Indefinite spin group

Definition

Let $(V,q)$ be a real quadratic space of signature $(p,q)$ , where this collection lists the $p$ negative and then the $q$ positive directions, and form its Clifford algebra using $v^2=-q(v)1$ . The indefinite spin group is the even part of the indefinite pin group:

\operatorname{Spin}(p,q)=\operatorname{Pin}(p,q)\cap\operatorname{Cl}^0(V,q).

Equivalently, it is generated by products of an even number of vectors $v$ satisfying $q(v)=\pm1$ . The ambient $\operatorname{Pin}(p,q)$ records products of arbitrary parity.

Orthogonal covering

The restriction of the twisted adjoint action is a surjective homomorphism

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

with kernel $\{\pm1\}$ . In indefinite signature, $\operatorname{Spin}(p,q)$ need not be connected. The full preimage of the identity component and its component convention are treated by the restricted spin group.

Convention warning

Authors also use Clifford relations $v^2=+q(v)$ or interchange the order of $p$ and $q$ . These choices can alter the notation for the Clifford algebra and its components but not the associated orthogonal group. The positive-definite construction is treated separately at spin group.

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.