Week 3

Session 1

Pre-class videos 37.16

Smooth homomorphisms, accessible version 14.13
One-parameter subgroups, accessible version 7.52
Linearity of homomorphisms in exponential charts, accessible version 15.11

Pre-class exercises

Exercise:

Prove that if $X$ if an $n$ -by- $n$ matrix with zeros everywhere except a 1 in the $i j$ th entry ( $i\neq j$ ) then $\det(\exp(X))=\exp(\mathrm{Tr}(X))$ .

Session 2

Pre-class videos 34.38

Lie's theorem, accessible version 12.42
Example: SU(2) to SO(3), accessible version 21.56

Pre-class exercises

Exercise:

We constructed a homomorphism $R\colon SU(2)\to O(3)$ . Can you see why $R$ lands in the subset $SO(3)$ of matrices with determinant 1?

Exercise:

Can you prove that $\mathfrak{su}(2)$ is the set of 2-by-2 matrices $X$ such that $X^{T}=-X$ and $\mathrm{Tr}(X)=0$ ?

Optional extras

Topology of Lie groups, accessible version