Let $k$ be a field and $\mathfrak{g}$ a nonnegatively graded Lie $k$-algebra. Let $\xi\in\mathfrak{g}^{\mathrm{even}}\backslash\{0\}$. How do you prove that $\xi$ is a left and right nonzero divisor on the universal enveloping algebra $U\mathfrak{g}$? I can see it if $\xi$ is central but in general I don't know how to prove it.
Non-zero divisors over universal enveloping algebras
2
$\begingroup$
abstract-algebra
lie-algebras
noncommutative-algebra
graded-rings
-
0Fir finite-dimensional Lie algebras one can use the Poincare-Birkhoff-Witt theorem, right? – 2017-01-13
-
0I'm not assuming $\mathfrak{g}$ to be finitely generated, just to be finitely generated degreewise. – 2017-01-14