Spin(2)

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

**spin geometry**, **string geometry**, **fivebrane geometry** …

The group $Spin(2)$ is the spin group in 2 dimensions, hence the double cover of SO(2)

$\array{
\mathbb{Z}/2
&\hookrightarrow&
Spin(2)
\\
&& \downarrow
\\
&& SO(2)
}$

In fact:

There is an isomorphism $Spin(2) \simeq SO(2) \simeq U(1)$ with the circle group which exhibits the double cover of SO(2) by Spin(2) as the real Hopf fibration

$\array{
\mathbb{Z}/2
&\hookrightarrow&
S^1
&\simeq&
Spin(2)
\\
&&
\downarrow^{\mathrlap{\cdot 2}}
&&
\downarrow^{\mathrlap{}}
\\
&&
S^1
&\simeq&
SO(2)
}$

Last revised on April 11, 2019 at 08:41:49. See the history of this page for a list of all contributions to it.