Editing Rotation matrix (section)

=== Spin group ===
{{main|Spin group|Rotation group SO(3)#Connection between SO(3) and SU(2)}}
The Lie group of {{math|''n'' × ''n''}} rotation matrices, {{math|SO(''n'')}}, is not [[simply connected space|simply connected]], so Lie theory tells us it is a homomorphic image of a [[universal covering group]]. Often the covering group, which in this case is called the [[spin group]] denoted by {{math|Spin(''n'')}}, is simpler and more natural to work with.<ref>{{Harvnb |Baker|2003|loc=Ch.&nbsp;5}}; {{Harvnb|Fulton|Harris|1991|pp=299–315}}</ref>

In the case of planar rotations, SO(2) is topologically a [[circle]], {{math|''S''<sup>1</sup>}}. Its universal covering group, Spin(2), is isomorphic to the [[real line]], {{math|'''R'''}}, under addition. Whenever angles of arbitrary magnitude are used one is taking advantage of the convenience of the universal cover. Every {{nowrap|2 × 2}} rotation matrix is produced by a countable infinity of angles, separated by integer multiples of 2{{pi}}. Correspondingly, the [[fundamental group]] of {{math|SO(2)}} is isomorphic to the integers, {{math|'''Z'''}}.

In the case of spatial rotations, [[Rotation group SO(3)|SO(3)]] is topologically equivalent to three-dimensional [[real projective space]], {{math|'''RP'''<sup>3</sup>}}. Its universal covering group, Spin(3), is isomorphic to the {{nowrap|3-sphere}}, {{math|''S''<sup>3</sup>}}. Every {{nowrap|3 × 3}} rotation matrix is produced by two opposite points on the sphere. Correspondingly, the [[fundamental group]] of SO(3) is isomorphic to the two-element group, {{math|'''Z'''<sub>2</sub>}}.

We can also describe Spin(3) as isomorphic to [[quaternion]]s of unit norm under multiplication, or to certain {{nowrap|4 × 4}} real matrices, or to {{nowrap|2 × 2}} complex [[special unitary group|special unitary matrices]], namely SU(2). The covering maps for the first and the last case are given by
:<math> \mathbb{H} \supset \{q \in \mathbb{H}: \|q\| = 1\} \ni w + \mathbf{i}x + \mathbf{j}y + \mathbf{k}z \mapsto
  \begin{bmatrix}
    1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\
    2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\
    2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2
  \end{bmatrix} \in \mathrm{SO}(3),
</math>
and
:<math>\mathrm{SU}(2) \ni \begin{bmatrix}
               \alpha &             \beta \\
    -\overline{\beta} & \overline{\alpha}
  \end{bmatrix} \mapsto 
  \begin{bmatrix}
    \frac{1}{2}\left(\alpha^2 - \beta^2 + \overline{\alpha^2} - \overline{\beta^2}\right) &
       \frac{i}{2}\left(-\alpha^2 - \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) &
      -\alpha\beta - \overline{\alpha}\overline{\beta} \\
    \frac{i}{2}\left(\alpha^2 - \beta^2 - \overline{\alpha^2} + \overline{\beta^2}\right) &
       \frac{i}{2}\left(\alpha^2 + \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) &
      -i\left(+\alpha\beta - \overline{\alpha}\overline{\beta}\right) \\
    \alpha\overline{\beta} + \overline{\alpha}\beta &
       i\left(-\alpha\overline{\beta} + \overline{\alpha}\beta\right) &
       \alpha\overline{\alpha} - \beta\overline{\beta}
  \end{bmatrix} \in \mathrm{SO}(3).
</math>

For a detailed account of the {{nowrap|SU(2)-covering}} and the quaternionic covering, see [[Rotation group SO(3)#Connection between SO(3) and SU(2)|spin group SO(3)]].

Many features of these cases are the same for higher dimensions. The coverings are all two-to-one, with {{math|SO(''n'')}}, {{math|''n'' > 2}}, having fundamental group {{math|'''Z'''<sub>2</sub>}}. The natural setting for these groups is within a [[Clifford algebra]]. One type of action of the rotations is produced by a kind of "sandwich", denoted by {{math|''qvq''<sup>∗</sup>}}. More importantly in applications to physics, the corresponding spin representation of the Lie algebra sits inside the Clifford algebra. It can be exponentiated in the usual way to give rise to a {{nowrap|2-valued}} representation, also known as [[projective representation]] of the rotation group. This is the case with SO(3) and SU(2), where the {{nowrap|2-valued}} representation can be viewed as an "inverse" of the covering map. By properties of covering maps, the inverse can be chosen ono-to-one as a local section, but not globally.