On the non-existence of sympathetic Lie algebras with dimension less than 25

TL;DR

This paper investigates the minimal dimension of sympathetic Lie algebras with simple Levi subalgebras, proving non-existence below dimension 15 and establishing lower bounds of 25 when the nilradical decomposes into four simple modules.

Contribution

It determines the structure and possible dimensions of sympathetic Lie algebras with simple Levi subalgebras, providing new lower bounds and structural constraints.

Findings

No sympathetic Lie algebra exists below dimension 15.

Sympathetic Lie algebras with four simple modules in the nilradical have dimension at least 25.

Explicit calculations of semisimple derivations for () Levi subalgebras.

Abstract

In this article we investigate the question of the lowest possible dimension that a sympathetic Lie algebra $\mathfrak{g}$ can attain, when its Levi subalgebra $\mathfrak{g}_L$ is simple. We establish the structure of the nilradical of a perfect Lie algebra $\mathfrak{g}$, as a $\mathfrak{g}_L$-module, and determine the possible Lie algebra structures that one such $\mathfrak{g}$ admits. We prove that, as a $\mathfrak{g}_L$-module, the nilradical must decompose into at least 4 simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra $\mathfrak{g}$ with Levi subalgebra $\mathfrak{g}_L = \mathfrak{sl}_2(\mathbb{C})$ and give necessary conditions for $\mathfrak{g}$ to be a sympathetic Lie algebra in terms of these semisimple derivations. We show that there is no sympathetic Lie algebra of dimension lower than 15, independently of the nilradical's decomposition. If the nilradical has 4 simple modules, we show that a sympathetic Lie algebra has dimension greater or equal than 25.