Introduction to Lie algebras

The Lie algebras are central to Lie theory. They are highly non-commutative and non-associative algebras. They were first introduced in 1870s by a Norwegian mathematician Marius Sophus Lie to study the concept of infinitesimal transformations. Their representations play an important role in theoretical Physics.

The course will introduce finite dimensional Lie algebras. We will begin with the basic definitions and properties of finite dimensional Lie algebras and prove some fundamental results about nilpotent and solvable Lie algebras. Then we will prove the Cartan’s criteria for solvability and semi simplicity. Finally, we will move on to the structure theory of semi-simple Lie algebras and prove their root space decomposition.