Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Infinite dimensional Lie algebras and vertex operator algebras» для направления
010100.68 «Математика» подготовки магистра
Правительство Российской Федерации
Федеральное государственное автономное образовательное учреждение высшего профессионального образования
"Национальный исследовательский университет
"Высшая школа экономики"
Факультет Математики
Программа дисциплины
Infinite dimensional Lie algebras and vertex operator algebras для направления 010100.68 «Математика» подготовки магистра магистерская программа «Математика» (англоязычная)
Автор программы:
Д.ф.м.н. Е.Б.Фейгин, evgfeig@hse.ru
Рекомендована секцией УМС по математике «___»____________ 2014 г
Председатель С.М. Хорошкин
Утверждена УС факультета математики «___»_____________2014 г.
Ученый секретарь Ю.М. Бурман_______________________________
Москва, 2014
Настоящая программа не может быть использована другими подразделениями университета и другими вузами без разрешения кафедры-разработчика программы.

Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Infinite dimensional Lie algebras and vertex operator algebras» для направления
010100.68 «Математика» подготовки магистра
This course program sets minimal requirements to the knowledge and skills of students and determines the contents and kinds of lectures and reporting. The program is intended for lecturers teaching this course, course assistants and students of 010100.68 specialization «Mathematics» who study the course “Infinite dimensional Lie algebras and vertex operator algebras”.
 The program has been elaborated in accordance with the Educational standard of HSE for training area 010100.68 «Mathematics» (Master level);
 In accordance with the working studying plan of the university for training area of 010100.68 specialization «Mathematics» (Master level), master program “Mathematics”, approved in 2012.
Studying the course «Infinite dimensional Lie algebras and vertex operator algebras» aims at making the students familiar with the main examples, constructions and techniques of the theory of infinite dimensional Lie algebras with applications in mathematical physics. In particular, the students will learn the basics of the theory of vertex operator algebras, playing the crucial role in the modern mathematical language of the quantum filed theory.
Course goals:
 To make students familiar with the main examples of the infinite dimensional Lie algebras.
 To describe the main constructions of the representation theory of Heisenberg and Virasoro Lie algebras.
 To study the bosonization procedure and to discuss its various applications.
 To make clear the role of the boson-fermion correspondence in the representation theory, combinatorics and mathematical physics.
 To educate students about major concepts of the theory of vertex operator algebras and their repre-
 sentation theory.
To make clear the connection between the Lie theory and the vertex operator algebras theory.
By the end of the course student is supposed to


Know: The basic concepts of the structure theory and representation theory of the infinite dimensional Lie algebras and vertex operator algebras.
Be able to: Understand the links between the representation theory and various problems in combinatorics and the theory of integrable systems. Solve problems of representation theoretic origin, arising in mathematical physics and combinatorics.
This course is a professional one. This is an elective course for the “Mathematics” specialization.

Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Infinite dimensional Lie algebras and vertex operator algebras» для направления
010100.68 «Математика» подготовки магистра
This course is based on knowledge and competences that were provided by the following disciplines:
 Алгебра (Algebra)


Дискретная математика (Discrete mathematics)
Группы и алгебры Ли (Lie groups and Lie algebras)
№ Название раздела
1
2
3
Heisenberg and Virasoro Lie algebras
Bosonization
Kac-Moody Lie algebras
Vertex operator algebras 4
Total:
Total hours
38
38
38
45
159
Contact hours
Lecture
Seminars s
14
14
14
15
57
4
4
4
5
17
Independent student’s work
20
20
20
25
85
Type of grading Type of work
Running
(week)
1 year Characteristics
1 2 3 4 test
Homework
4 4
6 8 written, 60 min
6
Running
Final
Test V wtitten, 90 min.
Examination V wtitten, 240 min
6.1
Knowledge and skills grading criteria
All work is graded on the scale from 1 to 10.
6.2
Calculating the grades for the course
The resulting grade for the running check evaluates the results of student’s work and is calculated according to the following formula:
Отекущий = 0.3·Ок/р + 0.7·Одз ;
The resulting grade for the final evaluation in the form of examination is calculated according to this formula (where Оэкзамен is the evaluation of the student’s performance at the exam itself):
Оитоговый = 0,4·Оэкзамен + 0,6·Отекущий
This grade which is the resulting grade for the course is written down in the student’s certificate (diploma).

Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Infinite dimensional Lie algebras and vertex operator algebras» для направления
010100.68 «Математика» подготовки магистра
№
Section 1. Heisenberg and Virasoro Lie algebras
Topic
1 Introduction. Generators, relations, vector fields.
2 Central extensions, two cocycles, Lie algebras cohomology, explicit computation for the Virasoro algebra.
3 Fock modules for the Heisenberg algebra, bases and characters, invariant forms,.
4 Representations of the Virasoro algebra, central charge, irreducibility and highest weight modules.
Total:
Section 2. Bosonization
Total hour s
8
Lectures Seminars
3 1
9
10
11
38
3
4
4
14
1
1
1
4
№
Topic
Total hours
8
9
Contact hours
Lecture
Seminars s
3 1
4 1
1 Normal ordering and infinite sums
2 Realization of the Virasoro algebra inside the completed emveloping algebra of the Heisenberg Lie algebra
3 Semi-infinite wedge products, Schur polynomials
4 Boson-fermion correspondence
Total:
10
11
38
4
4
14
1
1
4
Section 3. Kac-Moody Lie algebras
Independent student’s work
4
5
5
6
20
Independent student’s work
4
5
5
6
20
№
Topic
1 Lie algebras of infinite matrices
2 Affine Kac-Moody Lie algebras
3 Integrable representations, characters and fur-
Total hours
10
11
9
Lecture s
3
4
4
Contact hours
Seminars
1
1
1
Independent student’s work
4
5
5

Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Infinite dimensional Lie algebras and vertex operator algebras» для направления
010100.68 «Математика» подготовки магистра ther properties
4 Tensor products and Sugawara construction
Total:
Section 4. Vertex operator algebras
8
38
4
14
1
4
6
20
№
Topic
1 Definitions and main examples
2 Associatiativity and operator product expansion
3 Representation theory
4 Fields states correspondence and minimal models
5 Borcheds identity and conformal algebras
Total:
Total hours
9
9
Contact hours
Lecture
Seminars s
3
3
1
1
Independent student’s work
5
5
9
9
9
45
3
3
15
1
1
1
5
5
5
5
25
8.1
Topics for Current Control
Question for the test:
1.
Compute the polynomials which are the images of the given wedge monomials under the boson-fermion correspondence..
2.
Find all n such that the elements L_n,L_{-n},L_0 span a Lie subalagebra in the Virasoro algebra. Identify 3-dimensional Lie algebras thus obtained.
3.
Describe all one-dimensional central extensions of the Lie algebra sl(2).
4.
Prove that any one-dimensional central extension of a simple Lie algebra is trivial.
5.
Find the character of the basic representation of the affine algebra of type A1.
6.
Compute the Goddard-Kent-Olive decomposition of the tensor product of two level one modules for the affine sl(n).
8.2
Questions for evaluating student’s performance
1.
Describe the highest weight irreducible representations of the Heisenberg algebra.
2.
Bosonize the Virasoro algebra.
3.
Construct the fundamental representation of the Lie algebra of infinite matrices..
4.
Does there exist a central extension of the Witt algebra different from the Virasoro algebra?
5.
Compute the image of the wedge monomials under the boson-fermion correspondence..
6.
Describe the fields-states correspondence.
7.
Find the operator product expansion of the Virasoro field with itself..
8.
Find the character of a Fock module.

Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Infinite dimensional Lie algebras and vertex operator algebras» для направления
010100.68 «Математика» подготовки магистра
9.1
Fundamental textbooks
V. G. Kac, A. K. Raina, Bombay lectures on highest weight representations, World Sci. (1987)
Kac, Victor (1998), Vertex algebras for beginners, University Lecture Series 10 (2nd ed.), American
Mathematical Society.
9.2
Required reading
Frenkel, Edward; Ben-Zvi, David (2001), Vertex algebras and Algebraic Curves, Mathematical Surveys and Monographs, American Mathematical Society
Kac, Victor (1990), Infinite dimensional Lie algebras (3 ed.), Cambridge University Press.
Di Francesco, P.; Mathieu, P.; Sénéchal, D. (1997), Conformal Field Theory, Springer-Verlag.
9.3
Further reading
V. G. Kac, Highest weight representations of infinite dimensional Lie algebras, Proc. Internat.
Congress Mathematicians (Helsinki, 1978).
Borcherds, Richard (1986), "Vertex algebras, Kac-Moody algebras, and the Monster", Proc. Natl.
Acad. Sci. USA. 83: 3068–3071.
Frenkel, Igor; Lepowsky, James; Meurman, Arne (1988), Vertex operator algebras and the Monster,
Pure and Applied Mathematics 134, Academic Press.