definition. Spin group [lawson2016spin] [spin-0002]
definition. Spin group [lawson2016spin] [spin-0002]
Let \(V\) be a vector space over the commutative field \(k\) and suppose \(q\) is a quadratic form on \(V\).
We now consider the multiplicative group of units in the Clifford algebra \(C \ell (V, q)\) associated to \(V\), which is defined to be the subset \[ C \ell ^{\times }(V, q) \equiv \left \{\varphi \in C \ell (V, q): \exists \varphi ^{-1} \text { with } \varphi ^{-1} \varphi =\varphi \varphi ^{-1}=1\right \} \]This group contains all elements \(v \in V\) with \(q(v) \neq 0\).
The group of units always acts naturally as automorphisms of the algebra. That is, there is a homomorphism \[ \mathrm {Ad}: \mathrm {C} \ell ^{\times }(V, q) \longrightarrow \operatorname {Aut}(\mathrm {C} \ell (V, q)) \] called the adjoint representation, which is given by \[ \operatorname {Ad}_{\varphi }(x) \equiv \varphi \times \varphi ^{-1} \] The Pin group of \((V, q)\) is the subgroup \(\operatorname {Pin}(V, q)\) of \(\mathrm {P}(V, q)\) generated by the elements \(v \in V\) with \(q(v)= \pm 1\). The associated Spin group of \((V, q)\) is defined by \[ \operatorname {Spin}(V, q)=\operatorname {Pin}(V, q) \cap C \ell ^0(V, q) . \]