Quantum affine vertex algebras associated to untwisted quantum affinization algebras

TL;DR

This paper constructs quantum vertex algebras associated with untwisted quantum affinization algebras, establishing correspondences with modules and linking to classical affine vertex algebras in the finite type case.

Contribution

It introduces a new class of quantum vertex algebras linked to quantum affine algebras and establishes module correspondences, extending the theory of affine vertex algebras into the quantum setting.

Findings

Constructed $ar$-adic quantum vertex algebra $V_{\u00a8g,}(ll,0)$.

Established one-to-one correspondence between modules of the quantum vertex algebra and restricted quantum affine algebra modules.

Showed that the classical limit of the quantum algebra recovers the simple affine vertex algebra.

Abstract

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