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.

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

Last revised on October 1, 2024 at 09:56:06. See the history of this page for a list of all contributions to it.