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arXiv · 2302.08306

Isotopisms of nilpotent Leibniz algebras and Lie racks

Gianmarco La Rosa·Manuel Mancini·Gábor P. Nagy
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Abstract

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.

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Gianmarco La Rosa, Manuel Mancini, Gábor P. Nagy. 2023-02-16. Isotopisms of nilpotent Leibniz algebras and Lie racks. https://doi.org/10.1080/00927872.2024.2330686

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