Abstract

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.

Highlights

  • Introduction to the YangBaxter Equation with Open Problems Florin NichitaInstitute of Mathematics of the Romanian Academy, P.O

  • The Yang-Baxter equation first appeared in theoretical physics, in a paper by Yang [1], and in the work of Baxter in Statistical Mechanics [2,3]

  • Many scientists have used the axioms of various algebraic structures (quasi-triangular Hopf algebras, Yetter-Drinfeld categories, quandles, group actions, Liealgebras,algebra structures, Jordan triples, Boolean algebras, relations on sets, etc.) or computer calculations in order to produce solutions for the Yang-Baxter equation

Read more

Summary

Introduction

The Yang-Baxter equation first appeared in theoretical physics, in a paper by Yang [1], and in the work of Baxter in Statistical Mechanics [2,3]. It turned out to be one of the basic equations in mathematical physics, and more precisely for introducing the theory of quantum groups. It plays a crucial role in: Knot theory, braided categories, 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, (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. In this paper we present qualitative results concerning the (set-theoretical) Yang-Baxter equation. We conclude with a short section about directions for future research

Preliminaries
Main Results and Discussion
Conclusions and Directions for Future Research
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.