Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913133039190016 |
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| author | Di Bartolo, Alfonso La Rosa, Gianmarco Mancini, Manuel |
| author_facet | Di Bartolo, Alfonso La Rosa, Gianmarco Mancini, Manuel |
| contents | In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with $\operatorname{char}(\mathbb{F}) \neq 2$. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by $L_n$, where $n=\dim_{\mathbb{F}} L_n$. This generalizes the result found in [11], which is only valid when $\mathbb{F}=\mathbb{C}$. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of $L_n$. Eventually, we solve the coquecigrue problem for $L_n$ by integrating it into a Lie rack. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09102 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra Di Bartolo, Alfonso La Rosa, Gianmarco Mancini, Manuel Rings and Algebras 16W25, 17A32, 17B30, 17B40, 20M99, 22A30 In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with $\operatorname{char}(\mathbb{F}) \neq 2$. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by $L_n$, where $n=\dim_{\mathbb{F}} L_n$. This generalizes the result found in [11], which is only valid when $\mathbb{F}=\mathbb{C}$. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of $L_n$. Eventually, we solve the coquecigrue problem for $L_n$ by integrating it into a Lie rack. |
| title | Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra |
| topic | Rings and Algebras 16W25, 17A32, 17B30, 17B40, 20M99, 22A30 |
| url | https://arxiv.org/abs/2307.09102 |