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 \,.

(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.

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 February 20, 2013 03:09:08 by Urs Schreiber (80.81.16.253)