Une conjecture sur les suites centrales d’une boucle de Moufang commutative libre

Abstract

The lower and upper central series ${Cⁱ(E)}$ and ${Z_j(E)}$ of a CML (commutative Moufang loop) E are defined just as the central series of a group, the associators $(x,y,z)=(x(yz))^[-1} ((xy)z)$ playing the same role as the commutators for groups.As was skown recently, if $E=?_n$ (resp.$L_n$) is the free CML (resp. exponent 3 CML) on $n≥ 3$ generators, the common length of the central series is exactly $n-1$. Besides $Z₁(L_n)$ contains a torsion-free abelian group $A_n$ of rank n such that $?_n=L_n/A_n$.In view of WITT's result about the central series of "the free nilpotent groups of bounded class" we conjecture that the inclusion: $Cⁱ⊂ Z_{n-1-i}$ is in fact an equality in $L_n$.In $?_n$, this would imply that $Z_{n-1-i}$ is the direct product of $Cⁱ$ by $Aⁿ$.The required equalities will be actually checked when either i=1 or $n≤ 4.