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**Reflection equation as a tool for studying solutions to the Yang-Baxter equation.**
*(English)*
Zbl 1504.16059

Given a set-theoretic solution \((X, r)\) of the Yang-Baxter equation, briefly YBE solutions (see [V. G. Drinfel’d, Lect. Notes Math. 1510, 1–8 (1992; Zbl 0765.17014)]), a map \(\kappa:X\to X\) is said to be a reflection of \((X,r)\) if
\begin{align*}
r\left(id_{X}\times \kappa\right)r\left(id_{X}\times \kappa\right) = \left(id_{X}\times \kappa\right)r\left(id_{X}\times \kappa\right)r
\end{align*}
is satisfied (see ]K. De Commer, Proc. Edinb. Math. Soc., II. Ser. 62, No. 4, 1089–1113 (2019; Zbl 1470.16068)] and [A. Doikou, A. Doikou and A. Smoktunowicz, Lett. Math. Phys. 111, No. 4, Paper No. 105, 40 p. (2021; Zbl 1486.16039)], and [A. Smoktunowicz, Lett. Math. Phys. 111, No. 4, Paper No. 105, 40 p. (2021; Zbl 1486.16039)]). There is a linear version of these equations in physics and, consequently, an interest in learning them: into the specific, in the study of \(n\) particle scattering on a half-line, a YBE solution represents a collision between two particles, and a reflection represents a collision between a particle and the wall delimiting the half-line.

In the paper under review, the authors turn reflections into a useful tool for studying and finding new YBE solutions. Into the specific, they show how to construct a whole family of YBE solutions \((X,r^{(\kappa)})\) starting from a given right non-degenerate one and its reflection \(\kappa\). Many examples are provided. Moreover, they proved that the structure monoids of \((X,r)\) and \((X,r^{(\kappa)})\) are related by an explicit bijective \(1\)-cocycle-like map.

In addition, in a different direction, they investigate the reflections for non-degenerate involutive YBE solutions, by showing that in this case they can be obtained in a more systematic way checking only one easier relation.

In the paper under review, the authors turn reflections into a useful tool for studying and finding new YBE solutions. Into the specific, they show how to construct a whole family of YBE solutions \((X,r^{(\kappa)})\) starting from a given right non-degenerate one and its reflection \(\kappa\). Many examples are provided. Moreover, they proved that the structure monoids of \((X,r)\) and \((X,r^{(\kappa)})\) are related by an explicit bijective \(1\)-cocycle-like map.

In addition, in a different direction, they investigate the reflections for non-degenerate involutive YBE solutions, by showing that in this case they can be obtained in a more systematic way checking only one easier relation.

Reviewer: Marzia Mazzotta (Lecce)

### MSC:

16T25 | Yang-Baxter equations |

20N02 | Sets with a single binary operation (groupoids) |

20F36 | Braid groups; Artin groups |

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\textit{V. Lebed} and \textit{L. Vendramin}, J. Algebra 607, 360--380 (2022; Zbl 1504.16059)

### References:

[1] | Catino, Francesco; Colazzo, Ilaria; Stefanelli, Paola, The matched product of set-theoretical solutions of the Yang-Baxter equation, J. Pure Appl. Algebra, 224, 3, 1173-1194 (2020) · Zbl 1435.16008 |

[2] | Caudrelier, V.; Crampé, N.; Zhang, Q. C., Set-theoretical reflection equation: classification of reflection maps, J. Phys. A, 46, 9, Article 095203 pp. (2013), 12pp · Zbl 1267.81208 |

[3] | Chow, Wei-Liang, On the algebraical braid group, Ann. Math. (2), 49, 654-658 (1948) · Zbl 0033.01002 |

[4] | Cedó, F.; Jespers, E.; Okniński, J., Primitive set-theoretic solutions of the Yang-Baxter equation (2020) |

[5] | Cedó, F.; Jespers, E.; Okniński, J., Set-theoretic solutions of the Yang-Baxter equation, associated quadratic algebras and the minimality condition, Rev. Mat. Complut. (2020) |

