Week 4

Session 1

Pre-class videos 25.36

Representations, accessible version 11.32
Complete reducibility, accessible version 14.04

Pre-class exercises

Exercise:

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 $S\colon SO(3)\to GL(n,\mathbf{C})$ , 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 $M\in SU(2)$ .

Exercise:

Suppose we have a Lie algebra $\mathfrak{g}\subset\mathfrak{gl}(n,\mathbf{R})$ consisting of real matrices. Consider the subspace $\mathfrak{g}\otimes\mathbf{C}\subset\mathfrak{gl}(n,\mathbf{C})$ consisting of matrices of the form $M+iN$ with $M$ and $N$ in $\mathfrak{g}$ . Show that:

this is a Lie subalgebra, i.e. that it is preserved by Lie bracket
if $f\colon\mathfrak{g}\to\mathfrak{gl}(m,\mathbf{C})$ is a real-linear Lie algebra homomorphism then $f^{\mathbf{C}}\colon\mathfrak{g}\otimes\mathbf{C}\to\mathfrak{gl}(m,\mathbf{C})$ defined by $f^{\mathbf{C}}(M+iN)=f(M)+if(N)$ for $M$ and $N$ in $\mathfrak{g}$ is also a Lie algebra homomorphism.

Session 2

Pre-class videos 23.20

Pre-class exercises

Exercise:

Check that $(\pm i,1)$ is an eigenvector of $\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$ with eigenvalue $e^{\pm i\theta}$ .

Exercise:

Show that if $\langle,\rangle$ is a Hermitian inner product on $\mathbf{C}^{n}$ and $R\colon U(1)\to GL(n,\mathbf{C})$ is a representation then $\langle u,v\rangle_{inv}=\int_{0}^{2\pi}\langle R(e^{i\theta})u,R(e^{i\theta})% v\rangle\frac{d\theta}{2\pi}$ is also a Hermitian inner product on $\mathbf{C}^{n}$ .