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Transactions of the American Mathematical Society

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On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation


Author: Agata Smoktunowicz
Journal: Trans. Amer. Math. Soc. 370 (2018), 6535-6564
MSC (2010): Primary 16N80, 16N40, 16P90, 16T25, 16T20, 20F45, 81R50
DOI: https://doi.org/10.1090/tran/7179
Published electronically: March 20, 2018
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Abstract: It is shown that over an arbitrary field there exists a nil algebra $ R$ whose adjoint group $ R^{o}$ is not an Engel group. This answers a question by Amberg and Sysak from 1997. The case of an uncountable field also answers a recent question by Zelmanov.

In 2007, Rump introduced braces and radical chains $ A^{n+1}=A\cdot A^{n}$ and $ A^{(n+1)}=A^{(n)}\cdot A$ of a brace $ A$. We show that the adjoint group $ A^{o}$ of a finite right brace is a nilpotent group if and only if $ A^{(n)}=0$ for some $ n$. We also show that the adjoint group $ A^{o}$ of a finite left brace $ A$ is a nilpotent group if and only if $ A^{n}=0$ for some $ n$. Moreover, if $ A$ is a finite brace whose adjoint group $ A^{o}$ is nilpotent, then $ A$ is the direct sum of braces whose cardinalities are powers of prime numbers. Notice that $ A^{o}$ is sometimes called the multiplicative group of a brace $ A$. We also introduce a chain of ideals $ A^{[n]}$ of a left brace $ A$ and then use it to investigate braces which satisfy $ A^{n}=0$ and $ A^{(m)}=0$ for some $ m, n$.

We also describe connections between our results and braided groups and the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. It is worth noticing that by a result of Gateva-Ivanova braces are in one-to-one correspondence with braided groups with involutive braiding operators.


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Additional Information

Agata Smoktunowicz
Affiliation: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
Email: A.Smoktunowicz@ed.ac.uk

DOI: https://doi.org/10.1090/tran/7179
Keywords: Engel group, nilpotent group, adjoint group of a ring, braces, nil rings, nil algebras, the Yang-Baxter equation
Received by editor(s): November 27, 2015
Received by editor(s) in revised form: November 21, 2016, and December 16, 2016
Published electronically: March 20, 2018
Additional Notes: This research was supported with ERC advanced grant 320974.
Article copyright: © Copyright 2018 American Mathematical Society

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