Introduction to the Yang-Baxter Equation with Open Problems

Axioms

关键词: Yang-Baxter equation;    set-theoretical Yang-Baxter equation;    algebra structures;    Hopf algebras;    quantum groups;    relations on sets;   

DOI  :  10.3390/axioms1010033

来源: mdpi

【 摘 要 】

The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have found solutions for the Yang-Baxter equation, obtaining qualitative results (using the axioms of various algebraic structures) or quantitative results (usually using computer calculations). However, the full classification of its solutions remains an open problem. In this paper, we present the (set-theoretical) Yang-Baxter equation, we sketch the proof of a new theorem, we state some problems, and discuss about directions for future research.

【 授权许可】

CC BY   
© 2012 by the authors; licensee MDPI, Basel, Switzerland.