Contents

group theory

Contents

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.

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)