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arXiv · 2307.09102

Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

Alfonso Di Bartolo·Gianmarco La Rosa·Manuel Mancini
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Abstract

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.

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Alfonso Di Bartolo, Gianmarco La Rosa, Manuel Mancini. 2023-07-18. Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra. https://doi.org/10.1007/s00009-024-02679-0

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