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In mathematics the spin group, denoted Spin(n), [1] [2] 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)
In other words, the group Spin C (n) is a central extension of SO(n) by S 1. Viewed another way, Spin C (n) is the quotient group obtained from Spin(n) × Spin(2) with respect to the normal Z 2 which is generated by the pair of covering transformations for the bundles Spin(n) → SO(n) and Spin(2) → SO(2) respectively.
special euclidean group: group of rigid body motions in n-dimensional space. N 0 se(n) n + n(n−1)/2 Spin(n) spin group: double cover of SO(n) Y 0 n>1 0 n>2 Spin(1) is isomorphic to Z 2 and not connected; Spin(2) is isomorphic to the circle group and not simply connected so(n) n(n−1)/2 Sp(2n,R) symplectic group: real symplectic matrices: N 0 Z
A frequently occurring case is a double covering group, a topological double cover in which H has index 2 in G; examples include the spin groups, pin groups, and metaplectic groups. Roughly explained, saying that for example the metaplectic group Mp 2 n is a double cover of the symplectic group Sp 2 n means that there are always two elements in ...
The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected, but the simply connected spin group is its double cover. So for every rotation there are two elements of the spin group that ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...