nLab
local Lie group

Idea

A local Lie group (rarely also called Lie groupuscule) is a local / stalkwise version of a Lie group, containing information about the product operation in arbitrary small open neighborhoods of the unit element; the multiplication is defined only when the factors are sufficiently close to the unit element, and if the consecutive products of triples in both ways are defined they are associative. To every local Lie group one functorially associates its Lie algebra.

Every real Lie algebra is a Lie algebra of some local Lie group. Or in more modern and precise phrasing, the category of real local Lie groups is equivalent to the category of finite-dimensional real Lie algebras. This has been proved by Sophus Lie as his famous third theorem. The extension to the global Lie theory has been possible only after works of Élie Cartan, who extended the equivalence to the equivalence between the category of real Lie algebras and connected simply connected Lie groups.

Related concepts

Examples of sequences of infinitesimal and local structures

first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
	$\leftarrow$ differentiation		integration $\to$
derivative		Taylor series		germ		smooth function
tangent vector		jet		germ of curve		curve
Lie algebra		formal group		local Lie group		Lie group
Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization