Quantum current algebras associated with rational $R$-matrix

Abstract

We study quantum current algebra $\text{A}(\overline{R})$ 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 $\text{A}(\overline{R})$ 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 $\text{A}(\overline{R})$-module of level $c$ and vice versa.

27 pages, minor changes and corrections, final version

Cited by in corpus (4)

• On the quantum affine vertex algebra associated with trigonometric $R$-matrix
• $h$-adic quantum vertex algebras in types $B$, $C$, $D$ and their $\varphi$-coordinated modules
• On the Heisenberg algebra associated with the rational $R$-matrix
• Principal subspaces for the quantum affine vertex algebra in type $A_1^{(1)}$