Groups, Radical Rings, and the Yang-Baxter Equation

This monograph provides the first comprehensive introduction to set-theoretic solutions of the YangBaxter equation and their deep connections with skew braces. In recent decades, set-theoretic solutions have emerged as an accessible yet rich domain of study, offering new algebraic structures and revealing unexpected connections between seemingly distant fields. A key breakthrough in this direction was the discovery of braces by Rump and the later generalization to skew braces, which provide an elegant algebraic framework for understanding and constructing set-theoretic solutions. This book offers a self-contained, structured, and pedagogically motivated treatment of the subject. Each chapter ends with a list of exercises to work on the presented topics, some open problems, and comments on the authorship and development of the results presented.

The primary audience consists of researchers and advanced graduate students working on or entering topics related to the YangBaxter equation, especially those with interests in algebra, set-theoretic solutions, and skew braces theory. Given its self-contained nature, the book is also suitable for graduate students seeking a pathway into current research, as it provides foundational material alongside recent developments.

Winner of the 2025 Ferran Sunyer i Balaguer Prize Provides the first systematic and unified treatment of set-theoretic solutions to the Yang-Baxter equation Bridges foundational theory and current research in the area

Inhalt

Introduction.- Preliminaries.- The Jacobson Radical.- The Yang-Baxter Equation.- Nilpotent Groups.- Solvable Groups.- Complements.- Skew Braces.- The Structure Skew Brace of a Solution.- Bierbach Groups.- Garside Groups.- Left Nilpotent Skew Braces.- Right Nilpotent Skew Braces.- Multipermutation Solutions.- Factorizations.- Transitive Groups.- Involutive Solutions.- Simple Skew Braces.