A Lie group corresponding to the free Lie algebra and its universality
Yury A. Neretin
TL;DR
This paper constructs a Lie group associated with the real free Lie algebra by passing to the completion $\overline{\mathfrak{fr}}_n$ and taking $\overline{\mathrm{Fr}}_n=\exp(\overline{\mathfrak{fr}}_n)$, with a quadratic-defining submanifold in the formal noncommutative series described by the coproduct $\Delta$. It then introduces dense Polish subgroups $Fr_n^\infty \subset \overline{\mathrm{Fr}}_n$ and $Fr_n^\xi$ via intersections with Banach space completions $\mathrm{Afr}_n^\infty$ and $\mathrm{Afr}_n^\xi$, and proves an explicit universality: any Lie algebra homomorphism $\pi:\mathfrak{fr}_n\to \mathfrak{g}$ integrates to a unique group homomorphism $\Pi:\mathrm{Fr}_n^\infty\to G$ (and similarly for $\mathrm{Fr}_n^\xi$) satisfying $\Pi(\exp(z))=\mathrm{Exp}(\pi(z))$. The approach uses ordered exponentials and Volterra-type integral equations to show density of the exponential image and yields a universal, contractible model from which linear groups can be obtained as quotients. The results relate to Pestov's separable Banach–Lie group construction and provide an explicit universal object for integrating finite-dimensional representations of the free Lie algebra.
Abstract
Consider the real free Lie algebra $\mathfrak{fr}_n$ with generators $ω_1$, \dots, $ω_n$. 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 $ω_1$, \dots, $ω_n$, 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 $π$ from $\mathfrak{fr}_n$ to a real finite-dimensional Lie algebra $\mathfrak{g}$ can be integrated in a unique way to a homomorphism $Π$ from $\mathrm{Fr}_n^\infty$ to the corresponding simply connected Lie group $G$. If $π$ is surjective, then $Π$ also is surjective. Note that Pestov (1993) constructed a separable Banach--Lie group such that any separable Banach--Lie group is its quotient.