Quantum and super-quantum group related to the Alexander-Conway polynomial

TL;DR

This paper constructs a quasitriangular quantum group related to the Alexander-Conway polynomial, revealing connections with super-Hopf algebras and expanding the algebraic framework for knot invariants.

Contribution

It introduces a new quantum group structure associated with the Alexander-Conway polynomial and links it to super-Hopf algebras through superization techniques.

Findings

Constructed a universal R-matrix for the non-standard quantum group

Established a connection between the quantum group and super-Hopf algebra $U_qgl(1|1)$

Extended the algebraic understanding of knot invariants via quantum groups

Abstract

We describe the quasitriangular structure (universal $R$-matrix) on the non-standard quantum group $U_q(H_1,H_2,X^\pm)$ associated to the Alexander-Conway matrix solution of the Yang-Baxter equation. We show that this Hopf algebra is connected with the super-Hopf algebra $U_qgl(1|1)$ by a general process of superization.