Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{\frak {sl}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis

TL;DR

This paper extends the two-parameter quantum affine algebra framework for rak{sl}_n, establishing a Drinfel'd realization and isomorphism theorem using a novel combinatorial approach to the Lyndon basis.

Contribution

It introduces a two-parameter quantum affine algebra for rak{sl}_n, develops its Drinfel'd realization, and proves the isomorphism theorem with a new combinatorial method.

Findings

Defined two-parameter quantum affine algebra as a Drinfel'd double.

Established the Drinfel'd realization for the two-parameter case.

Developed an explicit algorithm for the quantum affine Lyndon basis.

Abstract

We further define two-parameter quantum affine algebra $U_{r,s}(\widehat{\frak {sl}_n})$ $(n>2)$ after the work on the finite cases (see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel'd double. Of importance for the quantum {\it affine} cases is that we can work out the compatible two-parameter version of the Drinfel'd realization as a quantum affinization of $U_{r,s}(\frak{sl}_n)$ and establish the Drinfel'd isomorphism Theorem in the two-parameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum {\it affine} Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel'd generators).