nLab
circle group

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.

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)U(1) \hookrightarrow \mathbb{C} is a complex line bundle.

Revised on July 2, 2014 00:59:15 by Urs Schreiber (89.204.130.184)