nLab circle group

Contents

Context

Group Theory

Definition
Properties
Related concepts

Definition

The circle group $\mathbb{T}$ is equivalently (isomorphically)

the quotient group $\mathbb{R}/\mathbb{Z}$ of the additive group of real numbers by the additive group of integers, induced by the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{R}$ ;
the unitary group $\mathrm{U}(1)$ ;
the special orthogonal group $SO(2)$ ;
the subgroup of the group of units $\mathbb{C}^\times$ of the field of complex numbers (its multiplicative group) 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

0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{T} \to 0 \,,

the “real exponential 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.

$U(1)$ is the compact real form of the multiplicative group $\mathbb{G}_m = \mathbb{C}^\times$ over the complex numbers, see at form of an algebraic group – Circle group and multiplicative group.

Related concepts

A principal bundle with structure group the circle group is a circle bundle. The associated bundle under the standard linear representation of $U(1) \hookrightarrow \mathbb{C}$ is a complex line bundle.
cyclic group, Q/Z
circle 2-group
circle, unit circle, circle type, minimal simplicial circle
torus

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)