On the nature of the classification of semisimple complex Lie algebras via root systems: Or how I learned to stop worrying and analyze root systems

Root systems arise from semisimple Lie algebras, from the structure of their subalgebras. We will look at Lie algebras, some of their properties, then how root systems arise from their subalgebras. Root systems can also be constructed axiomatically, studied as objects in and of themselves. In this paper, we will discuss their construction, properties, and classify all root systems of rank 1 and 2. Finally, we will look at a relation between root systems and semisimple complex Lie algebras, the Serre Relations.