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On maximal solvable extensions of nilpotent Lie algebras

Bakhrom Omirov, Gulkhayo Solijanova

TL;DR

The paper addresses the problem of classifying maximal solvable extensions of complex nilpotent Lie algebras, showing that for $d$-locally diagonalizable nilpotents the maximal extension is unique up to isomorphism and takes the form $\\mathcal{N}\\rtimes\\mathcal{T}$, where $\\mathcal{T}$ is a maximal torus of derivations. The approach leverages Mubarakzjanov-style constructions, root space decompositions, and the conjugacy of maximal tori to establish uniqueness; this resolves Šnobl's conjecture under the stated conditions and extends the framework to Lie superalgebras with an analogous result. The paper also analyzes innerness of derivations, demonstrates that maximal solvable extensions are complete (Der$(\\mathcal{N}\\rtimes\\mathcal{T})=\\mathcal{N}\\rtimes\\mathcal{C}(\\mathcal{T})$ with $\\mathcal{C}(\\mathcal{T})=\\mathcal{T}$), and discusses outer derivations in non-maximal cases, along with open questions and conjectures for broader algebras including Leibniz algebras. A constructive method for obtaining maximal solvable extensions via maximal tori is provided, and the results are extended to the Lie superalgebra setting with a corresponding uniqueness theorem. Overall, the work narrows the landscape of maximal solvable extensions to a concrete, torus-driven description in the $d$-locally diagonalizable regime, with clear directions for future research.

Abstract

In this paper, we provide a complete description of complex maximal solvable extensions for a certain class of nilpotent Lie algebras. In particular, we show that, up to isomorphism, a solvable extension of a $d$-locally diagonalizable nilpotent Lie algebra is unique and is realized as the semidirect product of its nilradical with a maximal torus. This result resolves a conjecture of Šnobl concerning the uniqueness of maximal solvable extensions under the condition $d$-locally diagonalizability on the nilradical. Moreover, we extend this description to the setting of Lie superalgebras and present an alternative method for constructing such maximal solvable extensions. Finally, we discuss further aspects and open questions related to maximal solvable extensions of nilpotent Lie algebras

On maximal solvable extensions of nilpotent Lie algebras

TL;DR

The paper addresses the problem of classifying maximal solvable extensions of complex nilpotent Lie algebras, showing that for -locally diagonalizable nilpotents the maximal extension is unique up to isomorphism and takes the form , where is a maximal torus of derivations. The approach leverages Mubarakzjanov-style constructions, root space decompositions, and the conjugacy of maximal tori to establish uniqueness; this resolves Šnobl's conjecture under the stated conditions and extends the framework to Lie superalgebras with an analogous result. The paper also analyzes innerness of derivations, demonstrates that maximal solvable extensions are complete (Der with ), and discusses outer derivations in non-maximal cases, along with open questions and conjectures for broader algebras including Leibniz algebras. A constructive method for obtaining maximal solvable extensions via maximal tori is provided, and the results are extended to the Lie superalgebra setting with a corresponding uniqueness theorem. Overall, the work narrows the landscape of maximal solvable extensions to a concrete, torus-driven description in the -locally diagonalizable regime, with clear directions for future research.

Abstract

In this paper, we provide a complete description of complex maximal solvable extensions for a certain class of nilpotent Lie algebras. In particular, we show that, up to isomorphism, a solvable extension of a -locally diagonalizable nilpotent Lie algebra is unique and is realized as the semidirect product of its nilradical with a maximal torus. This result resolves a conjecture of Šnobl concerning the uniqueness of maximal solvable extensions under the condition -locally diagonalizability on the nilradical. Moreover, we extend this description to the setting of Lie superalgebras and present an alternative method for constructing such maximal solvable extensions. Finally, we discuss further aspects and open questions related to maximal solvable extensions of nilpotent Lie algebras
Paper Structure (7 sections, 13 theorems, 62 equations)

This paper contains 7 sections, 13 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal solvable extensions of some nilpotent Lie algebras
  4. Further discussion and open directions
  5. Maximal solvable extensions of $d$-locally diagonalizable nilpotent Lie superalgebras
  6. On innerness of solvable Lie algebras
  7. Further examples, conjectures and open direction

Key Result

Proposition 2.4

Leger1 Let $\mathcal{L}$ be a Lie algebra over a field of characteristic $0$ such that $Der(\mathcal{L})=\operatorname{InDer}(\mathcal{L}).$ If the center of the Lie algebra $\mathcal{L}$ is non-trivial, then $\mathcal{L}$ is not solvable and the radical of $\mathcal{L}$ is nilpotent.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Example 3.1
  • Definition 3.2
  • Example 3.3
  • Remark 3.4
  • Example 3.5
  • ...and 25 more