$\phi$-coordinated modules for quantum vertex algebras and associative algebras

TL;DR

This paper explores $ $-graded $$-coordinated modules for quantum vertex algebras, establishing a correspondence with associative algebra modules and analyzing properties like rationality and fusion rules.

Contribution

It introduces a bijection between irreducible $ $-graded $$-coordinated modules and associative algebra modules, extending understanding of module structures in quantum vertex algebras.

Findings

Bijection between irreducible modules of quantum vertex algebra and associative algebra.

Rationality, regularity, and fusion rules are independent of conformal vector choice.

Framework applies to general quantum vertex algebras and vertex operator algebras.

Abstract

We study $\N$-graded $\phi$-coordinated modules for a general quantum vertex algebra $V$ of a certain type in terms of an associative algebra $\widetilde{A}(V)$ introduced by Y.-Z. Huang. Among the main results, we establish a bijection between the set of equivalence classes of irreducible $\N$-graded $\phi$-coordinated $V$-modules and the set of isomorphism classes of irreducible $\widetilde{A}(V)$-modules. We also show that for a vertex operator algebra, rationality, regularity, and fusion rules are independent of the choice of the conformal vector.