In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a one-parameter group or one-parameter subgroup usually means a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
group homomorphism
:
from the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
(as an
additive group) to some other
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
.
If
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
then
, the image, will be a subgroup of
that is isomorphic to
as an additive group.
One-parameter groups were introduced by
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius S ...
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
that is used to describe a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
of any dimension.
The
action of a one-parameter group on a set is known as a
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
. 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.
Examples
Such one-parameter groups are of basic importance in the theory of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, for which every element of the associated
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, with
being the group of
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
s on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. 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
.
Physics
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, one-parameter groups describe
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
. 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
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
the use of the
unit hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...
to calibrate spatio-temporal measurements has become common since
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
discussed it in 1908. The
principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.
For example, in the framework of special relativity the Maxwell equations ha ...
was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a
world-line
The world line (or worldline) of an object is the path that an object traces in 4- dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from ...
. Using the parametrization of the hyperbola with
hyperbolic angle, the theory of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
provided a calculus of relative motion with the one-parameter group indexed by
rapidity
In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with d ...
. 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
Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...
in 1910, and named by
Alfred Robb
Alfred Arthur Robb FRS (18 January 1873 in Belfast – 14 December 1936 in Castlereagh) was a Northern Irish physicist.
Biography
Robb studied at Queen's College, Belfast (BA 1894) and at St John's College, Cambridge (Tripos 1897, MA 1901). ...
the next year. The rapidity parameter amounts to the length of a
hyperbolic versor
In mathematics, a versor is a quaternion of norm one (a '' unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by W ...
, a concept of the nineteenth century. Mathematical physicists
James Cockle
Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.
Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charterh ...
,
William Kingdon Clifford, and
Alexander Macfarlane had all employed in their writings an equivalent mapping of the Cartesian plane by operator
, where
is the hyperbolic angle and
.
In GL(n,ℂ)
An important example in the theory of Lie groups arises when
is taken to be
, the group of invertible
matrices with complex entries. In that case, a basic result is the following:
:Theorem: Suppose
is a one-parameter group. Then there exists a unique
matrix
such that
::
:for all
.
It follows from this result that
is differentiable, even though this was not an assumption of the theorem. The matrix
can then be recovered from
as
:
.
This result can be used, for example, to show that any continuous homomorphism between matrix Lie groups is smooth.
[ Corollary 3.50]
Topology
A technical complication is that
as a
subspace of
may carry a topology that is
coarser than that on
; this may happen in cases where
is injective. Think for example of the case where
is a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
, and
is constructed by winding a straight line round
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
In mathematics, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary diffe ...
*
Noether's theorem
References
* .
{{Reflist
Lie groups
1 (number)
Topological groups