On the quantum affine vertex algebra associated with trigonometric $R$-matrix

Abstract

We apply the theory of $\varphi$-coordinated modules, developed by H.-S. Li, to the Etingof--Kazhdan quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$. We prove, for a certain associate $\varphi$ of the one-dimensional additive formal group, that any $\varphi$-coordinated module for the level $c\in \mathbb{C}$ quantum affine vertex algebra is naturally equipped with a structure of restricted level $c$ module for the quantum affine algebra in type $A$ and vice versa. Moreover, we show that any $\varphi$-coordinated module is irreducible with respect to the action of the quantum affine vertex algebra if and only if it is irreducible with respect to the corresponding action of the quantum affine algebra. In the end, we discuss relation between the centers of the quantum affine algebra and the quantum affine vertex algebra.

arXiv comment34 pages. Main Theorem extended to ${\mathfrak{sl}}_N$. Subsect.3.4 and Sect.4 added. Other minor changes. Comments are welcome

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Cited by in corpus (3)

• Quantum affine vertex algebras associated to untwisted quantum affinization algebras
• $h$-adic quantum vertex algebras in types $B$, $C$, $D$ and their $\varphi$-coordinated modules
• On two families of quantum vertex algebras of FRT-type