Can solvable extensions of a nilpotent subalgebra be useful in the classification of solvable algebras with the given nilradical?

TL;DR

This paper classifies all solvable Lie algebras with a specific nilradical, showing that the problem reduces to known cases and analyzing their invariants, which simplifies understanding their structure.

Contribution

It provides a complete classification of solvable Lie algebras with a particular nilradical and relates it to previously studied cases, introducing a reduction method.

Findings

Classification reduces to known nilradical cases

Invariants often have polynomial bases

Explicit invariants of solvable extensions are derived

Abstract

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_{n,3} which contains the previously studied filiform nilpotent algebra n_{n-2,1} as a subalgebra but not as an ideal. Rather surprisingly it turns out that the classification of such solvable algebras can be reduced to the classification of solvable algebras with the nilradical n_{n-2,1} together with one additional case. Also the sets of invariants of coadjoint representation of n_{n,3} and its solvable extensions are deduced from this reduction. In several cases they have polynomial bases, i.e. the invariants of the respective solvable algebra can be chosen to be Casimir invariants in its enveloping algebra.