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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 ...
In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R) is cyclic of order 2, [4] and the spin group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group Spin(2) is the unique connected 2-fold cover).
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 ...