In an earlier video, we constructed a map $R:SU(2)\to SO(3)$ . Check that $R({M}_{1})=R({M}_{2})$ if and only if ${M}_{1}=\pm {M}_{2}$ . Given a representation $S:SO(3)\to GL(n,\mathbf{C})$ , we get a representation $R\circ S:SU(2)\to SO(3)$ . Show that a representation $T:SU(2)\to SO(3)$ has this form if and only if $T(M)=T(M)$ for all $M\in SU(2)$ .
Week 4
Session 1
Preclass videos
Preclass exercises
Suppose we have a Lie algebra $\U0001d524\subset \U0001d524\U0001d529(n,\mathbf{R})$ consisting of real matrices. Consider the subspace $\U0001d524\otimes \mathbf{C}\subset \U0001d524\U0001d529(n,\mathbf{C})$ consisting of matrices of the form $M+iN$ with $M$ and $N$ in $\U0001d524$ . Show that:

this is a Lie subalgebra, i.e. that it is preserved by Lie bracket

if $f:\U0001d524\to \U0001d524\U0001d529(m,\mathbf{C})$ is a reallinear Lie algebra homomorphism then ${f}^{\mathbf{C}}:\U0001d524\otimes \mathbf{C}\to \U0001d524\U0001d529(m,\mathbf{C})$ defined by ${f}^{\mathbf{C}}(M+iN)=f(M)+if(N)$ for $M$ and $N$ in $\U0001d524$ is also a Lie algebra homomorphism.
Session 2
Preclass videos
Preclass exercises
Check that $(\pm i,1)$ is an eigenvector of $\left(\begin{array}{cc}\hfill \mathrm{cos}\theta \hfill & \hfill \mathrm{sin}\theta \hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill \mathrm{cos}\theta \hfill \end{array}\right)$ with eigenvalue ${e}^{\pm i\theta}$ .
Show that if $\u27e8,\u27e9$ is a Hermitian inner product on ${\mathbf{C}}^{n}$ and $R:U(1)\to GL(n,\mathbf{C})$ is a representation then $${\u27e8u,v\u27e9}_{inv}={\int}_{0}^{2\pi}\u27e8R({e}^{i\theta})u,R({e}^{i\theta})v\u27e9\frac{d\theta}{2\pi}$$ is also a Hermitian inner product on ${\mathbf{C}}^{n}$ .