Quantum affine vertex algebras associated to untwisted quantum affinization algebras

Abstract

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

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