Remarks on $\varphi$-coordinated modules for quantum vertex algebras

This paper is about $\varphi$-coordinated modules for weak quantum vertex algebras. Among the main results, several canonical connections among $\varphi$-coordinated modules for different $\varphi$ are established. For vertex operator algebras, a reinterpretation of Frenkel-Huang-Lepowsky’s theorem on contragredient module is given in terms of $\varphi$-coordinated modules.