Exercise 13, § 2

(In this exercise, it is assumed that every projective module over K is free; this is for example the case if K is a principal ideal domain.)

Let $l = \sum_{n=1}^\infty l_n$ be a graded Lie algebra admitting a basic family B formed of homogeneous elements, and let $h$ be a graded Lie subalgebra of $l$, which is a direct factor in $l$ as a module. (a) For every $i \geq 0$, let $l^{(i)}$ be the graded subalgebra of $l$ such that

We have $l = l^{(0)} \supset l^{(1)} \supset \ldots \supset h$. Now prove, by reasoning by induction on $i$, the existence of a basic family $B^{(i)}$ of $l^{(i)}$, formed of homogeneous elements, and such that the elements of $B^{(i-1)}$ and of $B^{(i)}$ of degree $< i - 1$ are the same. (Suppose that $B^{(i-1)}$ has been constructed. Let $m_i$ be the intersection of $l_i = l_i^{(i-1)}$ with the subalgebra generated by the $h_j$ for $j < i$, and let $b_i$ be the submodule of $l_i$ generated by the elements of degree $i$ of $B^{(i-1)}$. The induction hypothesis implies that $l_i = m_i \oplus b_i$. Since $m_i \subset h_i$, one can decompose $b_i$ into $b_i = y_i \oplus z_i$, in such a way that

¹ When g is not graded, it is not known whether the condition

implies that g is a free Lie algebra.

$\mathfrak{h}_i = m_i \oplus \mathfrak{z}_i;$ in view of the assumption made on $K$, the modules $\eta_i$ and $\mathfrak{z}_i$ are free; by changing, if necessary, the elements of degree $i$ of $B^{(i-1)}$, one can therefore suppose that $\mathfrak{z}_i$ is generated by a subset of $B^{(i-1)}$. Applying Exercise 11 to $l^{(i-1)}$ and to its ideal $l^{(i)}$, we deduce from it a basic family $B^{(i)}$ of $l^{(i)}$ having the desired properties.) (b) Deduce from (a) the fact that $\mathfrak{h}$ possesses a basic family formed of homogeneous elements.