definition. Spin group [hitzer2012introduction] [spin-000I]
definition. Spin group [hitzer2012introduction] [spin-000I]
A versor refers to a Clifford monomial (product expression) composed of invertible vectors. It is called a rotor, or spinor, if the number of vectors is even. It is called a unit versor if its magnitude is 1.
Every versor \(A=a_1 \ldots a_r, \quad a_1, \ldots , a_r \in \mathbb {R}^2, r \in \mathbb {N}\) has an inverse \[ A^{-1}=a_r^{-1} \ldots a_1^{-1}=a_r \ldots a_1 /\left (a_1^2 \ldots a_r^2\right ), \] such that \[ A A^{-1}=A^{-1} A=1 . \]This makes the set of all versors in \(C l(2,0)\) a group, the so called Lipschitz group with symbol \(\Gamma (2,0)\), also called Clifford group or versor group. Versor transformations apply via outermorphisms to all elements of a Clifford algebra. It is the group of all reflections and rotations of \(\mathbb {R}^2\).
The normalized subgroup of versors is called pin group \[ \operatorname {Pin}(2,0)=\{A \in \Gamma (2,0) \mid A \widetilde {A}= \pm 1\} . \] In the case of \(C l(2,0)\) we have \[ \begin {aligned} & \operatorname {Pin}(2,0) \\ & =\left \{a \in \mathbb {R}^2 \mid a^2=1\right \} \cup \left \{A \mid A=\cos \varphi +e_{12} \sin \varphi , \varphi \in \mathbb {R}\right \} . \end {aligned} \] The pin group has an even subgroup, called spin group \[ \operatorname {Spin}(2,0)=\operatorname {Pin}(2,0) \cap C l^{+}(2,0) . \] The spin group has in general a spin plus subgroup \[\operatorname {Spin}_{+}(2,0)=\{A \in \operatorname {Spin}(2,0) \mid A \widetilde {A}=+1\}.\]