Fall 2025
Yi-Zhi Huang
Subtitle:
Associative algebras and vertex operator algebras
Course Description:
The representation theory of Lie algebras is equivalent to the representation theory of universal enveloping algebras of Lie algebras. In particular, the universal enveloping algebras of Lie algebras and their quotients play an important role in the representation theory of Lie algebras.
For vertex operator algebras, though we also have universal enveloping algebras constructed by Frenkel and Zhu, they are not very useful since we consider only lower-bounded generalized modules. It is therefore important to find an associative algebra associated to a vertex operator algebra such that the category of modules for the associative algebra is equivalent to the category of lower-bounded generalized modules for the vertex operator algebra. In the case all lower-bounded generalized modules are completely reducible, such an algebra was found by Zhu. In the general case, such an algebra was found by Huang. In this course, we will study these algebras and their applications in the representation theory of vertex operator algebras. We will also study other related associative algebras.
Here is a list of topics to be covered in the course:
1. Vertex operator algebras and modules.
2. Universal enveloping algebras of vertex operator algebras.
3. Zhu algebra and zero-mode algebra.
4. The higher level generalizations of Zhu algebra by Dong-Li-Mason.
5. Huang's associative algebras and lower-bounded generalized modules.
6. Twisted generalizations of the associative algebras above.
7. Applications of the associative algebras above.
Text:
No textbook. Papers will be discussed in the classes.
Prerequisites:
Some knowledge of first year algebra and complex analysis for graduate students
Schedule of Sections:
16:640:557 Schedule of Classes
Previous Semesters
PLEASE NOTE: The course information changes from semester to semester for this course number. Fall 2021 is the first semester that Math 557 is offered with the course title "Topics in Vertex Operator Algebra Theory." In very many earlier semesters, through Spring 2021,
courses on vertex operator algebra theory and related subjects were given as sections of the courses "16:640:554 - Selected Topics in
Algebra" and "16:640:555 - Selected Topics in Algebra," with subtitles indicating the content.
Fall 2025 - Huang, Yi-Zhi
Fall 2024 - Huang, Yi-Zhi
Spring 2024 - Huang, Yi-Zhi
Fall 2023 - Lepowsky, James
Spring 2023 - Huang, Yi-Zhi
Fall 2022 - Lepowsky, James
Spring 2022 - Huang, Yi-Zhi
Fall 2021 - Lepowsky, James