Spin group
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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
For n > 2, Spin(n) is simply connected and so coincides with the universal cover of 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.
Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cℓ(n).
[edit] Accidental isomorphisms
In low dimensions, there are isomorphisms among the classical Lie groups called accidental isomorphisms. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups. Specifically, we have
- Spin(1) = O(1)
- Spin(2) = U(1)
- Spin(3) = Sp(1) = SU(2)
- Spin(4) = Sp(1) × Sp(1)
- Spin(5) = Sp(2)
- Spin(6) = SU(4)
There are certain vestiges of these isomorphisms left over for n = 7,8 (see Spin(8) for more details). For higher n, these isomorphisms disappear entirely.
