All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n-1

TL;DR

This paper classifies all solvable Lie algebras with a specific nilradical of dimension n and degree of nilpotency n-1, providing explicit bases for their Casimir invariants and demonstrating uniqueness up to isomorphism.

Contribution

It constructs all such solvable Lie algebras with the given nilradical and derives bases for their Casimir invariants using the method of moving frames.

Findings

Unique solvable algebra for each n with specified nilradical

Explicit basis for Casimir invariants of the nilradical

Rational function basis for invariants of the solvable extension

Abstract

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_(n,2) (of degree of nilpotency (n-1) and with an (n-2)-dimensional maximal Abelian ideal). We find that for given n such a solvable algebra is unique up to isomorphisms. Using the method of moving frames we construct a basis for the Casimir invariants of the nilradical n_(n,2). We also construct a basis for the generalized Casimir invariants of its solvable extension s_(n+1) consisting entirely of rational functions of the chosen invariants of the nilradical.