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 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 above as the real Hopf fibration

/2 S 1 2 S 1 \array{ \mathbb{Z}/2 &\hookrightarrow& S^1 \\ && \downarrow^{\mathrlap{\cdot 2}} \\ && S^1 }

rotation groups in low dimensions:

sp. orth. groupspin grouppin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
Spin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)

see also

Last revised on March 22, 2019 at 09:13:08. See the history of this page for a list of all contributions to it.