From Lie algebras to Chevalley groups

We follow Humphreys, studying the structure theory of semisimple Lie algebras (over algebraically closed fields of characteristic zero) in detail, proving the existence of a Chevalley basis and constructing Chevalley groups of adjoint type. We provide elementary definitions and results about Lie algebras. We take the perspective of toral subalgebras to show the root space decomposition with respect to a maximal torus. We utilise representation theory to prove that the set of roots forms a root system. Studying root systems in their own right then gives us further structural results for semisimple Lie algebras. These enable us to prove the existence of a Chevalley basis, which allows us to transfer the Lie algebra structure to finite fields. We conclude by using this to construct Chevalley groups of adjoint type.