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An obstruction to represent abelian Lie algebras by unipotent matrices.

J. C. BenjumeaF. J. EcharteNúñez, J.,Tenorio, A. F. — 2004

Extracta Mathematicae

The aim of this paper is the study of abelian Lie algebras as subalgebras of the nilpotent Lie algebra gn associated with Lie groups of upper-triangular square matrices whose main diagonal is formed by 1. We also give an obstruction to obtain the abelian Lie algebra of dimension one unit less than the corresponding to gn as a Lie subalgebra of gn. Moreover, we give a procedure to obtain abelian Lie subalgebras of gn up to the dimension which we think it is the maximum.

A constructive method to determine the variety of filiform Lie algebras

F. J. EcharteM. C. MárquezJ. Núñez — 2006

Czechoslovak Mathematical Journal

In this paper we use cohomology of Lie algebras to study the variety of laws associated with filiform Lie algebras of a given dimension. As the main result, we describe a constructive way to find a small set of polynomials which define this variety. It allows to improve previous results related with the cardinal of this set. We have also computed explicitly these polynomials in the case of dimensions 11 and 12.

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