This course provides a basic introduction to the theory of Lie groups and Lie algebras.
Lie groups are differentiable manifolds with a compatible group structure. They are omnipresent in the study of symmetry in geometry, analysis, physics and number theory. The group structure induces an algebraic structure on the tangent space of the Lie group at its unit element, turning it into a so-called Lie algebra. To a large extent Lie groups are determined by their Lie algebras. The Lie algebraic viewpoint allows to develop Lie theory independent of the ground field and to study deformation theory. This paves the way to, for instance, algebraic groups and quantum groups.
We start by introducing real Lie groups and their associated Lie algebras. We give basic examples and shortly discuss Lie's fundamental theorems. We treat the Killing-Cartan classification of semisimple Lie algebras. This includes a detailed discussion
of root systems. We treat universal enveloping algebras, highest weight modules, and we classify the finite dimensional irreducible representations of a semisimple Lie algebra. At the end of the course we treat Serre's presentation of the universal enveloping algebra of a semisimple Lie algebra and discuss some of its applications.
- Group and ring theory as covered by the standard algebra courses in the bachelor Mathematics.
- Basic notions from differential geometry, as covered by a bachelor mathematics course on Differential Geometry.
- Knowledge on representation theory of finite groups is helpful but not necessary.