Editing One-parameter group (section)

{{Short description|Lie group homomorphism from the real numbers}}
In [[mathematics]], a '''one-parameter group''' or '''one-parameter subgroup''' usually means a [[continuous (topology)|continuous]] [[group homomorphism]]

:<math>\varphi : \mathbb{R} \rightarrow G</math>

from the [[real line]] <math>\mathbb{R}</math> (as an [[Abelian group|additive group]]) to some other [[topological group]] <math>G</math>. 
If <math>\varphi</math> is [[injective]] then <math>\varphi(\mathbb{R})</math>, the image, will be a subgroup of <math>G</math> that is isomorphic to <math>\mathbb{R}</math> as an additive group.

One-parameter groups were introduced by [[Sophus Lie]] in 1893 to define [[infinitesimal transformation]]s. According to Lie, an ''infinitesimal transformation'' is an infinitely small transformation of the one-parameter group that it generates.<ref>[[Sophus Lie]] (1893) [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie-_infinite_continuous_groups_-_i.pdf Vorlesungen über Continuierliche Gruppen], English translation by D.H. Delphenich, §8, link from Neo-classical Physics</ref> It is these infinitesimal transformations that generate a [[Lie algebra]] that is used to describe a [[Lie group]] of any dimension.

The [[action (group theory)|action]] of a one-parameter group on a set is known as a [[flow (mathematics)|flow]]. A smooth vector field on a manifold, at a point, induces a ''local flow'' - a one parameter group of local diffeomorphisms, sending points along [[Integral curve#Generalization to differentiable manifolds|integral curves]] of the vector field. The local flow of a vector field is used to define the [[Lie derivative]] of tensor fields along the vector field.