Exercise 10, § 2

Let g be a finite-dimensional Lie algebra over a field, U its enveloping algebra and $U_n$ the set of elements of U of filtration $\leq n$. (a) Let $x \in U_n, y \in U_n$ be two non-zero elements. Show that there exists $u \in U, v \in V$ such that $ux = vy$. (Compare $\dim(U_p x) = \dim U_p, \dim(U_p y) = \dim U_p$ and $\dim U_{p+n}$. Deduce that $U_p x \cap U_p y \neq \{0\}$ for $p$ sufficiently large.) (b) Show that U admits a field of left quotients (Algebra, Chapter I, § 9, Exercise 8) which is at the same time a field of right quotients.