Set-theoretic solutions of the Yang-Baxter equation and new classes of R-matrices

TL;DR

This paper explores methods for constructing parameter-dependent R-matrices, characterizes certain set-theoretic solutions of the Yang-Baxter equation, and establishes a link between braces and these solutions, contributing new classes of R-matrices.

Contribution

It introduces new construction techniques for R-matrices, characterizes indecomposable set-theoretic solutions, and links braces to solutions of the quantum Yang-Baxter equation.

Findings

Constructed R-matrices with prescribed singular values.

Established a correspondence between braces and set-theoretic solutions.

Identified special forms of R-matrices related to involutive solutions.

Abstract

We describe several methods of constructing R-matrices that are dependent upon many parameters, for example unitary R-matrices and R-matrices whose entries are functions. As an application, we construct examples of R-matrices with prescribed singular values. We characterise some classes of indecomposable set-theoretic solutions of the quantum Yang-Baxter equation (QYBE) and construct R-matrices related to such solutions. In particular, we establish a correspondence between one-generator braces and indecomposable, non-degenerate involutive set-theoretic solutions of the QYBE, showing that such solutions are abundant. We show that R-matrices related to involutive, non-degenerate solutions of the QYBE have special form. We also investigate some linear algebra questions related to R-matrices.