Yi-Zhi Huang
Associative algebras and vertex operator algebras
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.
No textbook. Papers will be discussed in the classes.
Some knowledge of first year algebra and complex analysis for graduate students