nLab Spin(2)

Idea

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:

Proposition

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

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

Related concepts

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification