[Paper Review] Vertex operator algebras and associative algebras

Chongying Dong, Haisheng Li+1 more|ArXiv.org|Dec 5, 1996

Algebraic structures and combinatorial models10 references4 citations

TL;DR

This paper introduces a sequence of associative algebras $A_n(V)$ associated with a vertex operator algebra $V$, establishing a categorical equivalence between admissible $V$-modules and modules over $A_n(V)$ that are not modules over $A_{n-1}(V)$. The key result shows that $V$ is rational if and only if all $A_n(V)$ are finite-dimensional semisimple algebras, providing a bridge between vertex operator algebra representation theory and associative algebra structure.

ABSTRACT

Let V be a vertex operator algebra. We construct a sequence of associative algebras A_n(V) (n=0,1,2,...) such that A_{n}(V) is a quotient of A_{n+1}(V) and a pair of functors between the category of A_n(V)-modules which are not A_{n-1}(V)-modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is rational if and only if all A_n(V) are finite-dimensional semisimple algebras.

Motivation & Objective

• To establish a systematic connection between the representation theory of vertex operator algebras and associative algebras.
• To address the classification of admissible modules in vertex operator algebras using algebraic structures.
• To provide a criterion for rationality of vertex operator algebras in terms of finite-dimensional semisimple associative algebras.
• To construct functors that induce a bijection between simple modules in the categories of $A_n(V)$-modules and admissible $V$-modules.

Proposed method

• Define a sequence of associative algebras $A_n(V)$ as quotients of each other, derived from the structure of a vertex operator algebra $V$.
• Construct a pair of functors between the category of $A_n(V)$-modules not lifted from $A_{n-1}(V)$ and the category of admissible $V$-modules.
• Use the grading and commutator relations in $V$ to define the multiplication in $A_n(V)$, ensuring compatibility with the vertex operator algebra axioms.
• Demonstrate that the functors are quasi-inverse, establishing an equivalence of categories between the specified module categories.
• Analyze the structure of $A_n(V)$ using the $G_n$-subalgebras and the $n$-th Zhu algebra construction.
• Prove that $V$ is rational if and only if all $A_n(V)$ are finite-dimensional and semisimple, using the equivalence of categories and the finiteness of composition series.

Experimental results

Research questions

• RQ1 How can the representation theory of vertex operator algebras be systematically related to associative algebras?
• RQ2 What algebraic structure emerges from the $n$-th Zhu algebra construction in a vertex operator algebra?
• RQ3 Under what conditions does the category of $A_n(V)$-modules classify admissible $V$-modules?
• RQ4 Can rationality of a vertex operator algebra be characterized purely algebraically through its associated associative algebras?
• RQ5 What is the precise relationship between the semisimplicity and finite-dimensionality of $A_n(V)$ and the rationality of $V$?

Key findings

• The construction yields a sequence of associative algebras $A_n(V)$ such that $A_n(V)$ is a quotient of $A_{n+1}(V)$, forming a nested structure.
• A pair of functors establishes a bijection between the simple modules of $A_n(V)$-modules not arising from $A_{n-1}(V)$ and the simple admissible $V$-modules.
• The functors are quasi-inverse, inducing an equivalence of categories between the specified module categories.
• A vertex operator algebra $V$ is rational if and only if all $A_n(V)$ are finite-dimensional and semisimple associative algebras.
• The result provides a purely algebraic criterion for rationality of $V$, linking representation theory to finite-dimensional associative algebra structure.
• The paper corrects a mistake in the original version, confirming the validity of the algebraic construction and the equivalence of categories.