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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)
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).
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 ...
The basic examples of central extensions as covering groups are: the spin groups, which double cover the special orthogonal groups, which (in even dimension) doubly cover the projective orthogonal group. the metaplectic groups, which double cover the symplectic groups. The case of SL 2 (R) involves a fundamental group that is infinite cyclic.