Here is the document
Dynkin diagram, root system is a big mystery of my undergraduate life. Here I present three examples tha place they work.
Since my main interest is not algebraic geometry, maybe there is something wrong or unacceptable, please inform me if you have good reference. This proof and concepts are collected from Shafarevich.
1.1 Integral Quadratic Forms
1.2 Abstract Root systems
1.3 Dynkin Diagrams
Exercises: Relative Concepts
1.4 Basis determine root system
1.5 More about Weyl groups
1.6 Classification of Dynkin diagram
1.7 Euclidean diagrams and Quadratic forms again
2. How they work
2.1 Complex semisimple Lie algebras
2.2 Hereditary algebras
2.3 Mckay correspondence
2.4 Classification of Du Val Singularities