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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.
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 ...
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean ... These groups, Spin(n), SO(n) ...
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
The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S 3 and can be understood as the group of versors (quaternions with absolute value 1).
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of the
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 ...