On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation
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- by Agata Smoktunowicz PDF
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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
- MR Author ID: 367000
- Email: A.Smoktunowicz@ed.ac.uk
- 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.
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3814340