Exercise 2, § 3

Let $g$ be a Lie algebra which is a free module; it is identified by means of $\sigma : g \to Ug$ with its image in the enveloping algebra $Ug$. Let $U^+g$ be the kernel of the canonical homomorphism $Ug \to K$. Let $i$ be a mapping of $X$ into $g$ such that $i(X)$ generates $g$ as a Lie algebra. (a) Show that $U^+g$ is generated by $i(X)$ as a left $Ug$-module. (b) Show the equivalence of the following properties: (i) $g$ is free with basic family $i : X \to g$. (ii) The family $i$ is a basis of the left $Ug$-module $U^+g$. (The implication (i) $\Rightarrow$ (ii) follows from Exercise 1 (a) using the isomorphism $Ug \to A(X)$. To show (ii) $\Rightarrow$ (i), apply Exercise 1 (b) to the kernel of the homomorphism $A(X) \to Ug$ defined by $i$.)