On Classification of Finite Dimensional Complex Filiform Leibniz Algebras (Part 2)

TL;DR

This paper presents an algebraic method based on invariants for classifying finite dimensional complex filiform Leibniz algebras, focusing on the second class related to naturally graded filiform Lie algebras, with implications for geometric classification.

Contribution

It introduces a new algebraic classification method for filiform Leibniz algebras using invariants, applicable to low-dimensional cases and aiding geometric orbit classification.

Findings

Classified low-dimensional filiform Leibniz algebras algebraically

Developed invariants-based method for algebraic classification

Results facilitate geometric orbit classification

Abstract

The paper is devoted to classification problem of finite dimensional complex none Lie filiform Leibniz algebras. Actually, the observations show there are two resources to get classification of filiform Leibniz algebras. The first of them is naturally graded none Lie filiform Leibniz algebras and the another one is naturally graded filiform Lie algebras. Using the first resource we get two disjoint classes of filiform Leibniz algebras. The present paper deals with the second of the above two classes, the first class has been considered in our previous paper. The algebraic classification here means to specify the representatives of the orbits, whereas the geometric classification is the problem of finding generic structural constants in the sense of algebraic geometry. Our main effort in this paper is the algebraic classification. We suggest here an algebraic method based on invariants. Utilizing this method for any given low dimensional case all filiform Leibniz algebras can be classified. Moreover, the results can be used for geometric classification of orbits of such algebras.