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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)
The Lie algebra so(n, C) is isomorphic to the complexified Lie algebra spin n C in Cl n C via the mapping induced by the covering Spin(n) → SO(n) [2] [,]. It follows that both S and S′ are representations of so(n, C). They are actually equivalent representations, so we focus on S.
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.
The symmetry group of the sphere (n=3) or hypersphere. so(n) n(n−1)/2 SO(n) special orthogonal group: real orthogonal matrices with determinant 1 Y 0 Z n=2 Z 2 n>2 Spin(n) n>2 SO(1) is a single point and SO(2) is isomorphic to the circle group, SO(3) is the rotation group of the sphere. so(n) n(n−1)/2 SE(n)
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, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics.
In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold (,), one defines the spinor bundle to be the complex vector bundle: associated to the corresponding principal bundle: of spin frames over and the spin representation of its structure group on the space of spinors.
If n is even, it splits further [clarification needed] into two irreducible representations Δ = Δ + ⊕ Δ − called the Weyl or half-spin representations. Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.