nLab
Spin(2)

Contents

Context

Group Theory

Spin geometry

Contents

Idea

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

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

In fact:

Proposition

There is an isomorphism Spin(2)SO(2)U(1)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

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

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