A class of Lie racks associated to symmetric Leibniz algebras

TL;DR

This paper constructs and classifies Lie racks associated with symmetric Leibniz algebras in low dimensions, revealing their algebraic properties and connections to topological quandles.

Contribution

It introduces a method to associate Lie racks to symmetric Leibniz algebras and classifies these structures in dimensions 3 and 4, exploring their properties.

Findings

Classification of symmetric Leibniz algebras in dimensions 3 and 4

Construction of associated Lie racks and topological quandles

Characterization of quasi-trivial quandles

Abstract

Given a symmetric Leibniz algebra $(\mathcal{L},.)$, the product is Lie-admissible and defines a Lie algebra bracket $[\;,\;]$ on $\mathcal{L}$. Let $G$ be the connected and simply-connected Lie group associated to $(\mathcal{L},[\;,\;])$. We endow $G$ with a Lie rack structure such that the right Leibniz algebra induced on $T_eG$ is exactly $(\mathcal{L},.)$. The obtained Lie rack is said to be associated to the symmetric Leibniz algebra $(\mathcal{L},.)$. We classify symmetric Leibniz algebras in dimension 3 and 4 and we determine all the associated Lie racks. Some of such Lie racks give rise to non-trivial topological quandles. We study some algebraic properties of these quandles and we give a necessary and sufficient condition for {them} to be quasi-trivial.