Circle group

In mathematics, the circle group, denoted by ${\displaystyle \mathbb {T} }$ or ⁠${\displaystyle S^{1}}$⁠, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers $${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}.}$$ The circle group plays a fundamental role in many areas of mathematics.

When a given unit complex number ${\displaystyle u}$ multiplies the other points of the circle, the effect is to rotate them through an angle determined by ${\displaystyle u}$. In this way, the circle group becomes the group of symmetries of the circle which preserve its orientation (do not flip it). Composition of two rotations is the ordinary multiplication of complex numbers. Multiplication is commutative, ⁠${\displaystyle z_{1}z_{2}=z_{2}z_{1}}$⁠, making the circle group commutative (an abelian group); correspondingly several rotations of the plane can be composed in any order with the same result. Rotations can alternately be parametrized by the angle measure ⁠${\displaystyle \theta }$⁠, which is related to ⁠${\displaystyle z}$⁠ by the complex exponential function: $${\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .}$$

The circle group is sometimes denoted by ${\displaystyle U(1)}$, which is the unitary group of ${\displaystyle 1\times 1}$ complex matrices. It is structurally the same as (i.e., isomorphic to) the group of 2-dimensional rotation matrices, i.e., the special orthogonal group ⁠${\displaystyle \mathrm {SO} (2)}$⁠. The angle measure gives a periodic parameterization of the circle group, so the group is often treated as a periodic interval ${\displaystyle [0,2\pi ]}$ with the endpoints glued together, and the group operation as addition modulo ${\displaystyle 2\pi }$: the usual addition of angles. The notation ${\displaystyle \mathbb {T} }$ for the circle group stems from the fact that a circle is a 1-dimensional torus. More generally, ${\displaystyle \mathbb {T} ^{n}}$ (the direct product of ${\displaystyle \mathbb {T} }$ with itself ${\displaystyle n}$ times) is geometrically an ${\displaystyle n}$-torus. The notation ⁠${\displaystyle S^{1}}$⁠ stands for 1-dimensional sphere.

The circle group has applications throughout mathematics, especially in advanced mathematics. It is the group underlying classical Fourier series. It is dual to the additive group of the integers. It also has applications throughout topology and mathematical physics. It is the group underlying electromagnetism. Electromagnetic theory can be formulated as a theory of bundles associated to the circle group, which are primary objects of study in homotopy theory and algebraic topology.