Spin group
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)} .