ABSTRACT
This chapter introduces as directly as possible some basic concepts of the theory of Lie algebras, oriented towards the proof of two basic results needed for the theory of reductive groups in characteristic zero. It develops the necessary tools leading to the some theorems, Engel's and Lie's theorems concerning the structure of nilpotent and solvable Lie algebras, Cartan's solvability criterion, Cartan's semisimplicity criterion, the concept of Casimir operator and Killing form and some elementary cohomological tools. The chapter presents the basic definitions and examples of nilpotent and solvable Lie algebras. It shows that Lie's and Engel's theorems concerning the triangularization of solvable and nilpotent Lie algebras. The chapter defines the radical and the concept of semisimple Lie algebra and prove the important Cartan's criterion characterizing the semisimplicity in terms of the non degeneracy of the Killing form. It provides a brief introduction of a few cohomological concepts in the category of Lie algebra representations, due basically to Chevalley and Eilenberg.
