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PBW gives an ordered monomial basis for the enveloping algebra

Statement

Let $x_{1}, \dots, x_{r}$ be an ordered basis of a finite-dimensional complex Lie algebra $g$ . Then the monomials

$x_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{r}^{a_{r}} (a_{i} \in N_{0})$

form a basis of $U (g)$ . In particular, multiplication identifies $gr U (g)$ with the symmetric algebra $S (g)$ on the symbols of the $x_{i}$ .

Facts & Assumptions

Given: A finite ordered basis $x_{1}, \dots, x_{r}$ of a complex Lie algebra $g$ .

[F1]

The defining relation in $U (g)$ is $x_{j} x_{i} = x_{i} x_{j} + [x_{j}, x_{i}]$ (The universal enveloping algebra as a tensor quotient).

Proof

technique · direct

1.1

Orient [F1] as a rewriting rule whenever $j > i$ . Order words first by length and then by their number of inverted index pairs. The swapped term has one fewer inversion and the bracket term has smaller length, so every sequence of reductions terminates in a linear combination of ordered words. Hence the ordered monomials span $U (g)$ .

F1givenalgebra

1.2

Reductions on disjoint adjacent pairs commute. The only overlapping ambiguity occurs in a word $x_{k} x_{j} x_{i}$ with $k > j > i$ . Reducing the left pair first and the right pair first gives expressions whose difference is $[x_{k}, [x_{j}, x_{i}]] + [x_{j}, [x_{i}, x_{k}]] + [x_{i}, [x_{k}, x_{j}]],$ which is zero by the Jacobi identity. Thus every overlap is resolvable, and termination from step 1.1 implies that every word has a unique ordered normal form.

F1step 1.1algebra

2.1

If a linear combination of distinct ordered monomials represented zero in the tensor quotient, its unique normal form from step 1.2 would be that same nonzero combination and also the zero normal form, a contradiction. The ordered monomials are therefore linearly independent and hence a basis.

step 1.1step 1.2

3.1

In the associated graded algebra the bracket correction in [F1] has one lower tensor degree, so the symbol of an ordered monomial depends only on the corresponding commutative monomial. Step 2.1 shows that these symbols form a basis in each degree. Therefore the natural graded map $S (g) \to gr U (g)$ is an isomorphism.

F1step 2.1algebra∎

Depends on

Used by

The center of the enveloping algebra is polynomial on rank-many generators Corollary
The Harish-Chandra projection Definition
The leading PBW symbol of a central element is invariant Lemma
Harish-Chandra isomorphism for the center Theorem
The enveloping algebra is free over its center Theorem
Triangular decomposition from a chosen positive root system Theorem

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Sources

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)