Faithful Cyclic Modules for Enveloping Algebras and Sklyanin Algebras

Let $U$ be the enveloping algebra of a finite dimensional nonabelian Lie algebra $\mathfrak{g}$ over a field of characteristic zero. We show that there is an open nonempty open subset $X$ of $U_1=\mathfrak{g}\oplus K$ such that $U\mathrm{/}Ux$ is faithful for all $x\in X$. We prove similar results for homogenized enveloping algebras and for the three dimensional Sklyanin algebras at points of infinite order. It would be interesting to know if there is a common generalization of these results.