2024/11/17 by Neretin, Yury A.
#17B01 #17B35 #22E15 #22E65 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2411.11184
Consider the real free Lie algebra \mathfrakfrn with generators ω1, …, ωn. Since it is positively graded, it has a completion \mathfrakfrn consisting of formal series. By the Campbell--Hausdorff formula, we have a corresponding Lie group Frn. It is the set exp(\mathfrakfrn) in the completed universal enveloping algebra of \mathfrakfrn. Also, the group Frn is a 'submanifold' in the algebra of formal associative noncommutative series in ω1, …, ωn, the 'submanifold' is determined by a certain system of quadratic equations. We consider a certain dense subgroup Frn^∞⊂ Frn with a stronger (Polish) topology and show that any homomorphism π from \mathfrakfrn to a real finite-dimensional Lie algebra \mathfrakg can be integrated in a unique way to a homomorphism Π from Frn^∞ 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.