Classification of a subclass of filiform Leibniz algebra

This study centers on isomorphism classes and invariants of a subclass of filiform Leibniz algebras over complex field. The subclass of filiform Leibniz algebras considered arises from naturally graded filiform Lie algebras. It has been denoted by TLbn in a fixed dimension n. It is noted that dimensional filiform Lie algebras are in TLbn. The intent with this study is to find the classification of third class of filiform Leibniz algebras in dimensions 7 and 8. The classification is carried out by first choosing adapted basis, then construct appropriate multiplication table of the said basis. From the multiplication table, isomorphism criterion is set up using adapted linear transformation and elementary base change. With respect to the condition on the structure constants in adapted basis, difierent disjoint subsets of algebras in TLb7 and TLb8 are obtained. Some of these subsets are single orbits while others are represented as a union of parametric family of orbits. In parametric families of case,the invariants that characterize the parameter in the orbits are given. The filiform Lie algebras in each dimension are specified