Abstract

In this work it is considered the vector space composed by the infinitesimal deformations of the model $\mathbb{Z}_3$-filiform Lie algebra $L^{n,m,p}$. By using these deformations all the $\mathbb{Z}_3$-filiform Lie algebras can be obtained, hence the importance of these deformations. The results obtained in this work together to those obtained in [Integrable deformations of nilpotent color Lie superalgebras, J. Geom. Phys. 61(2011)1797-1808] and [Corrigendum to Integrable deformations of nilpotent color Lie superalgebras, J. Geom. Phys. 62(2012)1571], leads to compute the total dimension of the mentioned space of deformations.

Highlights

  • The concept of filiform Lie algebras was firstly introduced in [18] by Vergne. This type of nilpotent Lie algebra has important properties; in particular, every filiform Lie algebra can be obtained by a deformation of the model filiform algebra Ln

  • In the same way as filiform Lie algebras, all filiform Lie superalgebras can be obtained by infinitesimal deformations of the model Lie superalgebra Ln,m [1], [4], [8] and [9]

  • We have studied the infinitesimal deformations of the model Z3-color Lie superalgebra, i.e. the model Z3-filiform Lie algebra Ln,m,p

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Summary

Introduction

The concept of filiform Lie algebras was firstly introduced in [18] by Vergne. This type of nilpotent Lie algebra has important properties; in particular, every filiform Lie algebra can be obtained by a deformation of the model filiform algebra Ln. A representation of a (G, β)-color Lie superalgebra is a mapping ρ : L −→ End(V ), where V = g∈G Vg is a graded vector space such that

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