Algèbres de Lie rigides dont le nilradical est filiforme

Abstract

In this article the authors study filiform nilpotent Lie algebras n which possess a given torus T of semisimple derivations. The solvable Lie algebras obtained by a semidirect product T⊕n depend, up to isomorphism, on one or many parameters (continuous family) or zero parameters (rigid object). When the dimension n of n grows, the number of Jacobi relations increases faster than the number of structure constants and the parameters, on which depend the continuous families for n, satisfy new equations for n+1. This phenomenon, well known since an example given by F. Bratzlavsky [J. Algebra 30 (1974), 305–316; is at the origin of the existence of the rigid Lie algebras which have nonvanishing second adjoint cohomology group. This paper gives new examples of such algebras, thus confirming the frequency of this phenomenon. The authors propose as well the first example of a rigid Lie algebra with nonrational structure constants.