definition. Spin group [renaud2020clifford] [spin-000G]
definition. Spin group [renaud2020clifford] [spin-000G]
The Clifford group \(\Gamma (p, q)\) is the (multiplicative) group generated by invertible 1-vectors in \(R^{p, q}\).
The Pin group \(\operatorname {Pin}(p, q)\). \[ \operatorname {Pin}(p, q)=\{g \in \Gamma (p, q): g \widetilde {g}= \pm 1\} . \] So if \(g \in \operatorname {Pin}(p, q), \widetilde {g}\) is a scalar multiple of \(g^{-1}\). This is not true for arbitrary elements of \(\Gamma (p, q)\). The Spin group \(\operatorname {Spin}(p, q)\). This is the subgroup of \(\operatorname {Pin}(p, q)\) consisting of even elements only, i.e. \[ \operatorname {Spin}(p, q)=\operatorname {Pin}(p, q) \cap C l^{+}(p, q) . \] The \(\operatorname {Spin}\) group \(\operatorname {Spin}(p, q)\) has the further subgroup \[ \operatorname {Spin}^{\dagger }(p, q)=\{g \in \operatorname {Spin}(p, q): g \widetilde {g}=+1\} . \] \(\operatorname {Pin}(p, q)\), \(\operatorname {Spin}(p, q)\) and \(\operatorname {Spin}^{\dagger }(p, q)\) are respectively the two-fold covering groups of \(O(p, q), S O(p, q)\) and \(S O^{\dagger }(p, q)\) (where \(S O^{\dagger }(p, q)\) is the connected component of \(S O(p, q)\) ).