The universal R-matrix and its associated quantum algebra as functionals of the classical r-matrix: the $sl_{2}$ case

TL;DR

This paper constructs the quantum algebra ${ { m U}}_{ ext{ extbackslash hbar}}(sl_{2})$ and its universal R-matrix explicitly as functionals of the classical r-matrix using a geometric approach, embedding it in the current algebra.

Contribution

It provides a novel geometric method to derive the quantum algebra and R-matrix directly from the classical r-matrix for $sl_{2}$, clarifying their functional relationship.

Findings

Explicit construction of ${ { m U}}_{ ext{ extbackslash hbar}}(sl_{2})$ as a functional of the classical r-matrix

Embedding of the quantum algebra into the $sl_{2}$ current algebra

Geometric approach to the quantum Yang-Baxter equation

Abstract

Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ and its universal quantum $R$-matrix are explicitely constructed as functionals of the associated classical $r$-matrix. In this framework, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ is naturally imbedded in the universal envelopping algebra of the $sl_{2}$ current algebra.