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Material discussed in lectures
Exercises
Aim:
The aim of this course is to give a thorough introduction to the
theory of finite dimensional semisimple Lie algebras, and the
infinite dimensional affine algebras.
Course description:
The theory of Lie groups was initiated by the Norwegian
mathematician Sophus Lie (1842 - 1892) with the purpose of
analyzing differential equations in the presence of symmetries.
Much about a Lie group can be understood from
its linearization at the identity, the so called
Lie algebra.
In the course we will systematically develop the structure
theory of these Lie algebras. In particular we will
study the semisimple algebras. Over the field of complex
numbers these are precisely the complexified Lie algebras
of the compact Lie groups with finite center.
The structure of semisimple Lie algebras can be understood
in terms of so called root systems and the associated reflection
(or Weyl) groups. We will discuss the classification of these
algebras in terms of the so-called Dynkin diagrams.
Important (i.e. in quantum physics) is the representation theory
of semisimple algebras. We will discuss the classification
of irreducible representations in terms
of weight theory. The beautiful character and dimension formulas
of Weyl will be discussed.
Towards the end of the course we will construct the (untwisted) affine Lie
algebras. They appear in physics, e.g. in Wess-Zumino-Witten models,
under the name current algebras. These are Deadline: Wednesday, November 28, 14:00. Hand in before lecture.central extensions of
a loop algebra, i.e., the Lie algebra of polynomial maps from the
circle to a simple finite dimensional Lie algebra. We will show that
a loop algebra admits a presentation which resembles
the presentation with Serre relations in the finite
dimensional case. The presentation will be generalized
to obtain Kac-Moody Lie algebras.
If time permits we will address the representation theory of these.
Organization:
Each meeting will consist of 2 lectures of 45 minutes followed
by an assisted exercise session.
Examination:
There will be 3 assignments with take home exercises (all counting for 25 percent of your grade),
followed by an oral exam of 40 minutes in January (also counting for 25 percent).
Literature:
Prerequisites:
A sound knowledge of linear algebra;
knowledge of the basic principles of algebra.
Last update: December 4, 2007