Universal Enveloping Algebras and Verma Modules

Abstract

After Lie algebras were introduced by Sophus Lie, Elie Cartan gave the classification of semi-simple Lie algebras and began studying their representations by the early 20th century. The aim of this thesis is to survey the advances made in the representation theory of semi-simple Lie algebras since then, culminating in the study of the theory of the BGG Category \mathcal{O}.