Introduction to lie algebras.

Abstract

The purpose of this thesis is to introduce the basic ideas of Lie algebras to the reader with some basic knowledge of abstract and elementary linear algebra. In this study, Lie algebras are considered from a purely algebraic point of view, without reference to Lie groups and differential geometry. Such a view point has the advantage of going immediately into the discussion of Lie algebras without first establishing the topological machineries for the sake of defining Lie groups from which Lie algebras are introduced. In Chapter I we summarize for the reader's convenience rather quickly some of the basic concepts of linear algebra with which he is assumed to be familiar. In Chapter II we introduce the language of algebras in a form designed for material developed in the later chapters. Chapters III and IV were devoted to the study of Lie algebras and the Lie algebra of derivations. Some definitions, basic properties, and several examples are given. In Chapter II we also study the Lie algebra of antisymmetric operators, Ideals and homomorphisms. In Chapter III we introduce a Lie algebra structure on DerF(A) and study the link between the group of automorphisms of A and the Lie algebra of derivations DerF(A).