[Paper Review] The regular representation, Zhu's $A(V)$-theory and induced modules

Haisheng Li|ArXiv.org|Sep 1, 1999

Algebraic structures and combinatorial models16 references4 citations

TL;DR

This paper establishes a connection between Zhu's A(V)-theory and the regular representation in vertex operator algebra theory, introducing a construction of induced modules from A(V)-modules to V-modules via the regular representation. It provides a new proof of Frenkel and Zhu's fusion rule theorem using this framework, offering deeper structural insight into the representation theory of vertex operator algebras.

ABSTRACT

The regular representation is related to Zhu's $A(V)$-theory and an induced module from an $A(V)$-module to a $V$-module is defined in terms of the regular representation. As an application, a new proof of Frenkel and Zhu's fusion rule theorem is obtained.

Motivation & Objective

• To clarify the role of the regular representation in the context of Zhu's A(V)-theory.
• To define a systematic construction of induced modules from A(V)-modules to V-modules using the regular representation.
• To provide a new, representation-theoretic proof of Frenkel and Zhu's fusion rule theorem.
• To unify structural aspects of vertex operator algebra representations through A(V)-module induction.

Proposed method

• Utilizes the regular representation of a vertex operator algebra V to define a functor from A(V)-modules to V-modules.
• Applies the structure of the Zhu algebra A(V) to lift modules from the associative algebra level to the vertex algebra level.
• Employs the universal property of the regular representation to construct induced modules canonically.
• Establishes a correspondence between A(V)-module homomorphisms and V-module homomorphisms via the induced construction.
• Uses the grading and filtration of V to analyze the induced module structure.
• Applies the theory of weak modules and Zhu's associative algebra to verify the module axioms in the induced construction.

Experimental results

Research questions

• RQ1 How can the regular representation be used to systematically construct V-modules from A(V)-modules?
• RQ2 What is the precise relationship between the representation theory of A(V) and that of V?
• RQ3 Can the fusion rule theorem be re-proven using a categorical or induced module framework?
• RQ4 What structural properties does the induced module construction preserve from the A(V)-module level?
• RQ5 How does the regular representation mediate between the associative algebra A(V) and the vertex algebra V?

Key findings

• The induced module construction from A(V)-modules to V-modules is well-defined and respects the module structure via the regular representation.
• The induced module functor is left adjoint to the restriction functor from V-modules to A(V)-modules.
• The construction provides a new, representation-theoretic proof of Frenkel and Zhu's fusion rule theorem.
• The regular representation serves as a universal object that encodes the structure of all weak V-modules.
• The induced module construction preserves irreducibility under certain conditions related to the A(V)-module being simple.
• The framework reveals a deeper categorical relationship between A(V)-modules and V-modules, clarifying the role of Zhu's algebra in vertex operator algebra representation theory.