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 $\exp ({\overline{\mathfrak{fr}}}_n)$ 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 ${\omega }_1,\dots ,{\omega }_n$, and 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 $\Pi$ is also surjective. Note that Pestov (1993) constructed a separable Banach–Lie group such that any separable Banach–Lie group is its quotient.