The intuition for this problem comes from $\S$17 Exercise 1 Humphreys' Introduction to Lie Algebras and Representation Theory which essentially asks us to use PBW in order to prove that if a Lie algebra $L$ is finite-dimensional, then the universal enveloping algebra $\mathfrak{U}=\mathfrak{U}(L)$ has no zero divisors. So my question is the following:
Why does Humphreys impose the condition that $L$ be finite-dimensional? My original guess was the problem arrises because of the ambiguity of the an polynomial ring with infinite variables. Is this correct?
In his construction of the universal enveloping algebra via a universal property (pg. 90), Humphreys parenthetically says that $L$ can be infinite-dimensional. Then when the problem of existence is addressed (pg. 91), Humphreys naturally identifies $\mathfrak{U}$ with the quotient of tensor algebra $\mathfrak{T}=\mathfrak{T}(L)$ by the two-sided ideal $J\subset\mathfrak{T}$ generated by elements in $T^0\oplus T^2$ of the form $x\otimes y-y\otimes x-[xy]$. But when Humphreys defines the tensor algebra $\mathfrak{T}(V)$ (pg. 89), he only does so for finite dimensional vector spaces $V$.
Does he imposes finite dimensional only because he only really talked about PBW for finite-dimensional Lie algebras? Or is there something more sinister that happen when we move to infinite-dimension Lie algebras?
