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Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation


Authors: D. Bachiller, F. Cedó, E. Jespers and J. Okniński
Journal: Trans. Amer. Math. Soc. 370 (2018), 4881-4907
MSC (2010): Primary 16T25, 20E22, 20F16
DOI: https://doi.org/10.1090/tran/7180
Published electronically: February 1, 2018
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Abstract: Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $ B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorphic, as a left brace, to $ B$. It is hence of fundamental importance to describe all simple objects in the class of finite left braces. In this paper we focus on the matched product decompositions of an arbitrary finite left brace. This is used to construct new families of finite simple left braces.


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Additional Information

D. Bachiller
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Email: dbachiller@mat.uab.cat

F. Cedó
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Email: cedo@mat.uab.cat

E. Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: eric.jespers@vub.be

J. Okniński
Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
Email: okninski@mimuw.edu.pl

DOI: https://doi.org/10.1090/tran/7180
Keywords: Yang-Baxter equation, set-theoretic solution, brace, simple, matched product
Received by editor(s): October 3, 2016
Published electronically: February 1, 2018
Additional Notes: The first and second authors were partially supported by grants MINECO-FEDER MTM2014-53644-P and MTM2017-83487-P (Spain)
The third author was supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium)
The fourth author was supported by National Science Centre grants 2013/09/B/ST1/04408 and 2016/23/B/ST1/01045 (Poland)
Article copyright: © Copyright 2018 American Mathematical Society

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