Representations of vertex operator algebras

Abstract

This paper is an exposition of the representation theory of vertex operator algebras in terms of associative algebras A n (V ) and their bimodules. A new result on the rationality is given. That is, a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V ) is a semisimple associative algebra and each irreducible admissible V -module is ordinary. 2000MSC:17B69

Bimodules and $g$-rationality of vertex operator algebras

“…That is, V is g-rational if and only if A g (V ) is semisimple and any irreducible admissible g-twisted V -module is ordinary. In the case g = 1, this result has been obtained previously in [DJ2] and [DJ3].…”

Bimodules associated to vertex operator algebras

“…For example, this explicit construction leads to a natural contravariant form on the Verma type admissible V -module such that the radical is exactly the maximal proper submodule [4][5][6]. The results in this paper and [4][5][6] have been extended in [6] to study the twisted representations of vertex operator algebras.…”

On certain vertex algebras and their modules associated with vertex algebroids

“…A major step forward in the development of quantum vertex algebra theory was Li's introduction of φ-coordinated modules [30] which, in particular, enabled the association of quantum vertex algebras with quantum affine algebras; see, e.g., the papers by Jing, Kong, Li and Tan [17] and Kong [23]. For a more detailed overview of the evolution of quantum vertex algebra theory and its most recent results see [5,7,8,16,24] and references therein.…”