Projective Representation - Projective Representations of Lie Groups

Projective Representations of Lie Groups

Studying projective representations of Lie groups leads one to consider true representations of their central extensions (see Group extension#Lie groups). In many cases of interest it suffices to consider representations of covering groups; for a connected Lie group G, this amounts to studying the representations of the Lie algebra of G. Notable cases of covering groups giving interesting projective representations:

• The special orthogonal group SO(n) is double covered by the Spin group Spin(n). In particular, the group SO(3) (the rotation group in 3 dimension) is double-covered by SU(2). This has important applications in quantum mechanics, as the study of representations of SU(2) leads naturally to the idea of spin.
• The orthogonal group O(n) is double covered by the Pin groups Pin_±(n).
• The symplectic group Sp(2n) is double covered by the metaplectic group Mp(2n).