Exercise 16, § 2

We suppose that $K$ is a $\mathbf{Q}$-algebra. If $g$ is a Lie algebra, we denote by $U(g)$ its enveloping algebra, by $\sigma$ the canonical mapping from $g$ into $U(g)$, and by $S(g)$ the symmetric algebra of the $K$-module $g$. There exists a linear mapping

and one such mapping only, such that $\eta(g)(x^n) = \sigma(x)^n$ for all $x \in g$ and all $n \in \mathbf{N}$. We have: We propose to show that $\eta(g)$ is bijective. (a) Prove that $\eta(g)$ is surjective, and that it is bijective when the $K$-module $g$ is free (use the Poincaré–Birkhoff–Witt theorem). (b) Let $\mathfrak{h}$ be an ideal of $g$. We denote by $s_{\mathfrak{h}}$ (resp. $u_{\mathfrak{h}}$) the ideal of $S(g)$ (resp. the two-sided ideal of $U(g)$) generated by $\mathfrak{h}$ (resp. by $\sigma(\mathfrak{h})$). The diagram is commutative, and its lines are exact. Deduce that the image $\tilde{u}_{\mathfrak{h}}$ of $s_{\mathfrak{h}}$ by $\eta(g)$ is contained in $u_{\mathfrak{h}}$ and that one has $\tilde{u}_{\mathfrak{h}} = u_{\mathfrak{h}}$ when $\eta(g)$ and $\eta(g/\mathfrak{h})$ are injective (in particular when the modules $g$ and $g/\mathfrak{h}$ are free). (c) Take for $g$ a free Lie algebra, with basic family $(X_1, \ldots, X_n, H)$, with $n \geq 0$, and take for $\mathfrak{h}$ the ideal of $g$ generated by $H$. Show that $g$ and $g/\mathfrak{h}$ are free modules. Deduce that, in this case, one has $\tilde{u}_{\mathfrak{h}} = u_{\mathfrak{h}}$. In particular, there exists $x \in s_{\mathfrak{h}}$ such that $(\eta(g))(x) = \sigma(X_1) \ldots \sigma(X_n) \sigma(H)$. (d) We return to the general case. Show that $u_{\mathfrak{h}}$ is generated over $K$ by the elements of the form $\sigma(x_1) \ldots \sigma(x_n) \sigma(h)$, with $n \geq 0$, $x_i \in g$, and $h \in \mathfrak{h}$ (notice that $u_{\mathfrak{h}}$ coincides with the left ideal generated by $\mathfrak{h}$). Show, using (c), that such an element belongs to $\tilde{u}_{\mathfrak{h}}$ (use a suitable homomorphism from a free Lie algebra into $g$). Deduce that $\tilde{u}_{\mathfrak{h}} = u_{\mathfrak{h}}$. (e) Show that, if $\eta(g)$ is bijective, the same is true of $\eta(g/\mathfrak{h})$. Deduce finally that, for every Lie algebra $g$, $\eta(g)$ is bijective (write $g$ as a quotient of a free Lie algebra), and that the canonical homomorphism is an isomorphism (« Poincaré–Birkhoff–Witt theorem » for Lie algebras over $\mathbf{Q}$-algebras).