A Lie group corresponding to the free Lie algebra and its universality

Abstract

Consider the real free Lie algebra $\mathfrak{fr}_n$ with generators $\omega_1$, \dots, $\omega_n$. Since it is positively graded, it has a completion $\overline{\mathfrak{fr}}_n$ consisting of formal series. By the Campbell--Hausdorff formula, we have a corresponding Lie group $\overline{\mathrm{Fr}}_n$. It is the set $\exp\bigl(\overline{\mathfrak{fr}}_n\bigr)$ in the completed universal enveloping algebra of $\mathfrak{fr}_n$. Also, the group $\overline{\mathrm{Fr}}_n$ is a 'submanifold' in the algebra of formal associative noncommutative series in $\omega_1$, \dots, $\omega_n$, the 'submanifold' is determined by a certain system of quadratic equations. We consider a certain dense subgroup $\mathrm{Fr}_n^\infty\subset \overline{\mathrm{Fr}}_n$ with a stronger (Polish) topology and show that any homomorphism $\pi$ from $\mathfrak{fr}_n$ to a real finite-dimensional Lie algebra $\mathfrak{g}$ can be integrated in a unique way to a homomorphism $\Pi$ from $\mathrm{Fr}_n^\infty$ to the corresponding simply connected Lie group $G$. If $\pi$ is surjective, then $\Pi$ also is surjective. Note that Pestov (1993) constructed a separable Banach--Lie group such that any separable Banach--Lie group is its quotient.