Filiform Lie algebras of order 3

TL;DR

This paper extends the concept of filiform Lie algebras to Lie algebras of order 3, providing classifications, deformations, and structural insights using the sl(2,C)-module method.

Contribution

It introduces the notion of filiform Lie algebras of order 3 and analyzes their infinitesimal deformations and dimensions, expanding the theory of Lie algebras.

Findings

Families of filiform elementary Lie algebras of order 3 constructed via deformations

Dimension formulas derived using the sl(2,C)-module method

Explicit bases for infinitesimal deformations in generic cases

Abstract

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e. filiform Lie (super)algebras, into the theory of Lie algebras of order F$. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F=3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras (Vergne, 1970). Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.