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): $$\tilde{\Gamma} := \{x \in \mathcal{G}^\times : x^{*} V x^{-1} \subseteq V \}$$
 * 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: $$\mathrm{Pin} := \{x \in \Gamma : x x^\dagger = \pm 1 \} $$
 * (Spin group) the group of even unit versors: $$\mathrm{Spin} := \mathrm{Pin} \cap \mathcal{G}^{+}$$
 * the rotor group: $$\mathrm{Spin+} := \{x \in \mathrm{Spin} : x x^{\dagger} = 1 \}$$

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)).