Restricted spin group

Core idea

Let

\rho:\operatorname{Spin}(p,q)\longrightarrow SO(p,q)

be the covering from the indefinite spin group. In this collection, the restricted spin group is the full preimage

\operatorname{Spin}^+(p,q) := \rho^{-1}\bigl(SO^+(p,q)\bigr)

of the identity component of the special orthogonal group.

Its restricted homomorphism

\rho:\operatorname{Spin}^+(p,q)\longrightarrow SO^+(p,q)

is a two-sheeted covering with kernel $\{\pm1\}$ .

Components

In the standard mixed signatures of total dimension at least $3$ , this full preimage is connected and is the identity component of $\operatorname{Spin}(p,q)$ . Signature $(1,1)$ is exceptional: the full preimage has two components, while restricting further to the identity component makes the map to $SO^+(1,1)$ one-to-one.

Thus some authors use $\operatorname{Spin}_0(p,q)$ for a group that can differ in low dimension from the full preimage denoted $\operatorname{Spin}^+(p,q)$ here.

Lorentzian four-space

For the $(-+++)$ convention in four dimensions,

\operatorname{Spin}^+(1,3) \cong SL(2,\mathbb C)_{\mathbb R},

and the covering becomes the $SL(2,\mathbb C)$ spin cover of $SO^+(1,3)$ .

References

H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989, Chapter I, §§2–4. Publisher record.
Ian R. Porteous, Clifford Algebras and the Classical Groups, Cambridge University Press, 1995, Chapters 13–15. Publisher record.