Evaluation-type deformed modules over the quantum affine vertex algebras of type $A$

TL;DR

This paper explores the structure of deformed modules over quantum affine vertex algebras of type A, linking them to representations of quantum groups and reflection algebras, and introduces q-analogues of quantum immanants.

Contribution

It establishes a new connection between deformed modules and quantum group representations, and constructs q-analogues of quantum immanants at the critical level.

Findings

Connected deformed modules with quantum group representations.

Constructed q-analogues of quantum immanants.

Extended results to the Yang R-matrix case.

Abstract

Let $\mathcal{V}^c(\mathfrak{gl}_N)$ be Etingof--Kazhdan's quantum affine vertex algebra associated with the trigonometric $R$-matrix. We establish a connection between suitably generalized deformed $\phi$-coordinated $\mathcal{V}^c(\mathfrak{gl}_N)$-modules and the representations of quantized enveloping algebra $U_h(\mathfrak{gl}_N)$ and reflection equation algebra $\mathcal{O}_h(Mat_N)$. As an application, we demonstrate how the elements of the center of $\mathcal{V}^c(\mathfrak{gl}_N)$ at the critical level $c=-N$ give rise to the $q$-analogues of quantum immanants for $U_h(\mathfrak{gl}_N)$, which were recently found by Jing, Liu and Molev. Finally, we derive the analogues of these results for the quantum affine vertex algebra associated with the normalized Yang $R$-matrix.