bilinear map that is a derivation in each argument)? $\endgroup$
    – darij grinberg
    Commented Jul 16, 2021 at 9:54
  • $\begingroup$ @darijgrinberg Thanks for the reference. About 2., you are right. The Poisson structure is the Lie bracket extended to $S(L)$ by Leibniz rule. You may be right about this not being true in general and it has more to do with Duflo isomorphism than PBW. I just want to understand the map between these two centers for the free module $L$. $\endgroup$
    – Cusp
    Commented Jul 16, 2021 at 10:37

0

Reset to default

You must log in to answer this question.