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.
Categories: math.RA