Spin group

Common card · No. 458466

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...

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