Contents

Definition

The circle group $𝕋$ is equivalently (isomorphically)

• the quotient group $ℝ/\mathbb{Z}$ of the additive group of real numbers by the additive group of integers, induced by the canonical embedding $\mathbb{Z}\hookrightarrow ℝ$;

• the unitary group $\mathrm{U}(1)$;

• the special orthogonal group $\mathrm{SO}(2)$;

• the subgroup of the group ${ℂ}^{\times }$ of units of the field of complex numbers given by those of any fixed positive modulus (standardly $1$).

Properties

For general abstract properties usually the first characterization is the most important one. Notably it implies that the circle group fits into a short exact sequence

(On the other hand, the last characterization is usually preferred when one wants to be concrete.)

A character of an abelian group $A$ is simply a homomorphism from $A$ to the circle group.

Related concepts

A principal bundle with structure group the circle group is a circle bundle. The canonically corresponding associated bundle under the standard representation of $U(1)\hookrightarrow ℂ$ is a complex line bundle.