Lie Groups [Fall 2019]

Prerequisites:

Bachelor mathematics or physics, basic knowledge of manifolds (tangent space, tangent map and vector fields) as given in John M. Lee, Introduction to Smooth Manifolds:

Chapter 1, Smooth Manifolds (pp 1-24);
Chapter 2, Smooth Maps (pp 31-37, 49 - 55);
Chapter 3, Tangent vectors (pp. 60-73,75 - 78);
Chapter 4, Vector fields (pp. 81-92).

In case of knowledge deficiencies showing up during the course, recommendations for further reading will be given by the lecturer. Aim: To give a thorough introduction to the theory of Lie groups, in particular compact Lie groups and actions, representations, roots and weights.

Description:

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. Their role in geometry through actions will be discussed.
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 structure of compact Lie algebras. The notion of root systems will be discussed, as well as their role in weight- and representation theory. Then a number of results will be described: the classification of compact Lie algebras in terms of Dynkin diagrams. 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 caracter formula.

Organization:

Weekly lectures (2 times 45 minutes) and exercise classes (45 minutes)

Literature:

Lecture notes Lie groups by E.P. van den Ban, downloadable from his webpage for the course

Instructors: E.P. van den Ban (Universiteit Utrecht)
E-mail:
E.P.vandenBan@uu.nl