For a complex Lie algebra 𝔤, we can define its universal enveloping algebra U(𝔤) as the complex associative algebra T(𝔤)/I where T(𝔤) is the tensor algebra over 𝔤 and I is the ideal generated by {X⊗Y−Y⊗X−[X,Y]:X,Y∈𝔤}. We have the natural map ι:𝔤→U(𝔤). The universal enveloping algebra satisfies the following universal property:
For any complex associative algebra A and linear map f:𝔤→A satisfying [f(X),f(Y)]=f(X)f(Y)−f(Y)f(X) for all X,Y∈𝔤, there is a unique algebra homomorphism f∗:U(𝔤)→A such that f=f∗∘ι.
Theorem (Poincare-Birkhoff-Witt). Let {X1,…,Xm} be a basis for 𝔤. Then {(ιX1)n1⋯(ιXm)nm:n1,…,nm≥0} is a basis for U(𝔤).