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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.06241 |
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| _version_ | 1866909800250474496 |
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| author | Arzhantsev, Ivan |
| author_facet | Arzhantsev, Ivan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of addition in Lie algebras revisited Arzhantsev, Ivan Rings and Algebras 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. |
| title | Uniqueness of addition in Lie algebras revisited |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2401.06241 |