I. Course Description RS “Affine Kac-Moody Lie Algebras ”

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I. Course description RS “Affine Kac-Moody Lie algebras ” prof. E. Feigin. 1. Prerequisites include familiarity with the basic notions of the theory of Lie algebras and representation theory as well. 2. The course is electory. 3. This course will be devoted to the theory of affine Kac-Moody Lie algebras. The main goal will be to classifiy such Lie algebras using the theory, developed by Victor Kac and Robert Moody. The course will be based on the books ”Infinite-dimensional Lie algebras” of V.Kac and “Lie Algebras of Finite and Affine Type”by R.Carter. II. The objectives and goals of the RS “Affine Kac-Moody Lie algebras” are as follows: 1. introducing the audience to the theory of Kac-Moody Lie algebras attached to Cartan matrices, 2. explaining the key notions of the theory of affine Kac-Moody Lie algebras, such as the root system and the Weyl group, associated with the root system III. After mastering the course, the student is expected to: 1. understand such fundamental notions as Cartan matrices, root systems and Weyl groups 2. know the classification of affine root systems IV. Plan: 1. Infinite-dimensional Lie algebras 2. Generalized Cartan matrices, Kac-Moody Lie algebras - definitions and first properties. 3. Finite, Affine and Indefinite Kac-Moody Lie algebras. 4. Classification of affine Kac-Moody Lie algebras. 5. Weyl group and invariant bilinear form. 6. Real and imaginary roots. 7. Canonical central element and imaginary roots for affine Lie algebras. 8. Loop algebras and central extensions. V. Reading lists: 1. Required 1) V. Kac, Infinite dimensional Lie algebras, Cambridge University Press. 2) R.Carter, Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics 2. Optional 1) W.Fulton, J. Harris – Representation theory 2) J.E. Humphreys – Introduction to Lie algebras and Representation theory VI. Current control grade equals the percentage of the number of solved problems (including bonus problems) to the total number of problems given throughout the semester. The exam consists of a written 4-hour test, containing 8 problems. For a 100% result it suffices to solve at least 6 of 8 problems. The total grade for the course is computed via the following formula: Max(150, E+H)/15 where E equals the mark for the written exam and H is the percentage of number of solved problems to the total number of the problems. VII. Guidelines for Knowledge Assessment: 1. Sample problems which will be used for knowledge assessment: 1) Compute the set of positive roots of the Lie algebra sp(2n) in terms of simple roots. 2) Assume that a GCM $A$ is of finite type. Prove that the possible values of $A_{i,j}$, $i\ne j$ form the following list: $0,-1,-2,-3$. 3) Find the ratio of lengths of simple roots for all $2\times 2$ generalized Cartan matrices. 2. A number of questions, which can be used for the examination: 1) Compute the action of the Casimir element in the $(n+1)$-dimensional irreducible representation of $sl(2)$. 2) Compute explicitly the central extension for the affine Kac-Moody algebra sl(n). 3) Draw affine walls and alcoves for the affine sl(2). 4) Compute the root systems of affine sp(4). 5) Compute all positive real roots for the GCM $A=\begin{pmatrix}2 & -4\\ -1 & 2\end{pmatrix}.$. VIII. The students are given home tasks, containing routine exercises, which assist in understanding theoretical material, and research problems, which require more effort to solve and motivate the students to study extra materials. The solutions are either submitted in written form to the lecturer and his assisstants or can be sent via email. Some of the more difficult topics are made into talks, which then are given by students. .

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