Spin group

In mathematics the spin group Spin(n) 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 (with n ≠ 2)

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). A distinct article discusses the spin representations.

Construction of the Spin group often starts with the construction of the Clifford algebra over a real vector space V. The Clifford algebra is the quotient of the tensor algebra TV of V by a two-sided ideal. The tensor algebra (over the reals) may be written at

The Clifford algebra Cl(V) is then the quotient space

where ${\displaystyle \Vert v\Vert }$ is the norm of a vector ${\displaystyle v\in V}$. The resulting space is naturally graded, and can be written as

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