Semisimple decompositions of Lie algebras and prehomogeneous modules

TL;DR

This paper characterizes disemisimple Lie algebras as those with a solvable radical that is a prehomogeneous module, and shows that their radical is abelian for certain types of Levi subalgebras, extending previous classifications.

Contribution

It provides a complete characterization of disemisimple Lie algebras using prehomogeneous modules and extends known results to broader classes.

Findings

Disemisimple Lie algebras have a solvable radical equal to their nilradical.

The solvable radical of such algebras is abelian when the Levi subalgebra is simple.

Extension of results to disemisimple Lie algebras without simple quotients of type A.

Abstract

We study {\em disemisimple} Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra $\mathfrak{g}$ is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous $\mathfrak{s}$-module for a Levi subalgebra $\mathfrak{s}$ of $\mathfrak{g}$. We use the classification of prehomogeneous $\mathfrak{s}$-modules for simple Lie algebras $\mathfrak{s}$ given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type $A$.