  In this paper we find minimal faithful representations of several classes
filiform Lie algebras by means of strictly upper-triangular matrices. We
investigate Leibniz algebras whose corresponding Lie algebras are filiform Lie
algebras such that the action $I \times L \to I$ gives rise to a minimal
faithful representation of a filiform Lie algebra. The classification up to
isomorphism of such Leibniz algebras is given for low-dimensional cases.
