Exercise 6, § 2

Let $E$ be a module. Let $ME$ be the free algebra of $E$ (\emph{Algebra}, Chapter III, § 2, Exercise 13). Recall that $ME = \bigoplus_{n \geq 1} E_n$, where

and the algebra structure on $ME$ is defined by means of the canonical mappings $E_n \otimes E_m \to E_{n+m}$.

(a) Let $J$ be the smallest two-sided ideal of $ME$ containing the elements of the form $xx$ and $(xy)z + (yz)x + (zx)y$ with $x, y, z$ in $E$. Let $LE = ME/J$. Show that $J$ is a graded ideal of $ME$; deduce a graduation $(L^n E)_{n \geq 1}$ of $LE$.

(b) Show that $LE$ is a Lie algebra; it is called the \emph{free Lie algebra of the module} $E$. Show that for every Lie algebra $g$ and every linear mapping $f : E \to g$ there exists one and only one Lie algebra homomorphism $F : LE \to g$ which extends $f$.

(c) Define isomorphisms $E \to L^1 E$ and $\wedge^2 E \to L^2 E$. Show that $L^3 E$ is identified with the quotient of $E \otimes \wedge^2 E$ by the submodule generated by the elements

(d) Show that, when $E$ is a free module of basis $X$, $LE$ is identified with the free Lie algebra $L(X)$ defined in no. 2.

(e) Show that the canonical mapping of $E$ into the enveloping algebra $U(LE)$ of $LE$ can be extended to an isomorphism of the tensor algebra $TE$ onto $U(LE)$.

(f) Let $\sigma$ be the linear mapping of $\wedge^2 E$ into $T^2 E$ such that

Construct a module $E$ for which $\sigma$ is not injective. Deduce that for this module the canonical mapping of $LE$ into $U(LE)$ is not injective (compare with Exercise 9 of Chapter I, § 2).