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Solvable compatible Lie algebras with a given nilradical

A. Fernández Ouaridi, R. M. Navarro, B. A. Omirov, G. O. Solijanova

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.

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 and , 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 and show that each is a linear deformation of the model filiform Lie algebra . Finally, we study the existence of solvable extensions of this family, within the framework developed above.
Paper Structure (16 sections, 19 theorems, 60 equations)

This paper contains 16 sections, 19 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Compatible Lie Algebras
  3. Examples of nilpotent compatible Lie algebras
  4. The nilradical of a compatible Lie algebra
  5. Solvable extensions of compatible Lie algebras.
  6. Solvable extensions of the nilpotent compatible Lie algebra with Lie parts $\mathrm{L}_n$ and $\mathrm R_n$.
  7. Solvable extensions of $\mathcal{N}$ via non-nilpotent derivations of $\mathrm L_n$.
  8. Solvable extensions of $\mathcal{N}$ via non-nilpotent derivations of $\mathrm R_n$.
  9. Solvable extensions of $\mathcal{N}$ via non-nilpotent derivations of $\mathcal{N}$.
  10. Solvable extensions of the nilpotent compatible Lie algebra with Lie parts $\mathrm{L}_n$ and $\mathrm W_n$.
  11. Solvable extensions of $\mathcal{N}$ via non-nilpotent derivations of $\mathrm L_n$.
  12. Solvable extensions of $\mathcal{N}$ via non-nilpotent derivations of $\mathrm W_n$.
  13. Solvable extensions of $\mathcal{N}$ via non-nilpotent derivations of $\mathcal{N}$.
  14. Filiform compatible Lie algebras
  15. Graded filiform compatible Lie algebras
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let ${{\mathfrak g}}_1=({\mathfrak g}, [-,-]_1)$ and ${{\mathfrak g}}_2=({\mathfrak g}, [-,-]_2)$ be Lie algebras over the vector space ${\mathfrak g}$. Then the following conditions are equivalent:

Theorems & Definitions (50)

  • Proposition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • ...and 40 more