The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is a flat deformation of $S(\mathfrak{g})$, so these algebras should be similar in many ways. Does at least this general similarity hold?

3$\begingroup$ The universal enveloping algebra of a finite dimensional Lie algebra is a socalled Galegbra, hence is left and right Noetherian (see e.g. singular.unikl.de/Manual/315/sing_510.htm). Note that this includes quantized enveloping algebra as well. $\endgroup$ – Adrien Jan 10 '13 at 10:47

$\begingroup$ Adrien, thank you very much for the reference. $\endgroup$ – Oleg Jan 10 '13 at 11:36
Yes, if a filtered ring $R$ has the property that its associated graded ring is Noetherian, then $R$ is Noetherian. Universal enveloping algebras have a PBW filtration such that the associated graded algebra is $S(\mathfrak{g})$. This is proved in Noncommutative Noetherian Rings by McConnell, Robson, Small  see sections 1.6 and 1.7.

$\begingroup$ You're welcome  do you have a reference for the flat deformation idea you mentioned in the question? It seems like an interesting point of view. $\endgroup$ – M T Jan 10 '13 at 12:59

1$\begingroup$ I don't have a reference, but I can just explain you how I see it. $U(g)$ is the quotient of the tensor algebra $T(g)$ modulo the relations $x\otimes y−y\otimes x−[x,y]$ for $x,y\in g$. Let $k$ be the base field, $t$ be a variable and $B$ be the quotient of the algebra $T(g)\otimes_k k[t]$ modulo the relations $x\otimes y−y\otimes x−t[x,y]$. Then $B$ is (I hope  I haven't checked) a free $k[t]$module with the usual PBW basis and an algebra over $k[t]$, the fiber of $B$ over the point $t=0$ is $S(g)$, and the fiber over $t−a$ for $a\in k^*$ is $U(g)$. $\endgroup$ – Oleg Jan 10 '13 at 13:34

$\begingroup$ I think I got the idea of the construction from Chapter 6 of Eisenbud's Commutative Algebra with a View toward Algebraic Geometry. $\endgroup$ – Oleg Jan 10 '13 at 13:41

$\begingroup$ And here is more to deformation: mathoverflow.net/questions/41142/… $\endgroup$ – Oleg Jan 10 '13 at 17:10