Isotopisms of nilpotent Leibniz algebras and Lie racks
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913429056389120 |
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| author | La Rosa, Gianmarco Mancini, Manuel Nagy, Gábor P. |
| author_facet | La Rosa, Gianmarco Mancini, Manuel Nagy, Gábor P. |
| contents | In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra $\mathfrak{g}$ with $\operatorname{dim}[\mathfrak{g},\mathfrak{g}]=1$ is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra $\mathfrak{l}_{2n+1}^{J_1}$, where $J_1$ is the $n \times n$ Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08306 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Isotopisms of nilpotent Leibniz algebras and Lie racks La Rosa, Gianmarco Mancini, Manuel Nagy, Gábor P. Rings and Algebras 17A32, 17A36, 17B30, 20M99, 22A30 In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra $\mathfrak{g}$ with $\operatorname{dim}[\mathfrak{g},\mathfrak{g}]=1$ is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra $\mathfrak{l}_{2n+1}^{J_1}$, where $J_1$ is the $n \times n$ Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. |
| title | Isotopisms of nilpotent Leibniz algebras and Lie racks |
| topic | Rings and Algebras 17A32, 17A36, 17B30, 20M99, 22A30 |
| url | https://arxiv.org/abs/2302.08306 |