# The existence of zero-divisors in the universal enveloping algebra of an infinite-dimensional Lie algebra

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?

• PBW = Poincaré-Birkhoff-Witt; en.wikipedia.org/wiki/Poincare-Birkhoff-Witt_theorem – YCor Dec 4 '15 at 19:30
• To answer your first question, I did try (not always successfully) to avoid full treatment of universal enveloping algebras in that old textbook. And there is nothing sinister about the case of infinite dimensional Lie algebras over a field (see Dixmier's book, or Bourbaki's Chapter 1 in their treatise on Lie groups and Lie algebras, or modern textbooks on Kac-Moody Lie algebras). Anyway, there are no zero-divisors in general. When looking at semisimple Lie algebras I tried to defer all these questions as much as possible. The exercises aren't always worded helpfully. – Jim Humphreys Dec 4 '15 at 20:19
• Thank you for the helpful clarifications and suggested readings. A professor at my university also pointed me in the direction of Kac-Moody Lie algebras, so that it is definitely something I will have to look at. – Stella Sue Gastineau Dec 7 '15 at 6:41