Classification of second class filiform leibniz algebras in dimension nine

This study is centered on part of structural property of Leibniz algebras which is classification of a subclass of filiform Leibniz algebras over complex field. The class of filiform Leibniz algebras has been divided into three disjoint classes. The first two subclasses denoted by FLbn and SLbn arise from naturally graded non- Lie filiform Leibniz algebras. The third class, TLbn arises from naturally graded filiform Lie algebras. In particular, the study is focused on the classification prob- lem of second class filiform Leibniz algebras, SLbn. The method employed in the classification is to first construct multiplication table of SLb9. Then, adapted transformation and elementary base change are used to create an isomorphism criterion. From the isomorphism criterion, the isomorphism classes of algebras from SLb9 are deduced. Some of these classes are single orbits while others are parametric family orbits. An invariant algebraic approach is used to describe the parametric family of orbits. As a result, it is concluded that there are 38 families of non-isomorphic classes of algebras in SLb9. This comprises 17 single orbits and 21 parametric families of orbits.

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