Root Systems, Cartan Matrix and Dynkin Diagrams in Classification of Lie Algebras

Abstract

This review paper deals with Lie algebras, with some concentration on root systems, which help in classification and many applications of symmetric spaces. We deal with the basic concept of a root system. First, its origins in the theory of Lie algebras are exposed, then an axiomatic definition is provided.  Bases, Weyl groups,  and the  transitive  action  of  the  latter  on  the  former  are  explained. Finally, the Cartan matrix and Dynkin diagram are exposed to suggest the multiple applications of root systems to other fields of study and  their classification.