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

Consider the real free Lie algebra \mathfrak{fr}_{n} with generators \omega_{1},\ldots,\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},\ldots,\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.