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:

sp. orth. groupspin grouppin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)
SO(9)Spin(9)

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.