I am having a little confusion with the different types of lattices involved with Lie algebras.
Root system: represented as euclidian vector arrows. However I have seen the same arrangement with vertices and line segments. I believe this represents the symmetries, or the action of root spaces on eachother? In addition sometimes the 0 root space/cartan is included and sometimes it is not, why? And I assume the Coxeter plane projection is a way of representing the higher dimension "closed segment" root systems as 2D?
Root lattice: a lattice generated by roots. Weight lattice: a lattice generated by the fudamental weights.
Why are these two lattices represented by finite diagrams when the lattice space is infinite? What is the significance of the root lattice over the weight lattice?
Sorry if this seems like a lot of questions.
Help is appreciated.