Isotopisms of nilpotent Leibniz algebras and Lie racks

TL;DR

This paper classifies two-step nilpotent Leibniz algebras with one-dimensional commutator, showing their isotopism classes correspond to certain Heisenberg algebras and Lie racks, and introduces new invariants for these structures.

Contribution

It provides a classification of isotopism classes of specific nilpotent Leibniz algebras and their associated Lie racks, introducing new invariants for these algebraic structures.

Findings

Nilpotent Leibniz algebras with one-dimensional commutator are isotopic to Heisenberg algebras

Isotopism classes of algebras correspond to isotopic Lie racks

New isotopism invariants for Leibniz algebras and Lie racks

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.