Double groups in representation theory and physics

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

${\displaystyle \chi _{j}(\alpha )={\frac {\sin[\alpha (j+1/2)]}{\sin[\alpha /2]}}}$

where ${\displaystyle \alpha }$ 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 ${\displaystyle 2\pi }$. However, if j is half-integral, one gets that

${\displaystyle \chi _{j}(\alpha +2\pi )=-\chi _{j}(\alpha )}$

${\displaystyle \chi _{j}(\alpha +4\pi )=\chi _{j}(\alpha )}$

Thus one has for the even-dimensional representations that the rotation by ${\displaystyle 2\pi }$ 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 ${\displaystyle 2\pi }$, 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.

SO(3) and SU(2)

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

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

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