Editing Clifford algebra (section)

== Spin and pin groups ==
{{details|Spin group|Pin group|Spinor}}

In this section we assume that {{math|''V''}} is finite-dimensional and its bilinear form is non-singular.

The [[pin group]] {{math|Pin<sub>''V''</sub>(''K'')}} is the subgroup of the Lipschitz group {{math|Γ}} of elements of spinor norm {{math|1}}, and similarly the [[spin group]] {{math|Spin<sub>''V''</sub>(''K'')}} is the subgroup of elements of [[Orthogonal group#The Dickson invariant|Dickson invariant]] {{math|0}} in {{math|Pin<sub>''V''</sub>(''K'')}}. When the characteristic is not {{math|2}}, these are the elements of determinant {{math|1}}. The spin group usually has index {{math|2}} in the pin group.

Recall from the previous section that there is a homomorphism from the Lipschitz group onto the orthogonal group. We define the [[special orthogonal group]] to be the image of {{math|Γ<sup>0</sup>}}. If {{math|''K''}} does not have characteristic {{math|2}} this is just the group of elements of the orthogonal group of determinant {{math|1}}. If {{math|''K''}} does have characteristic {{math|2}}, then all elements of the orthogonal group have determinant {{math|1}}, and the special orthogonal group is the set of elements of Dickson invariant {{math|0}}.

There is a homomorphism from the pin group to the orthogonal group. The image consists of the elements of spinor norm {{math|1 ∈ ''K''<sup>×</sup>{{px2}}/{{px2}}(''K''<sup>×</sup>)<sup>2</sup>}}. The kernel consists of the elements {{math|+1}} and {{math|−1}}, and has order {{math|2}} unless {{math|''K''}} has characteristic {{math|2}}. Similarly there is a homomorphism from the Spin group to the special orthogonal group of&nbsp;{{math|''V''}}.

In the common case when {{math|''V''}} is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when {{math|''V''}} has dimension at least {{math|3}}. Further the kernel of this homomorphism consists of {{math|1}} and {{math|−1}}. So in this case the spin group, {{math|Spin(''n'')}}, is a double cover of {{math|SO(''n'')}}. Note, however, that the simple connectedness of the spin group is not true in general: if {{math|''V''}} is {{math|'''R'''<sup>''p'',''q''</sup>}} for {{math|''p''}} and {{math|''q''}} both at least {{math|2}} then the spin group is not simply connected. In this case the algebraic group {{math|Spin<sub>''p'',''q''</sub>}} is simply connected as an algebraic group, even though its group of real valued points {{math|Spin<sub>''p'',''q''</sub>('''R''')}} is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.{{which|date=July 2019}}