Generating Functions with $\tau$-Invariance and Vertex Representations of Quantum Affine Algebras $U_{r,s}(\widehat{\mathfrak{g}})$ (I): Simply-laced Cases

TL;DR

This paper introduces a unified definition of two-parameter generating functions with $ au$-invariance for quantum affine algebras and constructs their level-one vertex representations for simply-laced types, advancing the understanding of two-parameter quantum groups.

Contribution

It provides an exact formulation of two-parameter generating functions with $ au$-invariance and constructs their vertex representations for simply-laced quantum affine algebras, unifying prior approaches.

Findings

Unified definition for two-parameter quantum affine algebras.

Construction of level-one vertex representations for simply-laced types.

Verification of the effectiveness of the new definitions.

Abstract

We put forward the exact version of two-parameter generating functions with $\tau$-invariance, which allows us to give a unified and inherent definition for the Drinfeld realization of two-parameter quantum affine algebras for all the untwisted types. As verification, we first construct their level-one vertex representations of $U_{r,s}(\widehat{\mathfrak{g}})$ for simply-laced types, which in turn well-detect the effectiveness of our definitions both for $(r,s)$-generating functions and $(r,s)$-Drinfeld realization in the framework of establishing the two-parameter vertex representation theory.