One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism

$\varphi : \mathbb{R} \rightarrow G$

from the real line $\mathbb{R}$ (as an additive group) to some other topological group $G$. That means that it is not in fact a group,[1] strictly speaking; if $\varphi$ is injective then $\varphi(\mathbb{R})$, the image, will be a subgroup of $G$ that is isomorphic to $\mathbb{R}$ as 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.[2] It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension.

Discussion

That is, we start knowing only that

$\varphi(s+t) = \varphi(s)\varphi(t)$

where $s$, $t$ are the 'parameters' of group elements in $G$. We may have

$\varphi(s) = e$, the identity element in $G$,

for some $s \neq 0$. This happens for example if $G$ is the unit circle and

$\varphi(s) = e^{is}_{}$.

In that case the kernel of $\varphi$ consists of the integer multiples of $2\pi$.

The action of a one-parameter group on a set is known as a flow.

A technical complication is that $\varphi(\mathbb{R})$ as a subspace of $G$ may carry a topology that is coarser than that on $\mathbb{R}$; this may happen in cases where $\varphi$ is injective. Think for example of the case where $G$ is a torus $T$, and $\varphi$ 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

1. it has a definite parametrization,
2. the group homomorphism may not be injective, and
3. 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 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 $G$ 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 $\mathfrak{R}$, or to $\mathfrak{T}$, the additive group of real numbers $\mod 1$. In particular, every 1-dimensional Lie group is locally isomorphic to $\mathbb{R}$.

Physics

In physics, one-parameter groups describe dynamical systems.[3] 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 $r^2_{} = \pm{1}$.