Poincaré-Birkhoff-Witt theorem

Let $𝔤$ be a Lie algebra over $𝕂$ and $𝑈 (𝔤)$ its universal enveloping algebra with the canonical Lie algebra homomorphism $𝜄 : 𝔤 \to 𝑈 (𝔤)$ . For some ordered basis $(𝑥 𝑗) 𝑗 \in 𝐽$ of $𝔤$ , the universal enveloping algebra $𝑈 (𝔤)$ has a basis consisting of ordered products $𝑥 𝑗 1 \dots 𝑥 𝑗 𝑛$ for $𝑛 \geq 1$ , $𝑗 ℓ \in 𝐽$ , $𝑗 1 \leq \dots \leq 𝑗 𝑛$ . #m/thm/lie It follows that $𝜄$ is injective.

Proof

#missing/proof

Corollaries

Let $𝔣, 𝔥 \leq 𝖫 𝗂 𝖾 𝕂 𝔤$ and $𝔤 = 𝖵 𝖾 𝖼 𝗍 𝕂 𝔣 \oplus 𝔥$ . Then the following defines a $𝕂$ -linear isomorphism:
$𝑈 (𝔣) \otimes 𝕂 𝑈 (𝔥) \to 𝑈 (𝔤) 𝑥 \otimes 𝑦 \mapsto 𝑥 𝑦$
Therewithal if $𝑉$ is an $𝔥$ -module then the following defines a $𝕂$ -linear isomorphism
$𝑈 (𝔣) \otimes 𝕂 𝑉 \to 𝑈 (𝔤) \otimes 𝑈 (𝔥) 𝑉 𝑥 \otimes 𝑣 \mapsto 𝑥 \otimes 𝑣$
where the codomain is the induced module $I n d 𝔤 𝔥 𝑉$ .¹
The associated graded algebra of $𝑈 (𝔤)$ is the symmetric algebra $𝑆 ∙ 𝔤$ .

#state/develop | #lang/en | #SemBr

1988. Vertex operator algebras and the Monster, §1.5, p. 16 ↩

Poincaré-Birkhoff-Witt theorem

Corollaries

Footnotes