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Ingo Runkel
Affine Lie Algebras - Summer term 2012
Announcements:
- [6.7] Die mündlichen Prüfungen finden an den Terminen
Donnerstag, 26.7., und Freitag, 28.9., statt.
- [5.7] In der Übungen am Fr, 6.7. wird auch eine Vorlesung statt finden;
dadurch soll die ausgefallene Vorlesung nachgeholt werden. Am letzten Freitag im
Semester ist dann wieder Fragestunde.
- [26.6] Wenn Sie eine Prüfung in diesem Modul wüschen und
Terminwüsche haben, schreiben Sie mir bitte bis zum Montag, 2.7, eine Email.
Ich versuche dann, Ihre Wüsche zu berücksichtigen.
Overview of material covered:
[PDF].
Proof of the Weyl-Kac charakter formula
[PDF].
Exercises:
This course is an advanced class which involves a significant component
of individual study. That applies in particular to solving the exercises
mentioned in the class; these will be dicussed in the exercise classes.
(Exercises are not handed in and not marked.)
The individual work may also consist of small reading assignments (like
a section in a book) to prepare the next class or to supplement some
technical details skipped in the presentation.
Exam: This course will have oral exams
(to take the exam you should be registered for the course).
Description:
Affine Lie algebras are infinite dimensional Lie algebras constructed
out of finite-dimensional simple Lie algebras via central extensions
of loop algebras. In theoretical physics, affine Lie algebras chiefly
occur in systems of space-time dimension two or three, for example in
the study of conformal field theory, topological field theory,
integrable models, or in the world sheet theory of strings.
Examples of applications in mathematics are that affine Lie algebra
give rise to modular forms, representations of mapping class groups,
and to knot invariants.
This course can be divided into three related parts:
- Affine Lie algebras and their representations
(construction via central extension of loop algebra,
generalised Cartan matrices, affine Weyl group, integrable
highest weight representations, characters).
- Theta functions and modular transformations
(theta series associated to a lattice, behaviour under
the action of the modular group SL(2,Z), relation to characters
of affine Lie algebras, vector valued modular functions).
- Vertex operator algebras (formal power series and fields,
vertex operator algebras, reconstruction theorem, examples
from affine Lie algebras, relation to two-dimensional conformal
field theory).
This course is aimed at Masters students in Mathematics and Mathematical
Physics. A course on quantum groups (by C. Schweigert) is planned for the
winter term 2012/13; these two courses would serve as a good preparation
for a Masters Thesis in the groups of C. Schweigert or I. Runkel.
Prerequisites:
Some familiarity with Lie algebras, for example with the
structure theory for finite-dimensional semi-simple complex
Lie algebras. The module "Algebra 2 (Lie algebras)" is recommended.
Literature:
- Kac, Infinite dimensional Lie algebras, Cambridge University Press 1994.
- Kac, Vertex algebras for beginners, American Mathematical Society 1998.
- Wakimoto, Infinite dimensional Lie algebras, American Mathematical Society 2001.
and also
- Di Francesco, Mathieu, Senechal, Conformal field theory, Springer 1997.
- Fuchs, Affine Lie algebras and quantum groups, Cambridge University Press 1992.
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