Hopf Algebras Quantum Groups and Yang-Baxter Equations

The Yang-Baxter equation first appeared in theoretical physics in a paper by the Nobel laureate C.N. Yang and in the work of R.J. Baxter in the field of Statistical Mechanics. At the 1990 International Mathematics Congress Vladimir Drinfeld Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. It turned out that this equation is one of the basic equations in mathematical physics; more precisely it is used for introducing the theory of quantum groups. It also plays a crucial role in: knot theory braided categories the analysis of integrable systems non-commutative descent theory quantum computing non-commutative geometry etc. Many scientists have used the axioms of various algebraic structures (quasi-triangular Hopf algebras Yetter-Drinfeld categories quandles group actions Lie (super)algebras brace structures (co)algebra structures Jordan triples Boolean algebras relations on sets etc.) or computer calculations (and Grobner bases) in order to produce solutions for the Yang-Baxter equation. However the full classification of its solutions remains an open problem. At present the study of solutions of the Yang-Baxter equation attracts the attention of a broad circle of scientists. The current volume highlights various aspects of the Yang-Baxter equation related algebraic structures and applications.