2024 Students Enrolled in or before 2015 School of Science Mathematics
Special courses on advanced topics in Mathematics B
- Academic unit or major
- Mathematics
- Instructor(s)
- Kazuya Kawasetsu / Hironori Oya
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- ZUA.E332
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 4Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The course description:
We explain the representation theory of vertex algebras and infinite-dimensional Lie algebras and then introduce recent advances in the representation theory of affine vertex algebras.
The aims:
The theories of vertex algebras and infinite-dimensional Lie algebras are related to various fields in mathematics and physics such as the theory of modular forms, tensor categories, quantum groups, combinatorics, quantum field theory and others. In this lecture, we introduce the representation theory of vertex algebras associated to affine Lie algebras (affine vertex algebras) and then explain the relationship between affine Lie algebras, vertex algebras, modular forms, quantum groups and others. We will not explain details of the proofs of the propositions but mainly explain the propositions based on concrete examples.
Course description and aims
・Understand the definitions of vertex algebras and affine Lie algebras. Be able to use generators to calculate the structure of examples of algebras and representations.
・Understand the relationship between the representation theories of affine Lie algebras and affine vertex algebras.
・Understand the relation between the theories of vertex algebras and modular forms.
Keywords
Vertex algebras, representation theory, modular forms, quantum groups
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | ・Fock representations of the Heisenberg Lie algebra and Heisenberg vertex algebra ・The theory of vertex algebras ・General theory of the representation theories of affine Lie algebras and affine vertex algebras ・The representation theory of affine vertex algebras of type A_1 | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to course material.
Textbook(s)
None in particular.
Reference books, course materials, etc.
Details will be provided during each class session.
Evaluation methods and criteria
Assignments (100%)
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
Prerequisites
Basic knowledge on algebra is expected.