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
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).
Read more about Spin Group: Accidental Isomorphisms, Indefinite Signature, Topological Considerations, Center, Quotient Groups, Discrete Subgroups
Famous quotes containing the words spin and/or group:
“Words can have no single fixed meaning. Like wayward electrons, they can spin away from their initial orbit and enter a wider magnetic field. No one owns them or has a proprietary right to dictate how they will be used.”
—David Lehman (b. 1948)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)