In mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)
1
→
Z
2
→
Spin
(
n
)
→
SO
(
n
)
→
1.
{\displaystyle 1\to \mathbb {Z} _{2}\to \operatorname {Spin} (n)\to \operatorname {SO} (n)\to 1.}
The group multiplication law on the double cover is given by lifting the multiplication on
SO
(
n
)
{\displaystyle \operatorname {SO} (n)}
.
As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group.
For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).
The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −I.
Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n). A distinct article discusses the spin representations.
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