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arXiv · 2401.06241

Uniqueness of addition in Lie algebras revisited

Ivan Arzhantsev
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Abstract

We obtain new and improve old results on uniqueness of addition in Lie rings and Lie algebras. A Lie ring $\mathfrak{R}$ is called a unique addition ring, or a UA-Lie ring, if any commutator-preserving bijection from $\mathfrak{R}$ to an arbitrary Lie ring is additive. We describe wide classes of Lie rings that are not UA-Lie ring. In the other direction, it is known that if a finite-dimensional Lie algebra $\mathfrak{g}$ contains two elements whose centralizers have trivial intersection, then $\mathfrak{g}$ is a UA-Lie ring. We use this result to characterize UA-Lie rings among seaweed Lie algebras. The paper includes many open problems and questions.

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Ivan Arzhantsev. 2024-01-11. Uniqueness of addition in Lie algebras revisited. https://doi.org/10.64785/mc.30.2.3

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