Algebra of the Yang–Baxter Equation Skew Braces, Quandles, and Cohomology

This monograph presents a lucid and approachable introduction to the algebraic structures central to the study of the Yang-Baxter equation, with particular emphasis on skew braces, Rota-Baxter groups, racks, and quandles. It highlights the deep connections among these structures and underscores their importance in knot theory and low-dimensional topology. Rather than striving for exhaustive coverage, the monograph adopts a thoughtfully structured, step-by-step approach that begins with basic concepts and illustrative examples, enabling readers to engage with meaningful results early on without the need for extensive prior specialization. It is intended for graduate students and researchers in algebra, topology, and related fields, and assumes a working knowledge of topics such as group and ring theory, homological algebra, algebraic topology, and knot theory. The content is divided into three interconnected parts, each addressing a distinct dimension of the subject. Part I develops the algebraic framework for general set-theoretic solutions of the Yang-Baxter equation, with a focus on skew braces and Rota-Baxter groups. Part II provides a thorough exposition of the algebraic theory of racks and quandles. Part III, the most advanced portion of the monograph, is devoted to the homology and cohomology theories associated with solutions of the Yang-Baxter equation.