Lie groups and Lie algebras - a combinatorial approach
Serre showed that finite dimensional semisimple Lie algebras can be defined by generators and relations. We investigate this combinatorial approach to the study of Lie algebras and we define Chevalley groups, which give an alternate construction of semisimple Lie groups using data from the root system, automorphisms of the underlying Lie algebra and highest weight representations. This leads to a combinatorial definition of semisimple groups in terms of generators and relations via the Steinberg presentation. We also discuss how this combinatorial approach to Lie theory generalizes to infinite dimensions.