We describe the class of complex filiform nilpotent Lie algebras provided with a not trivial external torus of derivations. We prove also that, for dimensions greater than 8, any algebraic irreducible component of the variety of complex nilpotent filiform laws of Lie algebra contains an open set whose elements are characteristically nilpotent laws.

Sur les algèbres de Lie nilpotentes admettant un tore de dérivations

On the Classification of Rigid Lie Algebras

The structure of a solvable Lie group admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).

Einstein solvmanifolds with a simple Einstein derivation

Poisson algebras in terms of non-associative algebras

The invariants of all complex solvable rigid Lie algebras up to dimension 8 are computed. Moreover we show, for rank 1 solvable algebras, some criteria to deduce the non-existence of nontrivial invariants or the existence of fundamental sets of invariants formed by rational functions of the Casimir invariants of the associated nilradical.

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Invariants of solvable rigid Lie algebras up to dimension 8

One knows that a solvable rigid Lie algebra is algebraic and can be written as a semidirect product of the form g=T⊕n if n is the maximal nilpotent ideal and T a torus on n . The main result of the paper is equivalent to the following: If g is rigid then T is a maximal torus on n . The authors then study algebras of this form where n is a filiform nilpotent algebra. A classification of this law is given in the case in which the weights of T are kα , with 1≤k≤n=dimn .

Bermudez Jose Maria Ancochea

Papers

Le rang du systeme lineaire des racines d'une algebre de lie rigide resoluble complexe