The PBW theorem states:
$\omega:\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras.
Where $\mathfrak {S} $ is the symmetric tensor algebra of a Lie algebra $ L $.
Where $\mathfrak {E} $ is the graded algebra of the universal enveloping algebra of $ L $.
Does anyone have an intuitive reason why this is true?
I know this stems from the map of $\phi:\mathfrak {T} \mapsto \mathfrak {E} $, where $\mathfrak {T} $ is tensor algrbra of $ L $. The kernel of this map is suppose to be the same ideal of the tensor algebra that is used to make the symmetric algebra. In other words ker $\phi $ is suppose to equal the kernel of the map $\mathfrak {T} \mapsto \mathfrak {S} $, but I don't see why this true intuitively.
Thanks. All help us appreciated.