Semisimple Lie algebras and root systems

Abstract

In this paper we introduce Lie algebras and list basic properties and results, including Weyl's theorem. We then introduce the important Lie algebra $\mathfrak{sl}(2)$, and move on to briefly introduce root systems. The main goal of the paper is to prove that semisimple Lie algebras are in one to one correspondance with chrystallographic root systems.