Editing Projective representation (section)

===Examples of covers, leading to projective representations===
Notable cases of covering groups giving interesting projective representations:

* The [[special orthogonal group]] SO(''n'', ''F'') is doubly covered by the [[Spin group]] Spin(''n'', ''F'').  
*In particular, the [[rotation group SO(3)|group SO(3)]] (the rotation group in 3 dimensions) is doubly covered by [[special unitary group|SU(2)]].  This has important applications in quantum mechanics, as the [[representation theory of SU(2)|study of representations of SU(2)]] leads to a nonrelativistic (low-velocity) theory of [[Spin (physics)|spin]].
* The group [[Lorentz group|SO<sup>+</sup>(3;1)]], isomorphic to the [[Möbius group]], is likewise doubly covered by [[special linear group|SL<sub>2</sub>]]('''C'''). Both are supergroups of aforementioned SO(3) and SU(2) respectively and form a [[special relativity|relativistic]] spin theory.
*The universal cover of the [[Poincaré group]] is a double cover (the [[semidirect product]] of SL<sub>2</sub>('''C''') with '''R'''<sup>4</sup>). The irreducible unitary representations of this cover give rise to projective representations of the Poincaré group, as in [[Wigner's classification]]. Passing to the cover is essential, in order to include the fractional spin case.
* The [[orthogonal group]] O(''n'') is double covered by the [[Pin group]] Pin<sub>±</sub>(''n'').
* The [[symplectic group]] Sp(2''n'')=Sp(2''n'', '''R''') (not to be confused with the compact real form of the symplectic group, sometimes also denoted by Sp(''m'')) is double covered by the [[metaplectic group]] Mp(2''n''). An important projective representation of Sp(2''n'') comes from the [[Oscillator representation|metaplectic representation]] of Mp(2''n'').