## 16:640:555 Select Topics Algebra

## Spring 2020

### Yi-Zhi Huang

### Subtitle:

Quantum vertex algebras

### Course Description:

Vertex algebras are important algebraic structures appearing naturally in both mathematics and physics, including, in particular, in the study of moonshine phenomena and in two-dimensional conformal field theory. There is a noncommutative version of vertex algebra called meromorphic open-string vertex algebra. Quantum vertex algebras are "quantizations" or deformations of vertex algebras in the space of meromorphic open-string vertex algebras. Examples of quantum vertex algebras include deformations of affine Lie algebras, the Virasoro algebra and W–algebras. These deformations of infinite dimensional algebras also appear naturally in many problems in mathematics and physics.

This is an introductory course on vertex algebras, meromorphic open-string vertex algebras and quantum vertex algebras. We shall start with the definition of vertex algebra and basic examples, introduce meromorphic open-string vertex algebras as "noncommutative vertex algebras" and study quantum vertex algebras as special examples of meromporphic open-string vertex algebras.

In this course I will cover the following topics:

1. Vertex operator algebras and examples.

2. Meromorphic open-string vertex algebras, quantum vertex algebras, and examples.

3. Quantum vertex algebras as deformations of vertex algebras in the space of meromorphic open-string vertex algebras.

4. Representation Theory of quantum vertex algebras.

5. Conhomology of meromorphic open-string vertex algebras and deformation theory.

### Textbook:

No.

### Prerequisites:

Some basic algebra and analysis at the level of first year graduate courses. We will start from the definition of vertex algebras and basic examples.