In an earlier video, we constructed a map R\colon SU(2)\to SO(3). Check that R(M_1)=R(M_2) if and only if M_1=\pm M_2. Given a representation S\colon SO(3)\to GL(n,\CC), we get a representation R\circ S\colon SU(2)\to SO(3). Show that a representation T\colon SU(2)\to SO(3) has this form if and only if T(-M)=T(M) for all M\in SU(2).
Suppose we have a Lie algebra little g inside little g l n R consisting of real matrices. Consider the subspace little g tensor C inside little g l n C consisting of matrices of the form M + i N with M and N in little g. Show that:
this is a Lie subalgebra, i.e. that it is preserved by Lie bracket
if f from little g to little g l m C is a real-linear Lie algebra homomorphism then f superscript C from little g tensor C to little g l m C defined by f superscript C of M + i N equals f of M plus i f of N for M and N in little g is also a Lie algebra homomorphism.
Check that (\pm i,1) is an eigenvector of \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix} with eigenvalue e^{\pm i\theta}.
Show that if \langle,\rangle is a Hermitian inner product on \CC^n and R\colon U(1)\to GL(n,\CC) is a representation then u angle brackets v subscript inv equals the integral from 0 to 2 pi of the angle bracket between R of e to the i theta applied to u and R of e to the i theta applied to v, d theta over 2 pi is also a Hermitian inner product on \CC^n.