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

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.