Spin group

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In mathematics the spin group Spin(n) is a particular double cover of the special orthogonal group SO(n). That is, there exists a short exact sequence of Lie groups

[1\to\mathbb_2\to\operatorname(n)\to\operatorname(n)\to 1]

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).

Accidental isomorphisms

In low dimensions, there are isomorphism among the classical Lie groups called accidental isomorphims. These isomorphisms give rise to isomorphism between the spin groups in low dimensions and classical Lie groups. Specifically we have

Spin(1) = O(1)

Spin(2) = U(1)

Spin(3) = Sp(1) = SU(2)

Spin(4) = Sp(1) x 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.

Spin group

Accidental isomorphisms

See also