Lie algebras MOC

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 →𝑈(𝔤)¹ such that given any K-monoid 𝐴 and Lie algebra homomorphism 𝑓 :𝔤 →𝐴, there exists a unique unital algebra homomorphism ¯𝑓 :𝑈(𝔤) →𝐴 such that the following diagram commutes: lie

𝑈 :𝖫𝗂𝖾𝕂 →𝖠𝗌𝖠𝗅𝗀𝕂 has a unique extension to a functor such that 𝜄 :1 ⇒𝑈 :𝖫𝗂𝖾𝕂 →𝖫𝗂𝖾𝕂 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⊗𝑦−𝑦⊗𝑥−[𝑥,𝑦]

for 𝑥,𝑦 ∈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 ∘𝑗.

Proof

proof

Graded structure

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

Filtered structure

complete

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Footnotes

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