One-parameter group

From Academic Kids

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

φ : RG

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. 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 element 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 of φ consists of the integer multiples of 2π.

The other 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 an 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.

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.

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