## Description

The goal of this thesis is to understand the classification of complex semisimple Lie algebras through the use of root systems. To begin, the basic ideas of Lie theory are established. We restrict our attention to matrix Lie groups and their associated Lie algebras. The existence and uniqueness is shown for an important structure, the Cartan subalgebra, in order to discuss the root space decomposition. The roots associated to the Cartan subalgebra are described geometrically as a subset of a finite dimensional vector space with additional restrictions on the lengths of the vectors and the angles between them. From these parameters, a Cartan matrix is formed and used as a way of storing the information about the root system. This information can be represented graphically by a Dynkin diagrams. The process of arriving at a Dynkin from the starting point of a complex semisimple Lie algebra is the first component of this thesis and is given as a concise survey of the references. The next component involves the classification of Dynkin diagrams. The types of Dynkin diagrams are limited by the angles occurring between two roots. With this limitation, we may entirely classify the possible Dynkin diagrams associated to root systems. The complete classification of Dynkin diagrams is given, including new lemmas and properties. From each Dynkin diagram a root system is constructed, showing the bijection from root systems to Dynkin diagrams. We show that every root system arises from a complex semisimple Lie algebra, and conversely that every complex semisimple Lie algebra has an associated root system, to obtain the classification.