Exercise 19, § 6

Let $g$ be a reductive Lie algebra, $U$ its enveloping algebra, $Z$ the centre of $U$ and $V$ the subspace of $U$ generated by the elements of the form $uv - vu$ ($u \in U, v \in U$).

(a) $U$ is the direct sum of $Z$ and $V$. (Apply Proposition 6 of § 3 to the representation $x \mapsto \mathrm{ad}_U x$ of $g$ on $U$, observing that, in the decomposition $U = \sum_n U^n$ of § 2, no. 7, Corollary 4 to Theorem 1, the $U^n$ are stable under this representation.)

(b) Let $u \mapsto u^h$ be the projector of $U$ onto $Z$ parallel to $V$. Show that $(uv)^h = (vu)^h, (zu)^h = zu^h$ for all $u \in U, v \in U, z \in Z$.

(c) Let R be a two-sided ideal of U. Show that $R = (R \cap Z) + (R \cap V)$. (To show that the component in V of an element r of R belongs to R, reduce it to the case where K is algebraically closed; observe that R is stable under the representation $\varphi$ of U on U which extends $x \mapsto \mathrm{ad}_U x$; decompose r into its components in the $U^n$; finally, apply Corollary 2 to Theorem 1 of Algebra, Chapter VIII, § 4, no. 2 to the restriction of $\varphi$ to $U^n$.)

(d) Let $R'$ be an ideal of Z. Let $R_1$ be the two-sided ideal of U generated by $R'$. Show that $R_1 \cap Z = R'$. (Use (b).)