The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analyzed, and their generalized Casimir invariants calculated. It is shown that rank one solvable algebras have a contact form, which implies the existence of an associated dynamical system. Moreover, due to the structure of the quadratic Casimir operator of the nilradical, these algebras contain a maximal non-abelian quasi-classical Lie algebra of dimension 2n − 1, indicating that gauge theories (with ghosts) are possible on these subalgebras.

It is shown that a semidirect sum g = s − → ⊕ R r of a semisimple Lie algebra s and a solvable Lie algebra r with respect to a representation of s which does not decompose into a direct sum of ideals cannot have a radical r associated to a filiform Lie algebra. This proves that this class of nilpotent Lie algebras has none interest for the structure theory of nonsolvable Lie algebras. Mathematics Subject Classification: 17B05, 17B10

Dans ce travail on décrit les algèbres de Lie quasi-filiformes de rang non nul. De plus, on rappelle et corrige la classification des algèbres de Lie filiformes admettant un tore de dérivations, ainsi que la liste des algèbres graduées naturellement et quasi-filiformes. *

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Nous présentons toutes les algèbres de Lie réelles, résolubles et algébriquement rigides de dimension inférieure ouégaleà 8. Nous soulignerons les différences qui distinguent cette classification de celle des algèbres complexes résolubles rigides.