Indefinite pin group

Definition

Let $(V,q)$ be a nondegenerate real quadratic space of signature $(p,q)$ , with negative directions listed first, and form its Clifford algebra using $v^2=-q(v)1$ . The indefinite pin group is the subgroup

\operatorname{Pin}(p,q) \subseteq\operatorname{Cl}(V,q)^\times

generated by the vectors $v\in V$ satisfying $q(v)=\pm1$ .

Orthogonal covering

The twisted adjoint action on $V$ gives a surjective homomorphism

\rho:\operatorname{Pin}(p,q)\longrightarrow O(p,q)

with kernel $\{\pm1\}$ . A generating unit vector acts as the reflection in its orthogonal hyperplane, and the Cartan–Dieudonné theorem supplies surjectivity.

The even part of the pin group is the indefinite spin group $\operatorname{Spin}(p,q)$ , which maps onto $SO(p,q)$ .

Convention

With the alternative Clifford convention $v^2=+q(v)$ , the names $\operatorname{Pin}(p,q)$ and $\operatorname{Pin}(q,p)$ may be interchanged. A pin-group statement should therefore specify both the signature order and the Clifford relation.

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.