definition. Spin group [rosen2019geometric] [spin-000L]
definition. Spin group [rosen2019geometric] [spin-000L]
Let \(V\) be an inner product space. We denote by \(\Delta V\) the standard Clifford algebra \((\wedge V,+, \Delta )\) defined by the Clifford product \(\Delta \) on the space of multivectors in \(V\).
Let \(V\) be an inner product space. The Clifford cone of \(V\) is the multiplicative group \(\widehat {\triangle } V \subset \triangle V\) generated by nonsingular vectors, that is, vectors \(v\) such that \(\langle v\rangle ^2 \neq 0\). More precisely, \(q \in \widehat {\triangle } V\) if there are finitely many nonsingular vectors \(v_1, \ldots , v_k \in V\) such that \[ q=v_1 \Delta \cdots \Delta v_k . \]Let \(w \in \triangle V\). Then \(w \in \widehat {\triangle } V\) if and only if \(w\) is invertible and \(\widehat {w} v w^{-1} \in V\) for all \(v \in V\).
In this case \(w\) can be written as a product of at most \(\operatorname {dim} V\) nonsingular vectors, and \(\bar {w} w=w \bar {w} \in \mathbf {R} \backslash \{0\}\).
Let \(V\) be an inner product space. Define the orthogonal, special orthogonal, pin, and spin groups \[ \begin {aligned} \mathrm {O}(V) & :=\{\text { isometries } T: V \rightarrow V\} \subset \mathcal {L}(V), \\ \mathrm {SO}(V) & :=\{T \in \mathrm {O}(V) ; \operatorname {det} T=+1\} \subset \mathcal {L}(V), \\ \operatorname {Pin}(V) & :=\left \{q \in \widehat {\triangle } V ;\langle q\rangle ^2= \pm 1\right \} \subset \triangle V, \\ \operatorname {Spin}(V) & :=\left \{q \in \operatorname {Pin}(V) ; q \in \triangle ^{\mathrm {ev}} V\right \} \subset \triangle ^{\mathrm {ev}} V . \end {aligned} \] We call \(T \in \operatorname {SO}(V)\) a rotation and we call \(q \in \operatorname {Spin}(V)\) a rotor.