# One-parameter group

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

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

from the [real line](/source/real_line) <math>\mathbb{R}</math> (as an [additive group](/source/Abelian_group)) to some other [topological group](/source/topological_group) <math>G</math>. 
If <math>\varphi</math> is [injective](/source/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. Despite its name, "a one-parameter group" is not actually a group, but a homomorphism ''between'' groups.

One-parameter groups were introduced by [Sophus Lie](/source/Sophus_Lie) in 1893 to define [infinitesimal transformation](/source/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](/source/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](/source/Lie_algebra) that is used to describe a [Lie group](/source/Lie_group) of any dimension.

The [action](/source/action_(group_theory)) of a one-parameter group on a set is known as a [flow](/source/flow_(mathematics)). 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](/source/Integral_curve) of the vector field. The local flow of a vector field is used to define the [Lie derivative](/source/Lie_derivative) of tensor fields along the vector field.
==Definition==
A curve <math> \phi:\mathbb{R} \rightarrow G </math> is called one-parameter subgroup of <math> G </math> if it satisfies the condition<ref>{{cite book |last1=Nakahara |title=Geometry, topology, and physics |date=4 June 2003 |publisher=CRC Press |isbn=9780750306065 |pages=232}}</ref> 

:<math> \phi(t)\phi(s) = \phi(s+t) </math>.

==Examples==
[[File:Hyperbolic_rotation.gif|thumb|right|250px|The group of [squeeze mapping](/source/squeeze_mapping)s has one parameter.]]
In [Lie theory](/source/Lie_theory), one-parameter groups correspond to one-dimensional subspaces of the associated [Lie algebra](/source/Lie_algebra). The [Lie group–Lie algebra correspondence](/source/Lie_group%E2%80%93Lie_algebra_correspondence) is the basis of a science begun by [Sophus Lie](/source/Sophus_Lie) in the 1890s.

Another important case is seen in [functional analysis](/source/functional_analysis), with <math>G</math> being the group of [unitary operator](/source/unitary_operator)s on a [Hilbert space](/source/Hilbert_space). See [Stone's theorem on one-parameter unitary groups](/source/Stone's_theorem_on_one-parameter_unitary_groups).

In his monograph ''Lie Groups'', [P. M. Cohn](/source/P._M._Cohn) gave the following theorem:
:Any connected 1-dimensional Lie group is analytically isomorphic either to  the additive group of real numbers <math>\mathfrak{R}</math>, or to <math>\mathfrak{T}</math>, the additive group of real numbers <math>\mod 1</math>. In particular, every 1-dimensional Lie group is locally isomorphic to <math>\mathbb{R}</math>.<ref>[Paul Cohn](/source/Paul_Cohn) (1957) ''Lie Groups'', page 58, Cambridge Tracts in Mathematics and Mathematical Physics #46</ref>

==Physics==
In [physics](/source/physics), one-parameter groups describe [dynamical systems](/source/dynamical_systems).<ref>Zeidler, E. (1995) ''Applied Functional Analysis: Main Principles and Their Applications'' Springer-Verlag</ref> Furthermore, whenever a system of physical laws admits a one-parameter group of [differentiable](/source/derivative) [symmetries](/source/symmetry_group), then there is a [conserved quantity](/source/Conservation_law_(physics)), by [Noether's theorem](/source/Noether's_theorem).

In the study of [spacetime](/source/spacetime) the use of the [unit hyperbola](/source/unit_hyperbola) to calibrate spatio-temporal measurements has become common since [Hermann Minkowski](/source/Hermann_Minkowski) discussed it in 1908. The [principle of relativity](/source/principle_of_relativity) was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a [world-line](/source/world-line). Using the parametrization of the hyperbola with [hyperbolic angle](/source/hyperbolic_angle), the theory of [special relativity](/source/special_relativity) provided a calculus of relative motion with the one-parameter group indexed by [rapidity](/source/rapidity). The ''rapidity'' replaces the ''velocity'' in kinematics and dynamics of relativity theory. Since rapidity is unbounded, the one-parameter group it stands upon is non-compact. The rapidity concept was introduced by [E.T. Whittaker](/source/E.T._Whittaker) in 1910, and named by [Alfred Robb](/source/Alfred_Robb) the next year. The rapidity parameter amounts to the length of a [hyperbolic versor](/source/versor), a concept of the nineteenth century. Mathematical physicists [James Cockle](/source/James_Cockle_(lawyer)), [William Kingdon Clifford](/source/William_Kingdon_Clifford), and [Alexander Macfarlane](/source/Alexander_Macfarlane) had all employed in their writings an equivalent mapping of the Cartesian plane by operator <math>(\cosh{a} + r\sinh{a})</math>, where <math>a</math> is the hyperbolic angle and <math>r^2 = +1</math>.

==In GL(n,C)==
{{see also|Stone's theorem on one-parameter unitary groups}}
An important example in the theory of Lie groups arises when <math>G</math> is taken to be <math>\mathrm{GL}(n;\mathbb C)</math>, the group of invertible <math>n\times n</math> matrices with complex entries. In that case, a basic result is the following:<ref>{{harvnb|Hall|2015}} Theorem 2.14</ref>
:'''Theorem''': Suppose <math>\varphi : \mathbb{R} \rightarrow\mathrm{GL}(n;\mathbb C)</math> is a one-parameter group. Then there exists a unique <math>n\times n</math> matrix <math>X</math> such that
::<math>\varphi(t)=e^{tX}</math>
:for all <math>t\in\mathbb R</math>.
It follows from this result that <math>\varphi</math> is differentiable, even though this was not an assumption of the theorem. The matrix <math>X</math> can then be recovered from <math>\varphi</math> as
:<math>\left.\frac{d\varphi(t)}{dt}\right|_{t=0} = \left.\frac{d}{dt}\right|_{t=0}e^{tX}=\left.(Xe^{tX})\right|_{t=0} = Xe^0=X</math>.
This result can be used, for example, to show that any continuous homomorphism between matrix Lie groups is smooth.<ref>{{harvnb|Hall|2015}} Corollary 3.50</ref>

==Topology==
A technical complication is that <math>\varphi(\mathbb{R})</math> as a [subspace](/source/subspace_topology) of <math>G</math> may carry a topology that is [coarser](/source/finer_topology) than that on <math>\mathbb{R}</math>; this may happen in cases where <math>\varphi</math> is injective. Think for example of the case where <math>G</math> is a [torus](/source/torus) <math>T</math>, and <math>\varphi</math> is constructed by [winding a straight line round <math>T</math> at an irrational slope](/source/Irrational_winding_of_a_torus).

In that case the induced topology may not be the standard one of the real line.

== See also ==
* [Exponential map (Lie theory)](/source/Exponential_map_(Lie_theory))
* [Integral curve](/source/Integral_curve)
* [One-parameter semigroup](/source/One-parameter_semigroup)
* [Noether's theorem](/source/Noether's_theorem)

==References==
{{Wikibooks|Abstract Algebra|3x3 real matrices#One-parameter subgroups of GL(3,'''R''')|One-parameter subgroups of GL(3,'''R''')}}
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-3319134666}}.
{{Reflist}}

Category:Lie groups
Category:1 (number)
Category:Topological groups

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Adapted from the Wikipedia article [One-parameter group](https://en.wikipedia.org/wiki/One-parameter_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/One-parameter_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.