Quantum affine vertex algebras associated to untwisted quantum affinization algebras

Let ${\mathcal{U}}_{\mathrm{\hslash }}(\widehat{\mathfrak{g}})$ be the untwisted quantum affinization of a symmetrizable quantum Kac-Moody algebra ${\mathcal{U}}_{\mathrm{\hslash }}(\mathfrak{g})$. For $\mathrm{\ell }\in \mathbb{C}$, we construct an $\mathrm{\hslash }$-adic quantum vertex algebra $V_{\widehat{\mathfrak{g}},\mathrm{\hslash }}(\mathrm{\ell },0)$, and establish a one-to-one correspondence between $\phi$-coordinated $V_{\widehat{\mathfrak{g}},\mathrm{\hslash }}(\mathrm{\ell },0)$-modules and restricted ${\mathcal{U}}_{\mathrm{\hslash }}(\widehat{\mathfrak{g}})$-modules of level $\mathrm{\ell }$. Suppose that $\mathrm{\ell }$ is a positive integer. We construct a quotient $\mathrm{\hslash }$-adic quantum vertex algebra $L_{\widehat{\mathfrak{g}},\mathrm{\hslash }}(\mathrm{\ell },0)$ of $V_{\widehat{\mathfrak{g}},\mathrm{\hslash }}(\mathrm{\ell },0)$, and establish a one-to-one correspondence between certain $\phi$-coordinated $L_{\widehat{\mathfrak{g}},\mathrm{\hslash }}(\mathrm{\ell },0)$-modules and restricted integrable ${\mathcal{U}}_{\mathrm{\hslash }}(\widehat{\mathfrak{g}})$-modules of level $\mathrm{\ell }$. Suppose further that $\mathfrak{g}$ is of finite type. We prove that $L_{\widehat{\mathfrak{g}},\mathrm{\hslash }}(\mathrm{\ell },0)\mathrm{/}\mathrm{\hslash }{L}_{\widehat{\mathfrak{g}},\mathrm{\hslash }}(\mathrm{\ell },0)$ is isomorphic to the simple affine vertex algebra $L_{\widehat{\mathfrak{g}}}(\mathrm{\ell },0)$.