Introduction to Kac-Moody Lie algebras
Dozent: Prof. Dr. Nicolas Perrin
Zeit: Mo. 14-16 u. Mi. 10-12.
Raum: Montag AVZ 2.018 Mittwoch N 0.008.
Lecture notes.
Chapter 1 and 2: reminders on the theory of finite dimensional Lie algebras and some factors on associative algebras. PDF
Chapter 3: first definition and properties of Kac-Moody Lie algebras. PDF
Chapter 4: Weyl group of a Kac-Moody Lie algebra. PDF
Chapter 5: Coxeter groups. PDF
Chapter 6: Invariant bilinear form. PDF
Chapter 7: Classification of generalised Cartan matrices. PDF
Chapter 8: Real and imaginary roots. PDF
Chapter 9: The category O. PDF
Chapter 10: Character formula. PDF
Chapter 11: Untwisted affine Lie algebras. PDF
Chapter 12: Explicit construction of finite dimensional Lie algebras. PDF
Chapter 13: Twisted affine Lie algebras. PDF
Abstract.
These lectures are intented to be a natural following of lectures on classical reprensentation theory. We shall not need more that basic notions on linear algebra and Lie algebras.
Kac-Moody Lie algebras where simultaneously introduced by Kac and Moody in the 1960’s. They naturally generalise finite dimensional semisimple Lie algebras. This generalisation, appart from its own interest, has shown many applications in the finite dimensional setting. We will deal with the following material during the lectures.
- Quick review on the classification of semisimple Lie algebras and Serre’s presentation Theorem.
- Construction of Kac-Moody Lie algebras.
- Intergrable representations of Kac-Moody Lie algebras.
- Classification of Kac-Moody Lie algebras.
- Affine Lie algebras.
- Character formula for the intergrable highest weight modules.
If time permit we shall also see some connections with theta functions.
Prerequisites: Basics of linear algebra and Lie algebras.
References:
- Bourbaki, Nicolas. Groupes et algebres de Lie 4,5,6, Hermann 1954.
- Humphreys, James E. Introduction to Lie algebras and representation theory. Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978.
- Kac, Victor G. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990.
- Kumar, Shrawan. Kac-Moody groups, their flag varieties and representation theory. Progress in Mathematics, 204. Birkhauser Boston, Inc., Boston, MA, 2002.
- Wan, Zhe Xian. Introduction to Kac-Moody algebra. World Scientific Publishing Co., Inc., Teaneck, NJ, 1991.