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