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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)
Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, ... Concretely, a linear map is determined by where it sends a basis: ...
The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators.It was proposed by Pascual Jordan and Eugene Wigner [1] for one-dimensional lattice models, but now two-dimensional analogues of the transformation have also been created.
The complex spin representations of so(n,C) yield real representations S of so(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types. There is an invariant complex antilinear map r: S → S with r 2 = id S.
A spin C structure exists when the bundle is orientable and the second Stiefel–Whitney class of the bundle E is in the image of the map H 2 (M, Z) → H 2 (M, Z/2Z) (in other words, the third integral Stiefel–Whitney class vanishes).
It follows that SL(2, R) is isomorphic to the spin group Spin(2,1) +. Elements of the modular group PSL(2, Z) have additional interpretations, as do elements of the group SL(2, Z) (as linear transforms of the torus), and these interpretations can also be viewed in light of the general theory of SL(2, R).
Every year, celebrities try to capitalize on the holiday season by releasing festive music. Singers like Mariah Carey, Ariana Grande, and Michael Bublé managed to perfect the cheesy art form ...
It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group is not surjective or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both.