Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-05-03T03:02:53.264Z Has data issue: false hasContentIssue false
  • English
  • Français

On the structure of some one-generator braces

Part of: Hopf algebras, quantum groups and related topics Other nonassociative rings and algebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  20 March 2024

L A Kurdachenko Open the ORCID record for L A Kurdachenko [Opens in a new window]  and
I Ya. Subbotin Open the ORCID record for I Ya. Subbotin [Opens in a new window]
L A Kurdachenko
Affiliation:
Department of Algebra, Dnipro National University, Dnipro 10, Ukraine ([email protected])
I Ya. Subbotin
Affiliation:
Department of Mathematics, National University, Los Angeles, CA, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the one-generator braces A satisfying the condition $A^3 = \langle 0 \rangle$.

Keywords

left braceone-generator bracenilpotent brace

MSC classification

Primary: 17D99: None of the above, but in this section
Secondary: 16T25: Yang-Baxter equations 81R50: Quantum groups and related algebraic methods
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 2 , May 2024 , pp. 566 - 576
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Cedó, F., Solutions of the Yang–Baxter Equation, Advances in Group Theory and Applications 5(1) (2018), 33–90.Google Scholar
Cedó, F., Gateva-Ivanova, T. A. and Smoktunowicz, A., On the Yang–Baxter equation and left nilpotent left braces, J. Pure Appl Algebra 221(4) (2017), 751–756.CrossRefGoogle Scholar
Cedó, F., Smoktunowicz, A. and Vendramin, L., Skew left braces of nilpotent type, Proc. London Math. Soc. (3) 118(6) (2019), 1367–1392.CrossRefGoogle Scholar
Jespers, E., van Antwerpen, A. Vendramin, L. Nilpotency of skew braces and multipermutation solution of the Yang–Baxter equation: 2022, preprint, arxiv:2205.01572v1.CrossRefGoogle Scholar
Rump, W., A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193(1) (2005), 40–55.CrossRefGoogle Scholar
Rump, W., Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra 307(1) (2007), 153–170.CrossRefGoogle Scholar
Rump, W., Classification of cyclic braces, J. Pure Appl. Algebra 209(3) (2007), 671–685.CrossRefGoogle Scholar
Rump, W., Generalized radical rings, unknotted biquandles, and quantum groups, Colloq. Math. 109(1) (2007), 85–100.CrossRefGoogle Scholar
Rump, W., Semidirect products in algebraic logic and solutions of the quantum Yang–Baxter equation, J. Algebra Appl. 7(4) (2008), 471–490.CrossRefGoogle Scholar
Rump, W., Addendum to “Generalized radical rings, unknotted biquandles, and quantum groups” (Colloq. Math. 109 (2007), 85–100), Colloq. Math. 117(2) (2009), 295–298.CrossRefGoogle Scholar
Smoktunowicz, A., On Engel groups, nilpotent groups, rings, braces and the Yang–Baxter equation, Trans. Amer. Math. Soc. 370(9) (2018), 6535–6564.CrossRefGoogle Scholar