### M1: Lie Groups and Lie Algebras - 8EC

Prerequisites

• Basic knowledge of analysis, topology and group theory, as taught in the Bachelor programme.
• Basic knowledge from the theory of differentiable manifolds: smooth manifold, tangent spaces, vector fields and flow. Immersion and submersion theorem. See also Prerequisites from differential geometry, for Lie groups.
• Aim: The aim of this course is to give a thorough introduction to the theory of Lie groups and algebras.

Course 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 has expanded enormously in the course of the previous century.  Nowadays, it plays a vital role in the description of symmetries in Physics (quantum physics, elementary particles), Geometry and Topology, and Number Theory (automorphic forms). In this course we will begin by studying the basic properties of Lie groups. Much of the structure of a connected Lie group is captured by its Lie algebra, which may be defined as tangent space at the identity, with a suitable bracket structure. The exponential map will be introduced, and the relation between the structure of a Lie group and its Lie algebra will be investigated. Actions of Lie groups will be studied. After this introduction we will focus on compact Lie groups and the integration theory on them. The groups SU(2) and SO(3) will be discussed as basic examples. We will study representation theory and its role in the harmonic analysis on a Lie group. The classifiction of the irreducible representations of SU(2) will be studied. 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.

Exam: The exam consists of two parts:

• Homework exercises.
• Written exam at the end of the course.
• Written exam:

This will be an open book exam. You are allowed to bring lecture notes and personal notes, but no  exercises, worked exercises or notes on the exercises.

The final grade will be based on the grades obtained for the homework (40 percent) and the concluding exam (60 percent).

Recommended literature:

• T. Br"ocker & T. tom Dieck Representations of compact Lie groups, Springer-Verlag, New York, 1985.
• W. Rossmann Lie groups: An Introduction Through Linear Groups Oxford Graduate Texts in Mathematics, Number 5. Oxford University Press, 2002; ISBN 0198596839
• J.J. Duistermaat & J.A.C. Kolk Lie groups Universitext serie, Springer-Verlag, New York, 2000. ISBN 3-540-15293-8, cat prijs DM 79.
• Th. Frankel The Geometry of Physics-an introduction. Cambridge University Press, 1997.
• E.J.N. Looijenga Smooth manifolds Lecture Notes, available as pdf file.
• Lecturer: E.P. van den Ban (UU)