Circle group

In mathematics, the circle group is the group of all complex numbers on the unit circle under multiplication. This group is often given additional topological or analytic structure, and indeed the circle group has fundamental importance in the study of Lie groups.

Definitions

If we think in terms of the polar representation of complex numbers, then the point

[itex](\cos \theta , \,\sin \theta ) \,\![itex]

is identified with

[itex] \cos \theta + i\sin \theta = e^{i\theta}. \,\![itex]

The group law is to take the sum of the angles, modulo integer multiples of 2π. Thus, the circle group is isomorphic to the group R/2πZ under the mapping

[itex] e^{i\theta} \mapsto \theta + 2\pi\mathbb{Z} \,\![itex]

which, after rescaling, is isomorphic simply to R/Z. If complex numbers are realised as 2 × 2 real matrices, (see complex number), then the circle group is seen to be isomorphic to the special orthogonal group SO(2, R), under the association

[itex] e^{i\theta} \mapsto \begin{pmatrix}

\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{pmatrix}. [itex]

Lie group structure

The circle group is also called U(1), the first unitary group. In fact, the circle group is a compact Lie group, and any compact Lie group G of dimension ≥ 1 has a subgroup isomorphic to it. That means that, thinking in terms of symmetry, a compact symmetry group acting continuously can be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen for example at rotational invariance, and spontaneous symmetry breaking.

The circle group has many subgroups, but its only closed subgroups consist of roots of unity: there is one such, that is cyclic of order n, for each integer n ≥ 1.

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