Yang-Baxter R operators and parameter permutations

TL;DR

This paper constructs a unified framework for solutions to the Yang-Baxter equation with $s ext{l}(2)$ symmetry and its deformations, using elementary permutation operators to describe parameter symmetries.

Contribution

It introduces a uniform construction of R-operators for $s ext{l}(2)$ and its deformations, explicitly expressing them through basic operators linked to permutation group elements.

Findings

Explicit construction of R-operators using elementary permutation operators.

Representation of elementary permutations of parameters in the RLL-relation.

Unified approach applicable to undeformed and deformed symmetry algebras.

Abstract

We present an uniform construction of the solution to the Yang- Baxter equation with the symmetry algebra $s\ell(2)$ and its deformations: the q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator acting in the tensor product of two representations of the symmetry algebra with arbitrary spins $\ell_1$ and $\ell_2$ is built in terms of products of three basic operators $\mathcal{S}_1, \mathcal{S}_2,\mathcal{S}_3$ which are constructed explicitly. They have the simple meaning of representing elementary permutations of the symmetric group $\mathfrak{S}_4$, the permutation group of the four parameters entering the RLL-relation.