nLab
circle group

Contents

Contents

Definition

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

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

0𝕋0, 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 AA is simply a homomorphism from AA to the circle group.

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

rotation groups in low dimensions:

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)

see also

Last revised on March 22, 2019 at 09:12:53. See the history of this page for a list of all contributions to it.