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From actions of an abelian group on itself to left braces

Part of: Radicals and radical properties of rings Hopf algebras, quantum groups and related topics Groups and algebras in quantum theory

Published online by Cambridge University Press:  09 January 2025

A. BALLESTER–BOLINCHES Open the ORCID record for A. BALLESTER–BOLINCHES [Opens in a new window] ,
R. ESTEBAN–ROMERO Open the ORCID record for R. ESTEBAN–ROMERO [Opens in a new window] ,
L. A. KURDACHENKO Open the ORCID record for L. A. KURDACHENKO [Opens in a new window]  and
V. PÉREZ–CALABUIG
A. BALLESTER–BOLINCHES
Affiliation:
Departament de Matemàtiques, Universitat de València Dr. Moliner, 50, 46100 Burjassot, València, Spain. e-mails: Adolfo.Ballester@uv.es, Ramon.Esteban@uv.es
R. ESTEBAN–ROMERO
Affiliation:
Departament de Matemàtiques, Universitat de València Dr. Moliner, 50, 46100 Burjassot, València, Spain. e-mails: Adolfo.Ballester@uv.es, Ramon.Esteban@uv.es
L. A. KURDACHENKO
Affiliation:
Department of Algebra and Geometry, Oles Honchar Dnipro National University Science ave., 72, Dnipro 49095, Ukraine. e-mail: lkurdachenko@gmail.com
V. PÉREZ–CALABUIG
Affiliation:
Departament de Matemàtiques, Universitat de València Dr. Moliner, 50, 46100 Burjassot, València, Spain. e-mail: Vicent.Perez-Calabuig@uv.es
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Abstract

We present a construction of left braces of right nilpotency class at most two based on suitable actions of an abelian group on itself with an invariance condition. This construction allows us to recover the construction of a free right nilpotent one-generated left brace of class two.

MSC classification

Primary: 16T25: Yang-Baxter equations
Secondary: 16N40: Nil and nilpotent radicals, sets, ideals, rings 81R50: Quantum groups and related algebraic methods
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 15
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Part of this research was carried out in the Departament de Matemàtiques, Universitat de València; Dr. Moliner, 50, 46100 Burjassot, València, Spain.

References

Bardakov, V. G., Neshchadim, M. V. and Yadav, M. K.. On ${\lambda}$ -homomorphic skew braces. J. Pure Appl. Algebra 226 (2022), 106961.CrossRefGoogle Scholar
Bonatto, M. and Jedlička, P.. Central nilpotency of skew braces. J. Algebra Appl. 22 (2023), 2350255.CrossRefGoogle Scholar
Caranti, A.. Bi-skew braces and regular subgroups of the holomorph. J. Algebra 562 (2020), 647–665.CrossRefGoogle Scholar
Caranti, A. and Stefanello, L.. From endomorphisms to bi-skew braces, regular subgroups, the Yang-Baxter equation, and Hopf-Galois structures. J. Algebra 587 (2021), 462–487.CrossRefGoogle Scholar
Cedó, F.. Left braces: solutions of the Yang-Baxter equation. Adv. Group Theory Appl. 5 (2018), 33–90.Google Scholar
Childs, L. N.. Bi-skew braces and Hopf-Galois structures. New York J. Math. 25 (2019), 574–588.Google Scholar
Gould, H. W.. Combinatorial Identities. (Morgantown Printing and Binding Co., Morgantown, WV, USA, 1972).Google Scholar
Guarnieri, L. and Vendramin, L.. Skew-braces and the Yang-Baxter equation. Math. Comp. 86 (2017), 2519–2534.CrossRefGoogle Scholar
Jespers, E., Van Antwerpen, A. and Vendramin, L.. Nilpotency of skew braces and multipermutation solutions of the Yang–Baxter equation. Commun. Contemp. Math. 25 (2023), 2250064.CrossRefGoogle Scholar
Koch, A.. Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures. Proc. Amer. Math. Soc. Ser. B 8 (2021), 189–203.CrossRefGoogle Scholar
Rump, W. Braces. radical rings, and the quantum Yang-Baxter equation. J. Algebra 307 (2007), 153–170.CrossRefGoogle Scholar
Rump, W.. One-generator braces and indecomposable set-theoretic solutions to the Yang-Baxter equation. Proc. Edinburgh Math. Soc. 63 (2020), 676–696.Google Scholar
Smoktunowicz, A.. On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation. Trans. Amer. Math. Soc. 370 (2018), 6535–6564.CrossRefGoogle Scholar
Smoktunowicz, A. and Smoktunowicz, A.. Set-theoretic solutions of the Yang-Baxter equation and new classes of R-matrices. Linear Algebra Appl. 546 (2018), 86–114.CrossRefGoogle Scholar
Stefanello, L. and Trappeniers, S.. On bi-skew braces and brace blocks. J. Pure Appl. Algebra 227 (2023), 107295.CrossRefGoogle Scholar