From braces to pre-Lie rings

Shalev, Aner; Smoktunowicz, Agata

doi:10.1090/proc/16693

From braces to pre-Lie rings
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by Aner Shalev and Agata Smoktunowicz;

Proc. Amer. Math. Soc. 152 (2024), 1545-1559

DOI: https://doi.org/10.1090/proc/16693

Published electronically: January 11, 2024

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Abstract:

Let $A$ be a brace of cardinality $p^{n}$ where $p>n+1$ is prime and let $ann (p^{2})$ be the set of elements of additive order at most $p^{2}$ in this brace. We construct a pre-Lie ring related to the brace $A/ann(p^{2})$.

In the case of strongly nilpotent braces of nilpotency index $k<p$ the brace $A/ann(p^{2})$ can be recovered by applying the construction of the group of flows to the resulting pre-Lie ring. We do not know whether or not our construction is related to the group of flows when applied to braces which are not right nilpotent.

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