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

TL;DR

This paper constructs a universal Lie group associated with the free Lie algebra, demonstrating its properties and universality in integrating Lie algebra homomorphisms into Lie group homomorphisms.

Contribution

It introduces a Lie group corresponding to the free Lie algebra and proves its universality for integrating homomorphisms into finite-dimensional Lie groups.

Findings

The group $ar{ ext{Fr}}_n$ is a submanifold in formal series algebra.

Any homomorphism from the free Lie algebra extends uniquely to the group.

Surjective Lie algebra homomorphisms induce surjective group homomorphisms.

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.