Universal enveloping algebra

Let $𝔤$ be a Lie algebra over $𝕂$ . The universal enveloping algebra $𝑈 (𝔤)$ is the most general $𝕂$ -monoid with the Lie bracket of $𝔤$ as its commutator, as formalized by the Universal property and the Poincaré-Birkhoff-Witt theorem. In particular, this means any Lie algebra representation of $𝔤$ uniquely corresponds to a $𝑈 (𝔤)$ -module, motivating the abuse of terminology module over a Lie algebra.

Universal property

Let $𝔤$ be a Lie algebra over $𝕂$ . The universal enveloping algebra is a pair consisting of a $𝕂$ -monoid $𝑈 (𝔤)$ and a Lie algebra homomorphism $𝜄 : 𝔤 \to 𝑈 (𝔤)$ ¹ such that given any $𝕂$ -monoid $𝐴$ and Lie algebra homomorphism $𝑓 : 𝔤 \to 𝐴$ , there exists a unique unital algebra homomorphism $¯ 𝑓 : 𝑈 (𝔤) \to 𝐴$ such that the following diagram commutes: #m/def/lie

$𝑈 : 𝖫 𝗂 𝖾 𝕂 \to 𝖠 𝗌 𝖠 𝗅 𝗀 𝕂$ has a unique extension to a functor such that $𝜄 : 1 \Rightarrow 𝑈 : 𝖫 𝗂 𝖾 𝕂 \to 𝖫 𝗂 𝖾 𝕂$ becomes a natural transformation.

It is not immediately clear from the universal property that $𝜄$ should be an injection, but this is guaranteed by the Poincaré-Birkhoff-Witt theorem, so indeed $𝑈 (𝔤)$ contains $𝔤$ as a Lie subalgebra, whence every Lie algebra is a Lie subalgebra of some unital associative algebra.

Construction

Let $𝑇 ∙ 𝔤$ be the tensor algebra of $𝔤$ with inclusion $𝑗 : 𝔤 ↪ 𝑇 ∙ 𝔤$ and let $𝐼$ be the (two-sided) ideal generated by any terms of the form

𝑥 \otimes 𝑦 - 𝑦 \otimes 𝑥 - [𝑥, 𝑦]

for $𝑥, 𝑦 \in 𝔤$ . We construct the universal enveloping algebra as the quotient module

𝑈 (𝔤) = 𝑇 ∙ 𝔤 / 𝐼

with its natural projection $𝜋 : 𝑇 ∙ 𝑉 ↠ 𝑈 (𝔤)$ . The map $𝜄 = 𝜋 \circ 𝑗$ .

Proof

#missing/proof

Graded structure

Let $𝔤$ be a $𝔄$ -graded Lie algebra. Then $𝑈 (𝔤)$ is a graded algebra such that $𝔤 𝛼 𝔤 𝛽 \leq 𝔤 𝛼 + 𝛽$ . This is the same as the gradation given by the quotient graded algebra in the construction above.

Filtered structure

#to/complete

#state/tidy | #lang/en | #SemBr

As usual we regard an associative algebra as a Lie algebra under its commutator. ↩