Exercise 10, § 2

Let l be a free Lie algebra admitting a finite basic family with n elements and let u be a surjective endomorphism of l. Show that u is an isomorphism. (Write $\operatorname{gr}^n(l) = \mathscr{C}^n/\mathscr{C}^{n+1}$ and denote by $\operatorname{gr}^n(u)$ the endomorphism of $\operatorname{gr}^n(l)$ derived from u; note that $\operatorname{gr}^n(u)$ is surjective; as $\operatorname{gr}^n(l)$ is a free K-module of finite rank, deduce that $\operatorname{gr}^n(u)$ is bijective since the kernel of u is contained in $\bigcap_n \mathscr{C}^n l$, which is $\{0\}$.)

† When g is not graded, we do not know whether the condition "H^2(g, M) = \{0\} for every g-module M" implies that g is a free Lie algebra.

If $(y_1, \ldots, y_m)$ is a generating family of $l$, show that $m \geq n$ and that we have equality if and only if $(y_1, \ldots, y_m)$ is a basic family.