Pre-derivations and description of non-strongly nilpotent filiform Leibniz algebras

In this paper we investigate pre-derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three non-intersected families. We describe the pre-derivation of filiform Leibniz algebras for the first and second families. We found sufficient conditions under which filiform Leibniz algebras are strongly nilpotent. Moreover, for the first and second families, we give the description of characteristically nilpotent algebras which are non-strongly nilpotent.