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Skew braces and the Yang-Baxter equation


Authors: L. Guarnieri and L. Vendramin
Journal: Math. Comp.
MSC (2010): Primary 16T25; Secondary 81R50
DOI: https://doi.org/10.1090/mcom/3161
Published electronically: November 28, 2016
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Abstract: Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the non-commutative setting and use this new structure to study not necessarily involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation. Based on results of Bachiller and Catino and Rizzo, we develop an algorithm to enumerate and construct classical and non-classical braces of small size up to isomorphism. This algorithm is used to produce a database of braces of small size. The paper contains several open problems, questions and conjectures.


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Additional Information

L. Guarnieri
Affiliation: Departamento de Matemática – FCEN, Universidad de Buenos Aires, Pab. I – Ciudad Universitaria (1428) Buenos Aires, Argentina
Email: leandroguarnieri@gmail.com

L. Vendramin
Affiliation: Departamento de Matemática – FCEN, Universidad de Buenos Aires, Pab. I – Ciudad Universitaria (1428) Buenos Aires, Argentina
Email: lvendramin@dm.uba.ar

DOI: https://doi.org/10.1090/mcom/3161
Received by editor(s): December 3, 2015
Received by editor(s) in revised form: February 21, 2016, and March 13, 2016
Published electronically: November 28, 2016
Additional Notes: This work was partially supported by CONICET, PICT-2014-1376, MATH-AmSud and ICTP
Article copyright: © Copyright 2016 American Mathematical Society