Solvable compatible Lie algebras with a given nilradical

Abstract

We extend the classical construction of solvable Lie algebras from a nilradical to compatible Lie algebras. Since the sum of nilpotent ideals may fail to be nilpotent, we replace the usual nilradical by a \emph{special nilradical} that behaves well with the mixed Jacobi identity. We use the maximal tori of diagonal derivations to build solvable extensions. The method is applied to the pairs $({\mathrm{L}}_n,{\mathrm{R}}_n)$ and $({\mathrm{L}}_n,{\mathrm{W}}_n)$, yielding explicit one-dimensional solvable extensions and proving nonexistence of higher-dimensional ones in these cases. We also study filiform compatible Lie algebras. We introduce the model family ${\mathcal{L}}_s$ and show that each ${\mathcal{L}}_s$ is a linear deformation of the model filiform Lie algebra ${\mathcal{L}}_k$. Finally, we study the existence of solvable extensions of this family, within the framework developed above.