Lie algebras with nilpotent length greater than that of each of their subalgebras

TL;DR

This paper investigates finite-dimensional solvable Lie algebras with a focus on those whose nilpotent length exceeds that of all their subalgebras, introducing classifications and characterizations for specific subclasses.

Contribution

It characterizes minimal non-${ m N}$ Lie algebras with abelian nilpotent subalgebras and solvability index up to 3, expanding understanding of their structure.

Findings

Characterization of minimal non-${ m N}$ Lie algebras

Identification of Lie algebras with nilpotent length greater than subalgebras

Analysis of extreme Lie algebras with maximal conjugacy classes

Abstract

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent length $\leq k$, and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-${\mathcal N}$ Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index $\leq 3$.