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}_{\ZZ} 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 little s u 2 given by the Pauli matrices sigma_1 equals the matrix i, 0, 0, minus i; sigma_2 equals the matrix 0, 1, minus 1, 0; and sigma_3 equals the matrix 0, i, i, 0 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