Introduction to Kac-Moody and related Lie Algebras

Course notes (version January 24, 2014). If you find any error, please let the author know.

This two week mini-course is a part of the first winter school for the thematic semester New Directions in Lie Theory taking place at the Centre de Recherches Mathématiques. It is aimed at graduate students and postdoctoral fellows. It does not assume any prior knowledge of Kac-Moody algebras. Its goal is to prepare the students to participate in the workshops of the program. It can also serve as a preparation for the mini-courses of the winter school II whose focus is representation theory. The mini-course will therefore be devoted to the structure theory of Kac-Moody algebras.

Course description: There are two definitions of Kac-Moody algebras, presented for example in the books by Kac and Kumar respectively. The two definitions agree for symmetrizable Kac-Moody algebras. In the first part of the course we will give an introduction to a general class of Lie algebras which includes both definitions, following essentially the approach in the book by Moody-Pianzola. The second part of the mini-course will focus on affine Kac-Moody algebras and their realizations in terms of loop algebras.

References: (an incomplete list)

• R. Carter, Lie Algebras of Finite and Affine Type,, sections 14-18
• V. Kac, Infinite-dimensional Lie algebras, Ch. 1-8
• S. Kumar, Kac-Moody Groups, their Flag Varieties, and Representation Theory, Ch. I-III
• R. Moody and A. Pianzola, Lie Algebras With Triangular Decompositions, Ch. 4.

Prerequisites: It is assumed that participants know the structure theory of finite-dimensional semisimple Lie algebras over algebraically closed fields of characteristic zero as well as the concept of a presentation of a Lie algebra and Serre's Theorem on the presentation of semisimple Lie algebras. These topics are for example covered in the books

• N. Bourbaki, Lie Groups and Lie Algebras, Ch. I and VIII.
• R. Carter, Lie Algebras of Finite and Affine Type,, sections 1-9, or
• J. Humphreys, Introduction to Lie Algebras and Representation Theory, sections 1-12, 14, 17 and 18.

• Lecture Tuesdays, 10:00-11:30 and 14:00-15:30
• Lecture Thursdays, 14:00-15:30
• Lecture Friday, 10:00-11:30 and
• Exercise session Friday 14:00-15:30