One-parameter group
Encyclopedia
In mathematics
, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
from the real line
R (as an additive group
) to some other topological group
G. That means that it is not in fact a group
, strictly speaking; if φ is injective then φ(R), the image, will be a subgroup of G that is isomorphic to R as additive group.
where s, t are the 'parameters' of group elements in G. We may have
for some s ≠ 0. This happens for example if G is the unit circle and
In that case the kernel
of φ consists of the integer multiples of 2π.
The action of a one-parameter group on a set is known as a flow
.
A technical complication is that φ(R) as subspace
of G may carry a topology that is coarser than that on R; this may happen in cases where φ is injective. Think for example of the case where G is a torus
T, and φ is constructed by winding a straight line round T at an irrational slope.
Therefore a one-parameter group or one-parameter subgroup has to be distinguished from a group or subgroup itself, for the three reasons
s, 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 G being the group of unitary operator
s 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:
, 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 spacio-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 hyperbolic versor, a concept of the nineteenth century. Mathematical physicists James Cockle, William Kingdon Clifford
, and Alexander Macfarlane
had all employed in their writings an equivalent mapping of the Cartesian plane by operator (cosh a + r sinh a), where a is the hyperbolic angle and r 2 = +1.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
- φ : R → G
from the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R (as an additive group
Additive group
An additive group may refer to:*an abelian group, when it is written using the symbol + for its binary operation*a group scheme representing the underlying-additive-group functor...
) to some other topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
G. That means that it is not in fact a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, strictly speaking; if φ is injective then φ(R), the image, will be a subgroup of G that is isomorphic to R as additive group.
Discussion
That is, we start knowing only that- φ (s + t) = φ(s)φ(t)
where s, t are the 'parameters' of group elements in G. We may have
- φ(s) = e, the identity elementIdentity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
in G,
for some s ≠ 0. This happens for example if G is the unit circle and
- φ(s) = eis.
In that case the kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of φ consists of the integer multiples of 2π.
The action of a one-parameter group on a set is known as a flow
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
.
A technical complication is that φ(R) as subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
of G may carry a topology that is coarser than that on R; this may happen in cases where φ is injective. Think for example of the case where G is a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
T, and φ is constructed by winding a straight line round T at an irrational slope.
Therefore a one-parameter group or one-parameter subgroup has to be distinguished from a group or subgroup itself, for the three reasons
- it has a definite parametrizationParametrizationParametrization is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object....
, - the group homomorphism may not be injective, and
- the induced topology may not be the standard one of the real line.
Examples
Such one-parameter groups are of basic importance in the theory of Lie groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, for which every element of the associated Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
defines such a homomorphism, the exponential map
Exponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
. In the case of matrix groups it is given by the matrix exponential
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
.
Another important case is seen in functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, with G being the group of unitary operator
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
s on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. See Stone's theorem on one-parameter unitary groups
Stone's theorem on one-parameter unitary groups
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators...
.
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 mod 1. In particular, every 1-dimensional Lie group is locally isomorphic to R.
Physics
In physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, one-parameter groups describe dynamical systems. Furthermore, whenever a system of physical laws admits a one-parameter group of differentiable
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
symmetries
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
, then there is a conserved quantity
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
, by Noether's theorem
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
.
In the study of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
the use of the unit hyperbola
Unit hyperbola
In geometry, the unit hyperbola is the set of points in the Cartesian plane that satisfies x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial lengthWhereas the unit circle surrounds its center, the unit hyperbola requires the...
to calibrate spacio-temporal measurements has become common since Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
discussed it in 1908. The principle of relativity
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....
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
Hyperbolic angle
In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...
, the theory of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
provided a calculus of relative motion with the one-parameter group indexed by rapidity
Rapidity
In relativity, rapidity is an alternative to speed as a framework for measuring motion. On parallel velocities rapidities are simply additive, unlike speeds at relativistic velocities. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The...
. 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
Alfred Robb
Alfred Arthur Robb or Alfred A. Robb FRS was a British physicist.Robb studied at Queen's College in Belfast and at St John’s College in Cambridge. He then proceeded to University of Göttingen, where guided by Woldemar Voigt, he wrote his dissertation on the Zeeman effect. He also worked under J. J...
the next year. The rapidity parameter amounts to the length of a hyperbolic versor, a concept of the nineteenth century. Mathematical physicists James Cockle, William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics...
, and Alexander Macfarlane
Alexander Macfarlane
Alexander Macfarlane was a Nova Scotia lawyer and political figure. He was a member of the Canadian Senate from 1870 to 1898. His surname also appears as McFarlane in some sources....
had all employed in their writings an equivalent mapping of the Cartesian plane by operator (cosh a + r sinh a), where a is the hyperbolic angle and r 2 = +1.