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.
Chapter 1 Introduction.
Chapter 2 Set-theoretic Solutions to the
YangBaxter Equation.
Chapter 3 Braces and Skew Braces.
Chapter 4 Skew
Braces and Associated Actions.
Chapter 5 RotaBaxter Operators on Groups.-
Chapter 6 RotaBaxter Groups and Skew Braces.
Chapter 7 Racks and Quandles.-
Chapter 8 Adjoint Groups of Quandles.
Chapter 9 Automorphisms of Quandles.-
Chapter 10 Residual Finiteness of Quandles.
Chapter 11 Orderability of
Quandles.
Chapter 12 Quandle Rings and Algebras.
Chapter 13 Zero-divisors
and Idempotents in Quandle Rings.
Chapter 14 Trunks and Rack Spaces.-
Chapter 15 Homology and Cohomology of Solutions to the YangBaxter Equation.-
Chapter 16 Computations of Homology and Cohomology of Solutions.
Chapter 17
Automorphism Groups, Second Cohomology and Extensions of Quandles.
Valeriy Bardakov is a Professor in the Laboratory of Inverse Problems in Mathematical Physics at the Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Russia. He is an internationally recognized expert in combinatorial group theory, braid theory, knot theory, quandle theory, rings and algebras, Rota-Baxter operators, and the Yang-Baxter equation. He earned his PhD from Novosibirsk State University, Russia, in 1994. Professor Bardakov has published more than 120 research articles in reputed international journals, and his work has received substantial recognition and citations over the years. He has also supervised and graduated several PhD students.
Mohamed Elhamdadi is a Professor in the Department of Mathematics and Statistics at the University of South Florida, USA. His research expertise includes K-theory, knot theory, quandle theory, quantum algebra, and the Yang-Baxter equation. He received his PhD from Université de Nice-Sophia Antipolis, France, in 1996. Professor Elhamdadi has authored more than 85 research papers in leading international journals and is the author of the book Quandles: An Introduction to the Algebra of Knots (AMS). His work has garnered significant recognition and citations within the mathematical community. Alongside his research work, he has mentored and successfully graduated several PhD students.
Mahender Singh is a Professor in the Department of Mathematical Sciences at the Indian Institute of Science Education and Research Mohali, India. His research expertise spans equivariant algebraic topology, knot theory, finite and combinatorial group theory, mapping class groups, and the Yang-Baxter equation. He earned his PhD from the Harish Chandra Research Institute, India, in 2010. Professor Singh has published more than 60 research articles in reputed international journals, and his work has garnered notable citation impact. He is the author of the monograph Automorphisms of Finite Groups (Springer) and has served as a volume editor for Algebraic Topology and Related Topics (Birkhäuser, Springer). His distinctions include the Swarna Jayanti Fellowship (2020) and the INSPIRE Faculty Award (2012) from the Government of India, along with the Fulbright-Nehru Academic and Professional Excellence Fellowship (2021). Alongside his research endeavors, he has supervised several graduate students and mentored numerous postdoctoral fellows.
