I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject.
It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also an interesting problemto determine when a certain algebra cannot satisfy any polynomial identity!
For instance, the Weyl algebra $A_n$ is simple, hence primitive. If it were a PI-algebra, by Kaplansky Theorem, it would be finite dimensional over its center. However, we know that $A_n$ is an infinite dimension vector space, and its center restrict to the scalars.
Fix a base field of characteristic 0., and let $\mathfrak{g}$ be a finite dimensional Lie algebra. I result I've heard a number of times is that the enveloping algebra can be a PI-algebra if, and only if $\mathfrak{g}$ is abelian - in which case $U(\mathfrak{g})$ is just the polynomial algebra.
Does someone have a reference for this? Or can indicate a sketch of a proof? Or, provide a counter-example to show that this folklric fact is, in fact, false?
