nLab set theoretic Yang-Baxter equation

Idea

The set-theoretic Yang-Baxter equation is the quantum Yang-Baxter equation in the cartesian monoidal category of Sets. Thus one may talk about set-theoretic solutions of (quantum) Yang-Baxter equation.

References

Pioneering work

• T. Gateva-Ivanova, M. Van den Bergh, Semigroups of I-type, J. Algebra 206 (1998) 97-112

based on some observations from

• Tatiana Gateva-Ivanova, Noetherian properties of skew polynomial rings with binomial relations, Trans. Amer. Math. Soc. 343 (1994) 203-219, Skew polynomial rings with binomial relations, J. Algebra 185 (1996) 710-753

eventually lead to modern theory of braces and skew-braces.

• Pavel Etingof, Travis Schedler, Alexandre Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100(2) (1999) 169-209 doi
• W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), no. 1, 153-170 doi
• Tatiana Gateva-Ivanova, Shahn Majid, Matched pairs approach to set theoretic solutions of the Yang–Baxter equation, J. Algebra 319:4, 15 (2008) 1462-1529 doi
• F. Cedó, E. Jespers, J. Okniński, Braces and the Yang–Baxter equation, Commun. Math. Phys. 327, 101–116 (2014) doi
• Aryan Ghobadi, Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat, J. Algebra 586 (2021) 607-642 doi

A dynamical version of set theoretic quantum Yang-Baxter is introduced using category theory in

• Ryosuke Ashikaga, Youichi Shibukawa, Dynamical reflection maps, arXiv:2209.10079
• Noriaki Kamiya, Youichi Shibukawa, Dynamical Yang-Baxter maps associated with homogeneous pre-systems, J. Gen. Lie Theory Appl. 5 (2011)

Review:

• Anastasia Doikou, Algebraic structures in set-theoretic Yang-Baxter & reflection equations, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2307.06140]

category: algebra