[6] | Caudrelier, V.; Zhang, Q. C., Yang-Baxter and reflection maps from vector solitons with a boundary, Nonlinearity, 27, 6 (2014) · Zbl 1291.35259 |

[7] | De Commer, K., Actions of skew braces and set-theoretic solutions of the reflection equation, Proc. Edinb. Math. Soc. (2), 62, 4, 1089-1113 (2019) · Zbl 1470.16068 |

[8] | Dehornoy, Patrick, Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs, Adv. Math., 282, 93-127 (2015) · Zbl 1326.20039 |

[9] | Doikou, Anastasia; Smoktunowicz, Agata, Set theoretic Yang-Baxter & reflection equations and quantum group symmetries (2020) · Zbl 1486.16039 |

[10] | Gateva-Ivanova, Tatiana, Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math., 338, 649-701 (2018) · Zbl 1437.16028 |

[11] | Guarnieri, L.; Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comput., 86, 307, 2519-2534 (2017) · Zbl 1371.16037 |

[12] | Jespers, Eric; Kubat, Łukasz; Van Antwerpen, Arne, The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation, Trans. Am. Math. Soc., 372, 10, 7191-7223 (2019) · Zbl 1432.16032 |

[13] | Jespers, Eric; Kubat, Łukasz; Van Antwerpen, Arne; Vendramin, Leandro, Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation (2020) · Zbl 1442.16036 |

[14] | Katsamaktsis, Kyriakos, New solutions to the reflection equation with braces (2019) |

[15] | Kuniba, Atsuo; Okado, Masato, Set-theoretical solutions to the reflection equation associated to the quantum affine algebra of type \(A_{n - 1}^{( 1 )}\), J. Integrable Syst., 4, 1, Article xyz013 pp. (2019), 10pp · Zbl 1481.37084 |

[16] | Lebed, Victoria; Vendramin, Leandro, Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation, Adv. Math., 304, 1219-1261 (2017) · Zbl 1356.16027 |

[17] | Lebed, Victoria; Vendramin, Leandro, On structure groups of set-theoretic solutions to the Yang-Baxter equation, Proc. Edinb. Math. Soc. (2), 62, 3, 683-717 (2019) · Zbl 1423.16034 |

[18] | Lu, Jiang-Hua; Yan, Min; Zhu, Yong-Chang, On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1, 1-18 (2000) · Zbl 0960.16043 |

[19] | Meng, H.; Ballester-Bolinches, A.; Esteban-Romero, R., Left braces and the quantum Yang-Baxter equation, Proc. Edinb. Math. Soc. (2), 62, 2, 595-608 (2019) · Zbl 1471.17030 |

[20] | Rump, Wolfgang, Classification of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation, Forum Math., 32, 4, 891-903 (2020) · Zbl 1446.16041 |

[21] | Schwiebert, Christian, Extended reflection equation algebras, the braid group on a handlebody, and associated link polynomials, J. Math. Phys., 35, 5288-5305 (1994) · Zbl 0866.57009 |

[22] | Soloviev, Alexander, Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation, Math. Res. Lett., 7, 5-6, 577-596 (2000) · Zbl 1046.81054 |

[23] | Sossinsky, A. B., Preparation theorems for isotopy invariants of links in 3-manifolds, (Quantum groups. Quantum groups, Leningrad, 1990. Quantum groups. Quantum groups, Leningrad, 1990, Lecture Notes in Math., vol. 1510 (1992), Springer: Springer Berlin), 354-362 · Zbl 0765.57009 |

[24] | Smoktunowicz, Agata; Smoktunowicz, Alicja, Set-theoretic solutions of the Yang-Baxter equation and new classes of R-matrices, Linear Algebra Appl., 546, 86-114 (2018) · Zbl 1384.16025 |

[25] | Smoktunowicz, Agata; Vendramin, Leandro; Weston, Robert, Combinatorial solutions to the reflection equation, J. Algebra, 549, 268-290 (2020) · Zbl 1444.16050 |

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