User:Jheald/sandbox/GA/Some groups found in GA

(From Lundholm, Svensson (2009) "Clifford algebra, geometric algebra, and applications", Archiv 0907.5356v1.pdf; p. 58) -- or Lounesto §17.2, p.220.

The following are some of the groups that can be identified in a Clifford Algebra ${\mathcal {G}}$:

• the group of all invertible elements: ${\mathcal {G}}^{\times }:=\{x\in {\mathcal {G}}:\exists y\in {\mathcal {G}}:xy=yx=1\}$
• the Lipschitz group (after Rudolf Lipschitz 1880/86): ${\tilde {\Gamma }}:=\{x\in {\mathcal {G}}^{\times }:x^{*}Vx^{-1}\subseteq V\}$
• the versor group: $\Gamma :=\{v_{1}v_{2}\dots v_{k}\in G:v_{i}\in V^{\times }\}$

  ... but these are blades, not versors -- has a line got lost?

• (Pin group) the group of unit versors: $\mathrm {Pin} :=\{x\in \Gamma :xx^{\dagger }=\pm 1\}$
• (Spin group) the group of even unit versors: $\mathrm {Spin} :=\mathrm {Pin} \cap {\mathcal {G}}^{+}$
• the rotor group: $\mathrm {Spin+} :=\{x\in \mathrm {Spin} :xx^{\dagger }=1\}$

where $V^{\times }:=\{v\in V:v^{2}\neq 0\}$ is the set of invertible vectors.

• It is an important property of Clifford algebras that $\Gamma ={\tilde {\Gamma }}$
  • (Really? surely a typical rotor is a member of the first, but it is not a blade (the second group)).