@incollection{talk_21,
author = {Colazzo, I.},
title = {The matched product of the solutions of the {Y}ang-{B}axter equation},
note = {Malta, 11–15/03/2018},
publisher = {Noncommutative and non-associative structures, braces and applications},
year = {2018-03-13},
keywords = {invited},
file = {Malta2018-IC.pdf}
}
The Yang-Baxter equation is one of the fundamental equations in mathematical-physics. The problem of finding and classifying the set theoretical solutions of the Yang-Baxter equation has originally been posed by Drinfeld in [4]. In particular, the class of non-degenerate solutions has received considerable attention [5, 6, 7, 8, 3]. Although interesting and remarkable results on classifying non-degenerate solutions have been presented, there are still many open related problems. One of these problems is how to construct new families of solutions. Initial contribution to this aspect has been given by Etingof, Schedler, and Soloviev, who present a method to obtain a new solution via retraction. Recently, Vendramin in [9] and Bachiller, Cedó, Jespers, Okniński in [1] provide methods to obtain families of solutions starting from others, which are known.
In this talk we introduce a novel construction technique, called matched product of solutions [2], which allows one to obtain new solutions from two generic (not necessarily non-degenerate) solutions. In addition, we prove that the matched product of two left non-degenerate involutive solutions is still left non-degenerate and involutive.
References
[1] D. Bachiller, F. Cedó, E. Jespers, J. Okniński: A family of irretractable square-free solutions of the Yang-Baxter equation, Forum Math., 29 (2017), 1291-1306.
[2] F. Catino, I. Colazzo, P. Stefanelli: The matched product of set-theoretic solutions of the Yang-Baxter equation, in preparation.
[3] F. Cedó, E. Jespers, J. Okniński: Braces and the Yang-Baxter equation, Comm. Math. Phys., 327 (2014), 101-116.
[4] V. G. Drinfeld: On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer, Berlin, (1992), 1-8.
[5] P. Etingof, T. Schedler, A. Soloviev: Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209.
[6] T. Gateva-Ivanova, M. Van den Bergh: Semigroups of I-type, J. Algebra, 206 (1998), 97-112.
[7] J.-H. Lu, M. Yan, Y.-C. Zhu: On the set-theoretical Yang-Baxter equation, Duke Math. J., 104 (1) (2000) 1-18.
[8] W. Rump: Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra 307 (2007), 153-170.
[9] L. Vendramin: Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra, 220 (2016) 2064-2076.