This book is an introduction to the theory of Lie groups and Lie algebras, with emphasis on the theory of semisimple Lie algebras. It can serve as a basis for a two-semester graduate course or – omitting some material – as a basis for a rather intensive one-semester course. The book includes a large number of exercises.

The material covered in the book ranges from basic definitions of Lie groups to the theory of root systems and highest weight representations of semisimple Lie algebras; however, to keep book size small, the structure theory of semisimple and compact Lie groups is not covered.

Exposition follows the style of famous Serre's textbook on Lie algebras: we tried to make the book more readable by stressing ideas of the proofs rather than technical details. In many cases, details of the proofs are given in exercises (always providing sufficient hints so that good students should have no difficulty completing the proof). In some cases, technical proofs are omitted altogether; for example, we do not give proofs of Engel's or Poincare–Birkhoff–Witt theorems, instead providing an outline of the proof. Of course, in such cases we give references to books containing full proofs.

It is assumed that the reader is familiar with basics of topology and differential geometry (manifolds, vector fields, differential forms, fundamental groups, covering spaces) and basic algebra (rings, modules).