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Left Braces and the Quantum Yang–Baxter Equation

Part of: Special aspects of infinite or finite groups Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  03 December 2018

H. Meng ,
A. Ballester-Bolinches  and
R. Esteban-Romero
H. Meng*
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain ([email protected]; [email protected]; [email protected])
A. Ballester-Bolinches
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain ([email protected]; [email protected]; [email protected])
R. Esteban-Romero
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain ([email protected]; [email protected]; [email protected])
*
*Corresponding author.
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Abstract

Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang–Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang–Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang–Baxter equation.

Keywords

p-nilpotent groupbracesYang–Baxter equation

MSC classification

Primary: 16T25: Yang-Baxter equations
Secondary: 20F16: Solvable groups, supersolvable groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 595 - 608
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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Footnotes

†

Permanent address: Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Camí de Vera, s/n, 46022 València, Spain, email: [email protected].

References

1.Bachiller, D., Cedó, F. and Jespers, E., Solutions of the Yang–Baxter equation associated with a left brace, J. Algebra 463 (2016), 80–102.Google Scholar
2.Cedó, F., Gateva-Ivanova, T. and Smoktunowicz, A., On the Yang–Baxter equation and left nilpotent left braces, J. Pure Appl. Algebra 221(4) (2017), 751–756.Google Scholar
3.Cedó, F., Jespers, E. and Okniński, J., Braces and the Yang–Baxter equation, Commun. Math. Phys. 327 (2014), 101–116.Google Scholar
4.Etingof, P., Schedler, T. and Soloviev, A., Set theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999), 169–209.Google Scholar
5.Kurzweil, H. and Stellmacher, B., The theory of finite groups. An introduction. Universitext (Springer-Verlag, New York, 2004).Google Scholar
6.Radford, D. E., Hopf algebras (World Scientific, 2012).Google Scholar
7.Rump, W., Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra 307 (2007), 153–170.Google Scholar
8.Smoktunowicz, A., A note on set-theoretic solutions of the Yang–Baxter equation, J. Algebra 500 (2018), 3–18.Google Scholar
9.Smoktunowicz, A., On Engel groups, nilpotent groups, rings, braces and Yang–Baxter equation, Trans. Amer. Math. Soc. 370 (2018), 6535–6564.Google Scholar
10.Sysak, Y., Products of groups and local nearrings, Note Mat. 28 (2008), 181–216.Google Scholar
11.Yang, C. N., Some exact results for many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312–1315.Google Scholar