Editing One-parameter group

{{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.
==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==
In [[Lie theory]], one-parameter groups correspond to one-dimensional subspaces of the associated [[Lie algebra]]. The [[Lie group–Lie algebra correspondence]] is the basis of a science begun by [[Sophus Lie]] in the 1890s.

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

In his monograph ''Lie Groups'', [[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]] (1957) ''Lie Groups'', page 58, Cambridge Tracts in Mathematics and Mathematical Physics #46</ref>

==Physics==
In [[physics]], one-parameter groups describe [[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 [[derivative|differentiable]] [[symmetry group|symmetries]], then there is a [[Conservation law (physics)|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. The [[principle of relativity]] was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a [[world-line]]. Using the parametrization of the hyperbola with [[hyperbolic angle]], the theory of [[special relativity]] provided a calculus of relative motion with the one-parameter group indexed by [[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]] in 1910, and named by [[Alfred Robb]] the next year. The rapidity parameter amounts to the length of a [[versor#Hyperbolic versor|hyperbolic versor]], a concept of the nineteenth century. Mathematical physicists [[James Cockle (lawyer)|James Cockle]], [[William Kingdon Clifford]], and [[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 topology|subspace]] of <math>G</math> may carry a topology that is [[finer topology|coarser]] 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]] <math>T</math>, and <math>\varphi</math> is constructed by winding a straight line round <math>T</math> at an irrational slope.

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

== See also ==
* [[Integral curve]]
* [[One-parameter semigroup]]
* [[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]]