Isomorphism classes and invariants of low dimensional filiform Leibniz algebras

In this study, we extend the result on classification of a subclass of filiform Leibniz algebras in low dimensions to dimensions seven and eight based on the technique used by Rakhimov and Bekbaev for classification of subclasses which arise from naturally graded non-Lie filiform Leibniz algebras. The subclass considered here arises from naturally graded filiform Lie algebras. This subclass contains the class of filiform Lie algebras and consequently, by classifying this subclass, we again re-examine the classification of filiform Lie algebras. Our resulting list of filifom Lie algebras is compared with that given by Ancoch\'ea-Bermudez and Goze in 1988 and by G\'{o}mez, Jimenez-Merchan and Khakimdjanov in 1998.