**Location and time:**

**Material discussed in lectures**

**Exercises**

**Web movies of lectures**

**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 developed vastly
in the course of the previous century. 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 the algebra of left invariant vector fields.
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.

**Text:**

**Prerequisites:**

**Recommended literature:**

Representations of compact Lie groups,

Springer-Verlag, New York, 1985.

Lie groups: An Introduction Through Linear Groups

Oxford Graduate Texts in Mathematics, Number 5.

Oxford University Press, 2002; ISBN 0198596839

Lie groups

Universitext serie, Springer-Verlag, New York, 2000.

ISBN 3-540-15293-8, cat prijs DM 79.

The Geometry of Physics-an introduction.

Cambridge University Press, 1997.

Meetkunde op varieteiten

diktaat, available as ps file of als pdf file.

**Exam:**
The exam consists of two parts:

E.P.vandenBan[*at*]uu.nl

Last update: April 27, 2012