SO(n) еселік жабыны: Spin(n) тобы

Double cover Lie group of the special orthogonal 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 1=SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)

The group multiplication law on the double cover is given by lifting the multiplication on
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.

The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the electron. Strictly speaking, the spin group describes a fermion in a zero dimensional space; however, space is not zero dimensional, and so the spin group is used to define spin structures on (pseudo )Riemannian manifolds: the spin group is the structure group of a spinor bundle. The affine connection on a spinor bundle is the spin connection; the spin connection can simplify calculations in general relativity. The spin connection in turn enables the Dirac equation to be written in curved spacetime (effectively in the tetrad coordinates), which in turn provides a footing for quantum gravity, as well as a formalization of Hawking radiation (where one of a pair of entangled, virtual fermions falls past the event horizon, and the other does not).

The spin groups can be constructed less explicitly but without appealing to Clifford algebras. As a manifold, is the double cover of Its multiplication law can be defined by lifting as follows. Call the covering map Then is a set with two elements, and one can be chosen without loss of generality to be the identity. Call this Then to define multiplication in , for choose paths satisfying , and These define a path in defined satisfying Since is a double cover, there is a unique lift of with Then define the product as
It can then be shown that this definition is independent of the paths , that the multiplication is continuous, and the group axioms are satisfied with inversion being continuous, making a Lie group.