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.
References
[1]
Yang, C.N. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 1967, 19, 1312–1315, doi:10.1103/PhysRevLett.19.1312.
[2]
Baxter, R.J. Partition function for the eight-vertex lattice model. Ann. Phys. 1972, 70, 193–228, doi:10.1016/0003-4916(72)90335-1.
Caenepeel, S.; Militaru, G.; Zhu, S. Frobenius and separable functors for generalized Hopf modules and Nonlinear equations. In Lecture Notes in Mathematics 1787; Springer: Berlin, Germany, 2002.
[5]
Gateva-Ivanova, T. A combinatorial approach to the set-theoretical solutions of the Yang-Baxter equation. J. Math. Phys. 2004, 45, 3828–3859, doi:10.1063/1.1788848.
[6]
Kassel, C. Quantum groups. In Graduate Texts in Mathematics; Springer Verlag: Berlin, Germany, 1995.
[7]
Lambe, L.; Radford, D. Introduction to the quantum Yang-Baxter equation and quantum groups: An algebraic approach. In Mathematics and Its Applications 423; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997.
Drinfeld, V.G. On some unsolved problems in quantum group theory. In Quantum Groups, Lecture Notes in Mathematics; Kulish, P.P., Ed.; Springer-Verlag: Berlin, Germany, 1992; Volume 1510, pp. 1–8.
[10]
Hobby, D.; Nichita, F.F. Solutions of the set-theoretical Yang-Baxter equation derived from relations. Acta Universitatis Apulensis 2012. in press.
Brzezinski, T.; Nichita, F.F. Yang-baxter systems and entwining structures. Commun. Algebra 2005, 33, 1083–1093, doi:10.1081/AGB-200053815.
[13]
Turaev, V. The Yang-Baxter equation and invariants of links. Invent. Math. 1988, 92, 527–553, doi:10.1007/BF01393746.
[14]
Massuyeau, G.; Nichita, F.F. Yang-Baxter operators arising from algebra structures and the Alexander polynomial of knots. Commun. Algebra 2005, 33, 2375–2385, doi:10.1081/AGB-200063495.
[15]
Nichita, F.F.; Parashar, D. Spectral-parameter dependent Yang-Baxter operators and Yang-Baxter systems from algebra structures. Commun. Algebra 2006, 34, 2713–2726, doi:10.1080/00927870600651661.
[16]
Andruskiewitsch, N.; Schneider, H.J. Lifting of quantum linear spaces and pointed Hopf algebras of order p3. J. Algebra 1998, 209, 658–691, doi:10.1006/jabr.1998.7643.
[17]
Beattie, M.; Dascalescu, S.; Grünenfelder, L. On the number of types of finite dimensional Hopf algebras. Invent. Math. 1999, 136, 1–7, doi:10.1007/s002220050302.
[18]
Gelaki, S. Pointed Hopf algebras and Kaplansky’s 10-th conjecture. J. Algebra 1998, 209, 635–657, doi:10.1006/jabr.1998.7513.
[19]
Nichita, F.F. Self-inverse Yang-Baxter operators from (co)algebra structures. J. Algebra 1999, 218, 738–759, doi:10.1006/jabr.1999.7915.
[20]
Hlavaty, L.; Snobl, L. Solutions of a Yang-Baxter system. Available online: http://arxiv.org/pdf/math/9811016.pdf (accessed on 24 April 2012). 1998, arXiv:math.QA/9811016v2. arXiv.org e-Print archive.
