Editing Geometric algebra (section)

=== Subgroups of the Lipschitz group ===
We denote the grade involution as {{tmath|1= \widehat{S} }} and reversion as {{tmath|1= \widetilde{S} }}.

Although the Lipschitz group (defined as {{tmath|1= \{ S \in \mathcal{G}^{\times} \mid \widehat{S} V S^{-1} \subseteq V \} }}) and the versor group (defined as {{tmath|1= \textstyle \{ \prod_{i=0}^{k} v_i \mid v_i \in V^{\times}, k \in \N \} }}) have divergent definitions, they are the same group.  Lundholm defines the {{tmath|1= \operatorname{Pin} }}, {{tmath|1= \operatorname{Spin} }}, and {{tmath|1= \operatorname{Spin}^{+} }} subgroups of the Lipschitz group.{{sfn|ps=|Lundholm|Svensson|2009|p=58}}

{| class="wikitable"
|-
 ! Subgroup !! Definition !! GA term
|-
 | <math>\Gamma</math> || <math> \{ S \in \mathcal{G}^{\times} \mid \widehat{S} V S^{-1} \subseteq V \} </math> || versors
|-
 | <math>\operatorname{Pin}</math> || <math> \{ S \in \Gamma \mid S \widetilde{S} = \pm 1 \} </math> || unit versors
|-
 | <math>\operatorname{Spin}</math> || <math> {\operatorname{Pin}} \cap \mathcal{G}^{[0]} </math> || even unit versors
|-
 | <math>\operatorname{Spin}^{+}</math> || <math> \{ S \in \operatorname{Spin} \mid S \widetilde{S} = 1 \} </math> || rotors
|-
|}

Multiple analyses of spinors use GA as a representation.{{sfn|ps=|Francis|Kosowsky|2008}}