definition. Spin group [figueroa2010spin] [spin-000E]
definition. Spin group [figueroa2010spin] [spin-000E]
The Pin group Pin \((V)\) of \((V, Q)\) is the subgroup of (the group of units of) \(C \ell (V)\) generated by \(v \in \mathrm {V}\) with \(\mathrm {Q}(v)= \pm 1\). In other words, every element of \(\operatorname {Pin}(\mathrm {V})\) is of the form \(u_1 \cdots u_r\) where \(u_i \in \mathrm {V}\) and \(\mathrm {Q}\left (u_i\right )= \pm 1\). We will write \(\operatorname {Pin}(s, t)\) for \(\operatorname {Pin}\left (\mathbb {R}^{s, t}\right )\) and \(\operatorname {Pin}(n)\) for \(\operatorname {Pin}(n, 0)\).
The spin group of \((V, Q)\) is the intersection \[ \operatorname {Spin}(V)=\operatorname {Pin}(V) \cap C \ell (V)_0 . \] Equivalently, it consists of elements \(u_1 \cdots u_{2 p}\), where \(u_i \in \mathrm {V}\) and \(\mathrm {Q}\left (u_i\right )= \pm 1\). We will write \(\operatorname {Spin}(s, t)\) for \(\operatorname {Spin}\left (\mathbb {R}^{s, t}\right )\) and \(\operatorname {Spin}(n)\) for \(\operatorname {Spin}(n, 0)\).