The cohomology of filiform Lie algebras of maximal rank

We describe the structure of the cohomology of the filiform Lie algebras Ln and Qn as a module over their (2-dimensional) torus of derivations. Our approach relies on the fact that both filiform algebras have an ideal h of codimension 1 for which the structure of its cohomology under the action of the Levi factor of the algebra of derivations of h is known.