In mathematics the spin group Spin(n) is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups
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).
Read more about Spin Group: Accidental Isomorphisms, Indefinite Signature, Topological Considerations, Center, Quotient Groups, Discrete Subgroups
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