Solvable Lie algebras with naturally graded nilradicals and their invariants

TL;DR

This paper classifies certain solvable Lie algebras with graded nilradicals, computes their invariants, and explores their structural properties, revealing implications for dynamical systems and gauge theories.

Contribution

It provides a detailed analysis of solvable Lie algebras with maximal nilindex and specific subalgebra structures, including explicit invariant calculations and structural insights.

Findings

Rank one solvable algebras possess a contact form.

These algebras contain a maximal non-abelian quasi-classical Lie algebra.

Casimir invariants enable potential applications in gauge theories.

Abstract

The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analyzed, and their generalized Casimir invariants calculated. It is shown that rank one solvable algebras have a contact form, which implies the existence of an associated dynamical system. Moreover, due to the structure of the quadratic Casimir operator of the nilradical, these algebras contain a maximal non-abelian quasi-classical Lie algebra of dimension $2n-1$, indicating that gauge theories (with ghosts) are possible on these subalgebras.