Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$, with an ordered basis $x_1 < x_2 < … < x_n$.
We define the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ to be the free noncommutative algebra $k\langle x_1,…,x_n\rangle$ modulo the relations $(x_ix_j – x_jx_i = [x_i,x_j])$.
The Poincaré–Birkhoff–Witt (PBW) theorem states that $U(\mathfrak{g})$ has a basis consisting of lexicographically ordered monomials i.e. monomials of the form $x_1^{e_1}x_2^{e_2}…x_n^{e_n}$. Checking that this basis spans $U(\mathfrak{g})$ is trivial, so the work lies in showing that these monomials are linearly independent.
One standard proof of PBW is to construct a $\mathfrak{g}$-action on the commutative polynomial ring $k[y_1,…,y_n]$ by setting $x_1^{e_1}x_2^{e_2}…x_n^{e_n}\cdot 1 = y_1^{e_1}y_2^{e_2}…y_n^{e_n}$ and verify algebraically that this gives rise to a well-defined representation of $\mathfrak{g}$. Details can be found in Dixmier's book on enveloping algebras.
What other proofs of PBW are there out there?
Are there nice reformulations of the above proof from a different perspective, such as one that emphasizes the universal property of $U(\mathfrak{g})$?
However, I would be especially interested in learning about proofs which are not just repackaged versions of the same algebraic manipulations used in the above proof (for example, geometric proofs which appeal to some property of $U(\mathfrak{g})$ as differential operators, etc.). If we allow ourselves more tools than just plain algebra, what other proofs of PBW can we get?
Best Answer
The nicest one I have ever seen uses a mix of universal algebra and combinatorial algebra, and was given by P. J. Higgins in Baer Invariants and the Birkhoff-Witt Theorem, Journal of Algebra 11, pp. 469-482 (1969) (free PDF linked).
Then there is the purely computational one which works over any $\mathbb Q$-algebra as base "field" and was given in the book by Deligne-Morgan. See I don't get a part of Bernstein's / Deligne-Morgan's proof of Poincaré-Birkhoff-Witt for details.
Emanuela Petracci gave in her thesis another computational proof, which uses the language of bialgebras to make the manipulations manageable.
There is also Cohn's A remark on the Poincaré-Birkhoff-Witt Theorem, J. London Math. Soc. (1963) s1-38(1): 197-203. It also has a discussion topic on MO.
If $\mathfrak g$ is the Lie algebra of a Lie group over $\mathbb R$, then you can indeed prove PBW using geometry: see, e. g., Proposition 1.9 in PDF 1 of Chapter 2 of Helgason's Lie Groups lecture notes. However, I don't think it is realistic to use this as a general proof for PBW; Lie's Third Theorem seems to be hard and require PBW itself.
Poincaré might have proven PBW himself (at least over a field of characteristic $0$), but I don't understand his proof (at least in a modern translation, which might itself be erroneous).
I hate to say but the only of the above references that I found easily readable is Higgins's paper...