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Concept
One-parameter group
Summary
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
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. 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.
Such one-parameter groups are of basic importance in the theory of Lie groups, for which every element of the associated Lie algebra defines such a homomorphism, the exponential map. In the case of matrix groups it is given by the matrix exponential.
Another important case is seen in functional analysis, with being the group of unitary operators on a Hilbert space. See Stone's theorem on one-parameter unitary groups.
In his 1957 monograph Lie Groups, P. M. Cohn gives the following theorem on page 58:
Any connected 1-dimensional Lie group is analytically isomorphic either to the additive group of real numbers , or to , the additive group of real numbers . In particular, every 1-dimensional Lie group is locally isomorphic to .
In physics, one-parameter groups describe dynamical systems. Furthermore, whenever a system of physical laws admits a one-parameter group of differentiable symmetries, then there is a conserved quantity, by Noether's theorem.
In the study of spacetime the use of the unit hyperbola to calibrate spatio-temporal measurements has become common since Hermann Minkowski discussed it in 1908.
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