circle group




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


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.

Last revised on February 13, 2019 at 07:24:08. See the history of this page for a list of all contributions to it.