definition. Spin group [woit2012lie] [spin-000C]
definition. Spin group [woit2012lie] [spin-000C]
There are several equivalent possible ways to go about defining the \(\operatorname {Spin}(n)\) groups as groups of invertible elements in the Clifford algebra.
1. One can define \(\operatorname {Spin}(n)\) in terms of invertible elements \(\tilde {g}\) of \(C_{\text {even }}(n)\) that leave the space \(V=\mathbf {R}^n\) invariant under conjugation \[ \tilde {g} V \tilde {g}^{-1} \subset V \] 2. One can show that, for \(v, w \in V\), \[ v \rightarrow v-2 \frac {Q(v, w)}{Q(w, w)} w=-w v w / Q(w, w)=w v w^{-1} \] is reflection in the hyperplane perpendicular to \(w\). Then \(\operatorname {Pin}(n)\) is defined as the group generated by such reflections with \(||w||^2=1\) . \(\operatorname {Spin}(n)\) is the subgroup of \(\operatorname {Pin}(n)\) of even elements. Any rotation can be implemented as an even number of reflections (Cartan-Dieudonné) theorem.3. One can define the Lie algebra of \(\operatorname {Spin}(n)\) in terms of quadratic elements of the Clifford algebra.