A Lie group is a group with the additional structure of a differentiable manifold for which the group operation is differentiable. The name Lie group comes from the Norwegian mathematician M. Sophus Lie (1842-1899) who was the first to study these groups systematically in the context of symmetries of partial differential equations.
The theory of Lie groups plays a central role in the description of symmetries in Physics (quantum physics, elementary particles), Geometry and Topology (principal bundles), and Number Theory (automorphic forms).
In the course we will begin by studying basic properties of a Lie group and its linearization, the Lie algebra. We will then focus on compact Lie groups, where SO(3) and SU(2) will be guiding examples.
In the second half of the course we will discuss the representation theory of compact Lie groups and its role in
harmonic analysis on these groups.
The final part of the course will be devoted to the classification of compact Lie algebras. Key words are: root system, finite reflection group, Cartan matrix, Dynkin diagram.
The course will be concluded with the formulation of the classification of irreducible representations by their highest weight and with the formulation of Weyl's character formula.