definition. Spin group [ruhe2024clifford] [spin-000M]
Motivation E. 39 (The problem of generalizing the definition of the Spin group). For a positive definite quadratic form \(\mathfrak {q}\) on the real vector space \(V=\mathbb {R}^n\) with \(n \geq 3\) the \(\operatorname {Spin}\) group \(\operatorname {Spin}(n)\) is defined via the kernel of the Spinor norm (=extended quadratic form on \(\mathrm {Cl}(V, \mathfrak {q})\) ) restricted to the special Clifford group \(\Gamma ^{[0]}(V, \mathfrak {q})\) :
\[
\operatorname {Spin}(n):=\operatorname {ker}\left (\overline {\mathfrak {q}}: \Gamma ^{[0]}(V, \mathfrak {q}) \rightarrow \mathbb {R}^{\times }\right )=\left \{w \in \Gamma ^{[0]}(V, \mathfrak {q}) \mid \overline {\mathfrak {q}}(w)=1\right \}=\left .\overline {\mathfrak {q}}\right |_{\Gamma ^{[0]}(V, \mathfrak {q})} ^{-1}(1) .
\]
\(\operatorname {Spin}(n)\) is thus a normal subgroup of the special Clifford group \(\Gamma ^{[0]}(V, \mathfrak {q})\), and, as it turns out, a double cover of the special orthogonal group \(\mathrm {SO}(n)\) via the twisted conjugation \(\rho \). The latter can be summarized by the short exact sequence:
\[
1 \longrightarrow \{ \pm 1\} \xrightarrow {\text { incl }} \operatorname {Spin}(n) \xrightarrow {\rho } \mathrm {SO}(n) \longrightarrow 1 .
\]
definition. Spin group [ruhe2024clifford] [spin-000M]
We intend to generalize this in several directions: 1. from Spin to Pin group, 2. from \(\mathbb {R}^n\) to vector spaces \(V\) over general fields \(\mathbb {F}\) with \(\operatorname {ch}(\mathbb {F}) \neq 2\), 3. from non-degenerate to degenerate quadratic forms \(\mathfrak {q}\), 4. from positive (semi-)definite to non-definite quadratic forms \(\mathfrak {q}\). This comes with several challenges and ambiguities.
Definition E. 40 (The real Pin group and the real Spin group). Let \(V\) be a finite dimensional \(\mathbb {R}\)-vector space \(V, \operatorname {dim} V=n<\infty \), and \(\mathfrak {q} a\) (possibly degenerate) quadratic form on \(V\). We define the (real) Pin group and (real) Spin group, resp., of \((V, \mathfrak {q})\) as the following subquotients of the Clifford group. \(\Gamma (V, \mathfrak {q})\) and its even parity part \(\Gamma ^{[0]}(V, \mathfrak {q})\), resp.: \[ \begin {aligned} \operatorname {Pin}(V, \mathfrak {q}) & :=\{x \in \Gamma (V, \mathfrak {q}) \mid \overline {\mathfrak {q}}(x) \in \{ \pm 1\}\} / \bigwedge ^{[*]}(\mathcal {R}) \\ \operatorname {Spin}_{\infty }(V, \mathfrak {q}) & :=\left \{x \in \Gamma ^{[0]}(V, \mathfrak {q}) \mid \overline {\mathfrak {q}}(x) \in \{ \pm 1\}\right \} / \bigwedge ^{[*]}(\mathcal {R}) \end {aligned} \] Corollary E.41. Let \((V, \mathfrak {q})\) be a finite dimensional quadratic vector space over \(\mathbb {R}\). Then the twisted conjugation induces a well-defined and surjective group homomorphism onto the group of radical preserving orthogonal automorphisms of \((V, \mathfrak {q})\) : \[ \rho : \operatorname {Pin}(V, \mathfrak {q}) \rightarrow \mathrm {O}_{\mathcal {R}}(V, \mathfrak {q}), \] with kernel: \[ \operatorname {ker}\left (\rho : \underset {\sim \text { in }}{\operatorname {Pin}}(V, \mathfrak {q}) \rightarrow \mathrm {O}_{\mathcal {R}}(V, \mathfrak {q})\right )=\{ \pm 1\} . \] Correspondingly, for the \(\operatorname {Spin}(V, \mathfrak {q})\) group. In short, we have short exact sequences: \[ \begin {aligned} & 1 \longrightarrow \{ \pm 1\} \xrightarrow {\text { incl }} \operatorname {Pin}(V, \mathfrak {q}) \xrightarrow {\rho } \mathrm {O}_{\mathcal {R}}(V, \mathfrak {q}) \longrightarrow 1, \\ & 1 \longrightarrow \{ \pm 1\} \xrightarrow {\text { incl }} \operatorname {Spin}(V, \mathfrak {q}) \xrightarrow {\rho } \operatorname {SO}_{\mathcal {R}}(V, \mathfrak {q}) \longrightarrow 1 . \end {aligned} \]