Quantum current algebras associated with rational $R$-matrix

TL;DR

This paper explores the structure of quantum current algebras linked to the rational R-matrix, providing explicit formulas for their centers and establishing a correspondence between modules of quantum vertex algebras and restricted modules of the algebra.

Contribution

It introduces explicit formulas for the centers of quantum current algebras at the critical level and establishes an equivalence between modules of quantum vertex algebras and restricted modules of the algebra.

Findings

Explicit formulas for the center elements at the critical level.

Quantum vertex algebra structure on the vacuum module for any complex level.

Equivalence between modules of quantum vertex algebra and restricted modules of the algebra.

Abstract

We study quantum current algebra $\textrm{A}(\overline{R})$ associated with the rational $R$-matrix of $\mathfrak{gl}_N$ and we give explicit formulae for the elements of its center at the critical level. Due to Etingof--Kazhdan's construction, the level $c$ vacuum module $\mathcal{V}_c(\overline{R})$ for the algebra $\textrm{A}(\overline{R})$ possesses a quantum vertex algebra structure for any complex number $c$. We prove that any module for the quantum vertex algebra $\mathcal{V}_c(\overline{R})$ is naturally equipped with a structure of restricted $\textrm{A}(\overline{R})$-module of level $c$ and vice versa.