Week 9

Session 1

Pre-class videos 26.53

Maximal tori: Overview, accessible version 13.20
Weights, accessible version 9.16

Pre-class exercises

Exercise:

Can you see why the tori we have been using in $SU(2)$ and $SU(3)$ are maximal?

Exercise:

Explicitly, what is the set of matrices $\mathfrak{h}_{\mathbf{Z}}$ for our usual diagonal choice of maximal torus in $SU(3)$ ?

Session 2

Pre-class videos 35.12

The Killing form, accessible version 13.33
The dual Killing form, accessible version 21.39

Pre-class exercises

Exercise:

Consider the basis of $\mathfrak{su}(2)$ given by the Pauli matrices $\sigma_{1}=\begin{pmatrix}i&0\\ 0&-i\end{pmatrix},\ \sigma_{2}=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},\ \sigma_{3}=\begin{pmatrix}0&i\\ i&0\end{pmatrix}$ Find all possible Killing dot products $K(\sigma_{i},\sigma_{j})$ .

Optional extras

Finding compact abelian subgroups, accessible version 13.46
Compact connected abelian groups are tori, accessible version 18.02
Any torus is contained in a maximal torus, accessible version 8.54