Week 10

Session 1

Pre-class videos 43.26

sl(2,C) subalgebras, 1, accessible version 19.37
sl(2,C) subalgebras, 2, accessible version 11.09
Root spaces are 1-dimensional, accessible version 12.40

Pre-class exercises

Exercise:

True or false: If we use a sub-maximal torus, is it still true that $\mathfrak{h}=\mathfrak{g}_{0}$ ? If so, why? If not, what do we get instead?

Session 2

Pre-class videos 42.00

Root systems, accessible version 16.05
Classification of root systems, 1, accessible version 9.40
Classification of root systems, 2, accessible version 16.15

Pre-class exercises

Exercise:

Just by looking at the root diagrams, can you figure out the dimensions of the Lie algebras $\mathfrak{so}(5)$ and $\mathfrak{g}_{2}$ ?

Exercise:

Sketch the (3-dimensional) root system corresponding to your choice of Dynkin diagram from amongst: $A_{3}$ , $B_{3}$ , $C_{3}$ and $D_{3}$ .

Optional extras

Dihedral angles, accessible version 6.43
Simple roots, accessible version 19.55
End of proof, accessible version 15.21