Solvable compatible Lie algebras with a given nilradical

TL;DR

This paper extends classical methods to construct solvable compatible Lie algebras using a special nilradical, providing explicit examples, nonexistence results, and a deformation analysis of filiform compatible Lie algebras.

Contribution

It introduces a new approach using a special nilradical for compatible Lie algebras and applies it to specific pairs, also analyzing deformations of filiform compatible Lie algebras.

Findings

Explicit one-dimensional solvable extensions for certain pairs

Nonexistence of higher-dimensional extensions in these cases

Deformation of model filiform Lie algebras

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.