Poincaré–Birkhoff–Witt theorem - Imported Knowlpedia Content

Statement

Let $\mathfrak g$ be a Lie algebra over a field $k$ , and let $(x_i)_{i\in I}$ be a totally ordered basis. The Poincaré–Birkhoff–Witt theorem states that the ordered monomials

1,\qquad x_{i_1}x_{i_2}\cdots x_{i_n} \quad(n\geq1,\ i_1\leq i_2\leq\cdots\leq i_n)

form a $k$ -basis of the universal enveloping algebra $U(\mathfrak g)$ . In particular, the canonical Lie-algebra map $\mathfrak g\to U(\mathfrak g)$ is injective, so $\mathfrak g$ may be identified with its image in the associative algebra $U(\mathfrak g)$ .

Filtration form

Filter $U(\mathfrak g)$ by word length. Commuting two adjacent generators changes their product by $[x,y]$ , which has lower filtration degree. Consequently the associated graded algebra is commutative, and PBW gives a canonical graded-algebra isomorphism

\operatorname{Sym}(\mathfrak g)\xrightarrow{\;\sim\;}\operatorname{gr}U(\mathfrak g).

This formulation is independent of the chosen ordered basis Dixmier, §2.1.

Symmetrization in characteristic zero

When $k$ has characteristic zero, the symmetrization map

x_1\cdots x_n\longmapsto \frac{1}{n!}\sum_{\sigma\in S_n} x_{\sigma(1)}\cdots x_{\sigma(n)}

is a canonical vector-space isomorphism $\operatorname{Sym}(\mathfrak g)\to U(\mathfrak g)$ . It is generally not an algebra homomorphism: PBW identifies the associated graded multiplication, not the original noncommutative multiplication in $U(\mathfrak g)$ .

Structural consequences

The theorem transfers polynomial-degree arguments from the symmetric algebra to $U(\mathfrak g)$ . It underlies the construction of central and Casimir elements, controls filtrations on $U(\mathfrak g)$ -modules, and ensures that a Lie-algebra representation extends uniquely to an action of $U(\mathfrak g)$ . Over a general commutative base ring, a PBW statement requires additional hypotheses such as projectivity of the underlying module.

References

J. Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996. DOI record. Relevant: §2.1 on PBW and the canonical filtration.
A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. DOI record. Relevant: Chapter V, §2.