One-parameter group
Lie group homomorphism from the real numbers / From Wikipedia, the free encyclopedia
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In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
- :\mathbb {R} \rightarrow G}
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from the real line (as an additive group) to some other topological group . If is injective then , the image, will be a subgroup of that is isomorphic to as an additive group.
One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations. According to Lie, an infinitesimal transformation is an infinitely small transformation of the one-parameter group that it generates.[1] It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension.
The action of a one-parameter group on a set is known as a 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 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.