Exercise 4, § 6

Let $X$ be a finite set, and let $\hat{L} = \hat{L}(X)$. Endow $\hat{L}$ with the Hausdorff composition law $(a, b) \mapsto a + b$, which we denote simply by $a . b$. If $t \in K$, $a \in \hat{L}$, put $a^t = ta$.

a) Let $\mathscr{H}$ be a Hall set relative to X. If $m \in \mathscr{H}$, denote by $m_L$ the corresponding element of the Lie algebra $L(X)$ (§ 2, No. 11), and by $m_{\mathscr{H}}$ the basic commutator of the group $\hat{L}$ defined by $m$ (§ 5, No. 4, Remark). If $m$ is of length $l$, prove that $m_{\mathscr{H}} = m_L + \mu$, where $\mu$ is of filtration $\geqslant l + 1$ (reason by induction on $l$). Deduce that every element $w$ of the group $\hat{L}$ can be written uniquely in the form

with $\alpha(m) \in K$, the product being convergent in the topological group $\hat{L}$.

b) Let $P$ be the set of prime numbers, and let $c$ be an integer $\geqslant 1$. Take $K = \mathbf{Q}$. Prove that the group $\hat{L}/\hat{L}_c$ is identified with the $P$-envelope $F_P^c$ of the group $F^c = F(X)/C^cF(X)$, cf. § 5, exerc. 6.

c) Take for X a set with two elements $\{U, V\}$. Prove the existence of two families $\alpha(m), \beta(m)$ of rational numbers such that

For example

d) Let $g$ be a nilpotent $\mathbf{Q}$-Lie algebra, endowed with the Hausdorff group law. Let $u, v$ be in $g$. Prove that one has

where $\alpha$ and $\beta$ are the families of rational numbers defined above (« inversion of the Hausdorff formula »). (Use the continuous homomorphism $\varphi : \hat{L} \to g$ such that $\varphi(U) = u$, $\varphi(V) = v$, and remark that it is also a homomorphism for the Hausdorff group law.)

e) Let G be a nilpotent divisible group without torsion (i.e. P-divisible and without P-torsion). If $u \in G$, $t \in \mathbf{Q}$, $u^t$ is defined (§ 4, exerc. 16). If $u \in G$, $v \in G$, define $u + v$ and $[u, v]$ by the formulas of d) above. Prove that G is thus endowed with a structure of nilpotent $\mathbf{Q}$-Lie algebra and that the corresponding Hausdorff law is the group law given on G (verify these assertions when G is a group $F_P^c$, cf. b), and pass from there to the general case by using homomorphisms of $F_P^c$ into G).

Let $f$ be a mapping from G into a nilpotent divisible torsion-free group $G'$. Show that $f$ is a homomorphism (of groups) if and only if it is a homomorphism of Lie algebras. (The Hausdorff law therefore defines an isomorphism of the “category” of nilpotent $\mathbf{Q}$-Lie algebras onto that of nilpotent divisible torsion-free groups.)