A method for obtaining quantum doubles from the Yang-Baxter R-matrices

TL;DR

This paper presents a method to derive quantum doubles from regular invertible solutions of the quantum Yang-Baxter equation, linking algebraic structures to solutions of fundamental equations in quantum algebra.

Contribution

It introduces a systematic approach to associate a quantum double Hopf algebra with each regular invertible Yang-Baxter R-matrix, extending previous theoretical frameworks.

Findings

Established a correspondence between R-matrices and quantum doubles.

Demonstrated the construction of Hopf algebras from Yang-Baxter solutions.

Extended the algebraic understanding of quantum groups.

Abstract

We develop the approach of Faddeev, Reshetikhin, Takhtajan [1] and of Majid [2] that enables one to associate a quasitriangular Hopf algebra to every regular invertible constant solution of the quantum Yang-Baxter equations. We show that such a Hopf algebra is actually a quantum double.