Yang-Baxter Equation and Related Algebraic Structures

TL;DR

This paper provides an introduction to algebraic structures related to solutions of the quantum Yang-Baxter equation, highlighting their properties, interrelations, and applications in knot theory.

Contribution

It systematically explores key algebraic structures like skew braces, quandles, racks, and Rota-Baxter groups in relation to the Yang-Baxter equation, offering a comprehensive reference.

Findings

Analysis of algebraic, combinatorial, and homological properties

Elucidation of interrelations among algebraic structures

Applications to knot theory and low-dimensional topology

Abstract

In the 1990s, Drinfel'd proposed the study of set-theoretical solutions to the quantum Yang-Baxter equation, initiating a line of research that has since garnered substantial attention and led to notable developments in algebra, low-dimensional topology, and related areas. This monograph offers a concise introduction to the algebraic theory of such solutions, focusing on key structures including skew braces, quandles, racks, and Rota-Baxter groups, which have emerged as central objects in this framework. We investigate the algebraic, combinatorial, and homological properties of these structures, with an emphasis on their interrelations and applications to knot theory. The monograph is intended as a reference for researchers interested in the deep interplay between these algebraic structures and the quantum Yang-Baxter equation.