User:Patrusfarr/Double Group

Double groups in representation theory and physics
Characters for a (2j + 1)-dimensional representation of the rotation group are given by

\chi_j(\alpha) = \frac{\sin[\alpha(j+1/2)]}{\sin[\alpha/2]} $$ where $$\alpha$$ is the angle of rotation. Respectively, the integral and half-integral values of j correspond to the odd and even-dimensional representations. Note that if j takes on integer values, the characters are invariant under a rotation of $$2\pi$$. However, if j is half-integral, one gets that

\chi_j(\alpha + 2\pi) = -\chi_j(\alpha) $$

\chi_j(\alpha + 4\pi) = \chi_j(\alpha) $$ Thus one has for the even-dimensional representations that the rotation by $$2\pi$$ is not equivalent to the identity element. This leads to a group extension of any subgroup of the rotation group by the group E formed from the identity element and the rotation by $$2\pi$$, 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.

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

\begin{bmatrix} z & x-iy \\ x + iy & -z \\ \end{bmatrix} $$

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