Prerequisites:

  1. Group and ring theory as covered by the standard algebra courses in the bachelor Mathematics.
  2. Basic notions from differential geometry, as covered by a bachelor mathematics course on Differential Geometry (in particular, familiarity with notions like: differentiable manifolds, tangent space, vector fields, integral curves, immersions and submersions). These basic notions are shortly discussed in Erik van den Ban's Prerequisites from differential geometry.

Aim/Description:

The course provides a basic introduction to the theory of Lie groups and Lie algebras. Lie groups are differentiable manifolds with a compatible group structure. This notion was introduced in the late nineteenth century by Sophus Lie to capture and analyse continuous symmetries of differential equations. The group structure of the Lie group induces an algebraic structure on the tangent space at its unit element, turning it in a so-called Lie algebra. Lie algebras provide a powerful algebraic tool in the study of Lie groups and their algebraic geometric counterparts, the so-called algebraic groups.

In the course we introduce real Lie groups and their associated Lie algebras. We give basic examples and discuss Lie's fundamental theorems on the correspondences between Lie groups and Lie algebras. We treat the Killing-Cartan classification of semisimple Lie algebras. We give a first introduction to the representation theory of compact Lie groups and semisimple Lie algebras.

Organization:

Weekly lectures (3 hours), which includes time to discuss exercises.

Lecturers:

Eric Opdam and Jasper Stokman