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 jth entry (i not equal to j) then det of exp X equals exp of trace 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 from 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 little s u 2 is the set of 2-by-2 matrices X such that X transpose equals minus X and trace of X equals 0?

Optional extras

Topology of Lie groups, accessible version