Lie Algebras

This chapter describes some of the main classical results on the structure of Lie algebras. Lie algebras are constructed as the tangent space at the identity of an algebraic group. They capture a lot of the structure of the corresponding algebraic group, but at the same time they are linear spaces, and hence the powerful tools of linear algebra can be employed to investigate them. The chapter provides an overview of some basic concepts such as algebra, representation, centre and derivation. It considers a decomposition of a vector npace relative to the action of nilpotent Lie algebra. Cartan subalgebras arise as those nilpotent subalgebras for which this decomposition is the most interesting. The classification of the semisimple Lie algebras over algebraically closed fields of characteristic 0 is viewed by many as one of the main results in modern mathematics. It has many applications in such diverse fields as group theory, geometry and physics.