I just finished a quick primer for my class on Advanced QFT on Representations of Lie algebras and useful facts about them. Here is a link to the lecture notes. It’s a hands on approach with no proofs, but at least it makes various concepts a little bit more accessible and can serve for a quick reminder of basic stuff like what is a Dynkin diagram and labels of various representations. The notes cover mostly SU(N), but various concepts are quite general.
My first serious encounter with Dynkin diagrams was for a paper I wrote together with Richard Corrado and Jacques Distler in 97, when I was working on Matrix theory in graduate school. The difference is that the Dynkin diagrams that were appearing there where those of the Affine Lie algebras (they have an extra node). There is some fun discrete group theory and geometry there. I was reminded of this stuff when I was preparing my notes.
The main issue for this class was to get enough group theory so that most computations that we are going to do for Grand Unified Theories become simple. I remember also reading various sections of the book on group theory by Hammermesh when I was an undergraduate. Although a lot of that was on Crystallographic groups. Nowadays I recommend the Fulton and Harris book on representation theory. However, I have not read other books recently that might contain more information on Dynkin diagrams (the book of Fulton and Harris does not treat these in a lot of detail).
Group theory appears all of the time in theoretical physics, both discrete and continuous. It’s probably one of the most useful tools to approach physical problems in general and any student who is seriously considering becoming a theorist should be well advised to learn Lie theory (continuous group theory). The one sound bite I can give about this is that if you understand well the simplest non-abelian group ( SU(2) angular momentum), you are 50% of the way there already.
I’m providing these notes as a service to whoever can use them. After all the work that goes into typing them, it would be wasteful if they do not become more widely available. readers beware: I can not guarantee that they are free of typos. Fortunately, there are not that many formulae.