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arXiv · 2102.00387

A class of Lie racks associated to symmetric Leibniz algebras

Abchir Hamid·Fatima-Ezzahrae Abid·Mohamed Boucetta
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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.

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Abchir Hamid, Fatima-Ezzahrae Abid, Mohamed Boucetta. 2021-01-31. A class of Lie racks associated to symmetric Leibniz algebras. https://arxiv.org/abs/2102.00387

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