A class of Lie racks associated to symmetric Leibniz algebras

2021/01/31 by Hamid, Abchir, Abid, Fatima-Ezzahrae, Boucetta, Mohamed
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2102.00387

Abstract

Given a symmetric Leibniz algebra (L,.), the product is Lie-admissible and defines a Lie algebra bracket [ , ] on L. Let G be the connected and simply-connected Lie group associated to (L,[ , ]). We endow G with a Lie rack structure such that the right Leibniz algebra induced on T_eG is exactly (L,.). The obtained Lie rack is said to be associated to the symmetric Leibniz algebra (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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