Exercise 16, § 2

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 of $g$ into $U(g)$ and by $S(g)$ the symmetric algebra of the $K$-module $g$. There exists one and only one linear mapping

such that $\eta(g)(x^n) = \sigma(x)^n$ for all $x \in g$ and all $n \in \mathbf{N}$. Then 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 $g$. Let $s_\mathfrak{h}$ (resp. $u_\mathfrak{h}$) denote the ideal of $S(g)$ (resp. the two-sided ideal of $U(g)$) generated by $\mathfrak{h}$ (resp. $\sigma(\mathfrak{h})$). The diagram

is commutative and its rows are exact. Deduce that the image $\tilde{u}_\mathfrak{h}$ of $s_\mathfrak{h}$ under $\eta(g)$ is contained in $u_\mathfrak{h}$ and that $\tilde{u}_\mathfrak{h} = u_\mathfrak{h}$ when $\eta(g)$ and $\eta(g/\mathfrak{h})$ are injective (in particular, when the module $g$ and $g/\mathfrak{h}$ are free).

(c) Take $g$ to be a free Lie algebra with basic family $(X_1, \ldots, X_n, H)$, where $n \geq 0$, and take $h$ to be the ideal of $g$ generated by $H$. Show that $g$ and $g/h$ are free modules. Deduce that, in that case, $\tilde{u}_h = u_h$. In particular, there exists $x \in s_h$ such that $(\eta(g))(x) = \sigma(X_1) \ldots \sigma(X_n) \sigma(H)$.

(d) We return to the general case. Show that $u_h$ 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 $h \in h$ (note that $u_h$ coincides with the left ideal generated by $h$). Show, using (c), that such an element belongs to $\tilde{u}_h$ (use a suitable homomorphism of a free Lie algebra into $g$). Deduce that $\tilde{u}_h = u_h$.

(e) Show that, if $\eta(g)$ is bijective, so is $\eta(g/h)$. Deduce finally that, for every Lie algebra $g$, $\eta(g)$ is bijective (write $g$ as the 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).