The aim of this course is to give a thorough introduction to the theory
of Lie groups and algebras.
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.
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
The exam consists of two parts:
The final grade will be based on the grades obtained for the homework (40 percent) and the concluding exam (60 percent). Out of the 8 HW scores, only the 7 best will count for the homework average H (calculated in one
decimal accuracy). The grade E for the concluding exam will be calculated with one decimal of accuracy.
The final grade G is calculated by G = 0.4 H + 0.6 E, rounded off to an integer in the usual way.
January 30, 2020, 13:30 - 16:30. Educatorium Megaron
Last update: 11/7 - 2019