Matrix factorization for solutions of the Yang-Baxter equation

TL;DR

This paper introduces a new factorized form for solutions to the Yang-Baxter equation across various algebraic deformations, simplifying their structure and providing explicit formulas for matrices with operator entries.

Contribution

It presents a novel factorization approach for rational, trigonometric, and elliptic solutions of the Yang-Baxter equation involving different algebraic structures.

Findings

New simplified factorized forms of solutions

Explicit formulas for matrix solutions with operator entries

Unified approach across different algebraic deformations

Abstract

We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finite-dimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sl_2, the modular double (trigonometric deformation) and the Sklyanin algebra (elliptic deformation). The solutions are matrices with operator entries. The matrix elements are differential operators in the case of sl_2, finite-difference operators with trigonometric coefficients in the case of the modular double or finite-difference operators with coefficients constructed out of Jacobi theta functions in the case of the Sklyanin algebra. We find a new factorized form of the rational, trigonometric, and elliptic solutions, which drastically simplifies them. We show that they are products of several simply organized matrices and obtain for them explicit formulae.