Skew braces and the braid equation on sets. (English) Zbl 07919706
Kielanowski, Piotr (ed.) et al., Geometric methods in physics XL, workshop, Białowieża, Poland, June 20–25, 2023. Cham: Birkhäuser. Trends Math., 145-152 (2024).
Summary: We discuss how skew braces allow us to construct solutions of the braid equation on the underlying set of a skew brace. First, we discuss methods introduced by W. Rump and then generalised by L. Guarnieri and L. Vendramin. Further, we present a recently introduced way of deforming solutions acquired from skew braces. We analyse skew braces of size 4. We linearise the deformed solutions and observe that the Jordan matrix of deformation is different to the original. That leads to the conclusion that deformations give us a tool to manipulate the solution not only up to the change of the basis.
For the entire collection see [Zbl 07882604].
For the entire collection see [Zbl 07882604].
References:
| [1] | Bachiller, D.: Classification of braces of order \(p^3\). Journal of Pure and Applied Algebra 219(8), 3568-3603 (2015). https://doi.org/https://doi.org/10.1016/j.jpaa.2014.12.013 · Zbl 1312.81099 |
| [2] | Brzeziński, T., Mereta, S., Rybołowicz, B.: From pre-trusses to skew braces. Publicacions Matemàtiques 66, 683-714 (2022). https://doi.org/10.5565/PUBLMAT6622206 · Zbl 1509.16057 |
| [3] | Cedó, F., Jespers, E., Okniński, J.: Braces and the Yang-Baxter Equation. Communications in Mathematical Physics 327(1), 101-116 (2014). https://doi.org/10.1007/s00220-014-1935-y · Zbl 1287.81062 |
| [4] | Doikou, A., Rybołowicz, B.: Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces (2022). ArXiv:2204.11580 |
| [5] | Doikou, A., Rybołowicz, B.: Near braces and \(p\)-deformed braided groups. Bulletin of the London Mathematical Society (2023). https://doi.org/10.1112/blms.12918 · Zbl 07798501 |
| [6] | Doikou, A., Smoktunowicz, A.: Set-theoretic Yang-Baxter & reflection equations and quantum group symmetries. Letters in Mathematical Physics 111(4), 105 (2021). https://doi.org/10.1007/s11005-021-01437-7 · Zbl 1486.16039 |
| [7] | Drinfeld, V.G.: On some unsolved problems in quantum group theory. In: P.P. Kulish (ed.) Quantum Groups, pp. 1-8. Springer Berlin Heidelberg, Berlin, Heidelberg (1992) · Zbl 0765.17014 |
| [8] | The GAP Group: GAP - Groups, Algorithms, and Programming, Version 4.12.2 (2022). https://www.gap-system.org |
| [9] | Guarnieri, L., Vendramin, L.: Skew braces and the Yang-Baxter equation. Mathematics of Computation 86(307), pp. 2519-2534 (2017). https://www.jstor.org/stable/90010152 · Zbl 1371.16037 |
| [10] | Lu, J.H., Yan, M., Zhu, Y.C.: On the set-theoretical Yang-Baxter equation. Duke Mathematical Journal 104(1), 1-18 (2000). https://doi.org/10.1215/S0012-7094-00-10411-5 · Zbl 0960.16043 |
| [11] | Rump, W.: Braces, radical rings, and the quantum Yang-Baxter equation. Journal of Algebra 307(1), 153-170 (2007). https://doi.org/10.1016/j.jalgebra.2006.03.040 · Zbl 1115.16022 |
| [12] | Vendramin, L., Konovalov, O.: YangBaxter - a GAP package, 0.10.2. https://www.gap-system.org/Packages/yangbaxter.html |
| [13] | Wolfram Research, Inc.: Wolfram \(|\) Alpha Notebook Edition. Champaign, IL (2023) |
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