Quantization of certain skew-symmetric solutions of the classical Yang-Baxter equation

TL;DR

This paper presents explicit quantizations of specific skew-symmetric solutions to the classical Yang-Baxter equation, resulting in new R-matrices that extend Jordanian matrices to higher dimensions using three different construction methods.

Contribution

It introduces novel quantization techniques for skew-symmetric solutions, expanding the class of R-matrices and providing multiple approaches for their construction.

Findings

Explicit family of R-matrices generalizing Jordanian matrices

Three distinct methods for constructing the quantized solutions

Connections established between classical and quantum Yang-Baxter solutions

Abstract

An explicit quantization is given of certain skew-symmetric solutions of the classical Yang-Baxter, yielding a family of $R$-matrices which generalize to higher dimensions the Jordanian $R$-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa-Ueno Yang-Baxter operators on meromorphic functions; as boundary solutions of the quantum Yang-Baxter equation; via a vertex-IRF transformation from solutions to the dynamical Yang-Baxter equation.