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

Lebed, Victoria; Vendramin, Leandro

J. Algebra 607, Part B, 360-380 (2022).

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.

Reviewer: Marzia Mazzotta (Lecce)

MSC:

16T25	Yang-Baxter equations
20N02	Sets with a single binary operation (groupoids)
20F36	Braid groups; Artin groups

Keywords:

Yang-Baxter equation; reflection equation; structure monoid; structure shelf; braid group

Citations:

Zbl 0765.17014; Zbl 1470.16068; Zbl 1486.16039

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