Fatima-ezzahrae Abid

Cadi Ayyad University | UCAM · Department of Mathematics

A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V$, \[ x\rhd y=y+\sum_{n=1}^\in...

A linear Lie rack structure on a finite dimensional vector space V is a Lie rack operation (x,y)↦x⊳y pointed at the origin and such that for any x, the left translation Lx:y↦Lx(y)=x⊳y is linear. A linear Lie rack operation ⊳ is called analytic if for any x,y∈V, x⊳y=y+∑n=1∞An,1(x,…,x,y), where An,1:V×⋯×V→V is an n + 1-multilinear map symmetric in th...

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 $(\mat...

We classify symmetric Leibniz algebras in dimensions 3 and 4 and we determine all associated Lie racks. Some of such Lie racks give rise to nontrivial topological quandles. We study some algebraic properties of these quandles and we give a necessary and sufficient condition for them to be quasi-trivial.