A note on the zeroth products of Frenkel-Jing operators

TL;DR

This paper explores the application of zeroth products of Frenkel-Jing operators within quantum vertex algebra theory to construct and analyze an infinite-dimensional module related to quantum groups.

Contribution

It introduces a new quantum analogue of the universal enveloping algebra and describes the structure of the resulting module.

Findings

Constructed an infinite-dimensional module for a quantum algebra.

Described the module's structure as a $U_q (\mathfrak{sl}_{n+1})_z$-module.

Linked Frenkel-Jing operators with quantum vertex algebra extensions.

Abstract

Quantum vertex algebra theory, developed by H.-S. Li, allows us to apply zeroth products of Frenkel-Jing operators, corresponding to Drinfeld realization of $U_q (\widehat{\mathfrak{sl}}_{n+1})$, on the extension of Koyama vertex operators. As a result, we obtain an infinite-dimensional space and describe its structure as a module for the associative algebra $U_q (\mathfrak{sl}_{n+1})_z$, a certain quantum analogue of $U(\mathfrak{sl}_{n+1})$ which we introduce in this paper.