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User:Patrusfarr/Double Group

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Double groups in representation theory and physics

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Characters for a (2j + 1)-dimensional representation of the rotation group are given by

where is the angle of rotation. Respectively, the integral and half-integral values of j correspond to the odd and even-dimensional representations. Note that if j takes on integer values, the characters are invariant under a rotation of . However, if j is half-integral, one gets that

Thus one has for the even-dimensional representations that the rotation by is not equivalent to the identity element. This leads to a group extension of any subgroup of the rotation group by the group E formed from the identity element and the rotation by , which is defined as its double group. The double group D has twice as many elements as their corresponding single group G and the quotient D/E is isomorphic to G.

SO(3) and SU(2)

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The homomorphism from SU(2) onto SO(3)
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To each vector r = (x, y, z) in R3, construct a 2x2 traceless Hermitian matrix M(r) whose components are defined as

If U ∈ SU(2), then UMU-1 is also a traceless Hermitian matrix.