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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 algebras and examples. 2. Quantum vertex algebras, meromorphic open-string 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 vertex algebras and quantum vertex algebras. 5. Conhomology of meromorphic open-string vertex algebras and deformation theory. Prerequisites: Basic courses in algebra and analysis. I will start from the very beginning of the theory of vertex operator algebras. Text: Lecture notes:
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